Properties

Label 1764.2.b.n.1567.14
Level $1764$
Weight $2$
Character 1764.1567
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1567,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + 4132 x^{6} - 16824 x^{5} + 27780 x^{4} - 26200 x^{3} + 14608 x^{2} + \cdots + 782 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.14
Root \(1.14224 + 0.405613i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1567
Dual form 1764.2.b.n.1567.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.05050 - 0.946809i) q^{2} +(0.207107 - 1.98925i) q^{4} +1.60804i q^{5} +(-1.66587 - 2.28580i) q^{8} +O(q^{10})\) \(q+(1.05050 - 0.946809i) q^{2} +(0.207107 - 1.98925i) q^{4} +1.60804i q^{5} +(-1.66587 - 2.28580i) q^{8} +(1.52250 + 1.68925i) q^{10} +2.67798i q^{11} +3.37849i q^{13} +(-3.91421 - 0.823973i) q^{16} +1.60804i q^{17} +4.30629 q^{19} +(3.19879 + 0.333036i) q^{20} +(2.53553 + 2.81322i) q^{22} -6.46521i q^{23} +2.41421 q^{25} +(3.19879 + 3.54911i) q^{26} +4.71179 q^{29} +10.3963 q^{31} +(-4.89203 + 2.84043i) q^{32} +(1.52250 + 1.68925i) q^{34} +0.242641 q^{37} +(4.52377 - 4.07724i) q^{38} +(3.67565 - 2.67878i) q^{40} +11.6464i q^{41} +7.95699i q^{43} +(5.32716 + 0.554628i) q^{44} +(-6.12132 - 6.79172i) q^{46} +9.04753 q^{47} +(2.53613 - 2.28580i) q^{50} +(6.72066 + 0.699709i) q^{52} -2.46148 q^{53} -4.30629 q^{55} +(4.94975 - 4.46117i) q^{58} -9.04753 q^{59} -3.37849i q^{61} +(10.9213 - 9.84332i) q^{62} +(-2.44975 + 7.61569i) q^{64} -5.43275 q^{65} -11.2529i q^{67} +(3.19879 + 0.333036i) q^{68} +8.03394i q^{71} -6.17733i q^{73} +(0.254894 - 0.229734i) q^{74} +(0.891862 - 8.56628i) q^{76} -3.29589i q^{79} +(1.32498 - 6.29420i) q^{80} +(11.0270 + 12.2346i) q^{82} -12.7951 q^{83} -2.58579 q^{85} +(7.53375 + 8.35883i) q^{86} +(6.12132 - 4.46117i) q^{88} +0.275896i q^{89} +(-12.8609 - 1.33899i) q^{92} +(9.50445 - 8.56628i) q^{94} +6.92468i q^{95} -12.9343i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 40 q^{16} - 16 q^{22} + 16 q^{25} - 64 q^{37} - 64 q^{46} + 40 q^{64} - 64 q^{85} + 64 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05050 0.946809i 0.742817 0.669495i
\(3\) 0 0
\(4\) 0.207107 1.98925i 0.103553 0.994624i
\(5\) 1.60804i 0.719136i 0.933119 + 0.359568i \(0.117076\pi\)
−0.933119 + 0.359568i \(0.882924\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.66587 2.28580i −0.588974 0.808152i
\(9\) 0 0
\(10\) 1.52250 + 1.68925i 0.481458 + 0.534187i
\(11\) 2.67798i 0.807441i 0.914882 + 0.403721i \(0.132283\pi\)
−0.914882 + 0.403721i \(0.867717\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i 0.883457 + 0.468513i \(0.155210\pi\)
−0.883457 + 0.468513i \(0.844790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.91421 0.823973i −0.978553 0.205993i
\(17\) 1.60804i 0.390007i 0.980803 + 0.195003i \(0.0624717\pi\)
−0.980803 + 0.195003i \(0.937528\pi\)
\(18\) 0 0
\(19\) 4.30629 0.987931 0.493966 0.869481i \(-0.335547\pi\)
0.493966 + 0.869481i \(0.335547\pi\)
\(20\) 3.19879 + 0.333036i 0.715270 + 0.0744690i
\(21\) 0 0
\(22\) 2.53553 + 2.81322i 0.540578 + 0.599781i
\(23\) 6.46521i 1.34809i −0.738690 0.674045i \(-0.764555\pi\)
0.738690 0.674045i \(-0.235445\pi\)
\(24\) 0 0
\(25\) 2.41421 0.482843
\(26\) 3.19879 + 3.54911i 0.627334 + 0.696038i
\(27\) 0 0
\(28\) 0 0
\(29\) 4.71179 0.874958 0.437479 0.899229i \(-0.355871\pi\)
0.437479 + 0.899229i \(0.355871\pi\)
\(30\) 0 0
\(31\) 10.3963 1.86723 0.933616 0.358275i \(-0.116635\pi\)
0.933616 + 0.358275i \(0.116635\pi\)
\(32\) −4.89203 + 2.84043i −0.864797 + 0.502121i
\(33\) 0 0
\(34\) 1.52250 + 1.68925i 0.261107 + 0.289703i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.242641 0.0398899 0.0199449 0.999801i \(-0.493651\pi\)
0.0199449 + 0.999801i \(0.493651\pi\)
\(38\) 4.52377 4.07724i 0.733852 0.661415i
\(39\) 0 0
\(40\) 3.67565 2.67878i 0.581171 0.423553i
\(41\) 11.6464i 1.81887i 0.415848 + 0.909434i \(0.363485\pi\)
−0.415848 + 0.909434i \(0.636515\pi\)
\(42\) 0 0
\(43\) 7.95699i 1.21343i 0.794920 + 0.606715i \(0.207513\pi\)
−0.794920 + 0.606715i \(0.792487\pi\)
\(44\) 5.32716 + 0.554628i 0.803100 + 0.0836133i
\(45\) 0 0
\(46\) −6.12132 6.79172i −0.902539 1.00138i
\(47\) 9.04753 1.31972 0.659859 0.751389i \(-0.270616\pi\)
0.659859 + 0.751389i \(0.270616\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.53613 2.28580i 0.358664 0.323261i
\(51\) 0 0
\(52\) 6.72066 + 0.699709i 0.931988 + 0.0970321i
\(53\) −2.46148 −0.338110 −0.169055 0.985607i \(-0.554072\pi\)
−0.169055 + 0.985607i \(0.554072\pi\)
\(54\) 0 0
\(55\) −4.30629 −0.580660
\(56\) 0 0
\(57\) 0 0
\(58\) 4.94975 4.46117i 0.649934 0.585780i
\(59\) −9.04753 −1.17789 −0.588944 0.808174i \(-0.700456\pi\)
−0.588944 + 0.808174i \(0.700456\pi\)
\(60\) 0 0
\(61\) 3.37849i 0.432572i −0.976330 0.216286i \(-0.930606\pi\)
0.976330 0.216286i \(-0.0693943\pi\)
\(62\) 10.9213 9.84332i 1.38701 1.25010i
\(63\) 0 0
\(64\) −2.44975 + 7.61569i −0.306218 + 0.951961i
\(65\) −5.43275 −0.673849
\(66\) 0 0
\(67\) 11.2529i 1.37476i −0.726299 0.687379i \(-0.758761\pi\)
0.726299 0.687379i \(-0.241239\pi\)
\(68\) 3.19879 + 0.333036i 0.387910 + 0.0403865i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.03394i 0.953453i 0.879052 + 0.476726i \(0.158177\pi\)
−0.879052 + 0.476726i \(0.841823\pi\)
\(72\) 0 0
\(73\) 6.17733i 0.723002i −0.932372 0.361501i \(-0.882264\pi\)
0.932372 0.361501i \(-0.117736\pi\)
\(74\) 0.254894 0.229734i 0.0296309 0.0267061i
\(75\) 0 0
\(76\) 0.891862 8.56628i 0.102304 0.982620i
\(77\) 0 0
\(78\) 0 0
\(79\) 3.29589i 0.370817i −0.982662 0.185409i \(-0.940639\pi\)
0.982662 0.185409i \(-0.0593608\pi\)
\(80\) 1.32498 6.29420i 0.148137 0.703713i
\(81\) 0 0
\(82\) 11.0270 + 12.2346i 1.21772 + 1.35109i
\(83\) −12.7951 −1.40445 −0.702225 0.711955i \(-0.747810\pi\)
−0.702225 + 0.711955i \(0.747810\pi\)
\(84\) 0 0
\(85\) −2.58579 −0.280468
\(86\) 7.53375 + 8.35883i 0.812385 + 0.901356i
\(87\) 0 0
\(88\) 6.12132 4.46117i 0.652535 0.475562i
\(89\) 0.275896i 0.0292449i 0.999893 + 0.0146224i \(0.00465463\pi\)
−0.999893 + 0.0146224i \(0.995345\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.8609 1.33899i −1.34084 0.139599i
\(93\) 0 0
\(94\) 9.50445 8.56628i 0.980309 0.883545i
\(95\) 6.92468i 0.710457i
\(96\) 0 0
\(97\) 12.9343i 1.31328i −0.754204 0.656640i \(-0.771977\pi\)
0.754204 0.656640i \(-0.228023\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.500000 4.80247i 0.0500000 0.480247i
\(101\) 16.1947i 1.61143i 0.592304 + 0.805714i \(0.298218\pi\)
−0.592304 + 0.805714i \(0.701782\pi\)
\(102\) 0 0
\(103\) 10.3963 1.02438 0.512189 0.858873i \(-0.328835\pi\)
0.512189 + 0.858873i \(0.328835\pi\)
\(104\) 7.72255 5.62813i 0.757259 0.551884i
\(105\) 0 0
\(106\) −2.58579 + 2.33055i −0.251154 + 0.226363i
\(107\) 6.46521i 0.625016i −0.949915 0.312508i \(-0.898831\pi\)
0.949915 0.312508i \(-0.101169\pi\)
\(108\) 0 0
\(109\) −7.41421 −0.710153 −0.355076 0.934837i \(-0.615545\pi\)
−0.355076 + 0.934837i \(0.615545\pi\)
\(110\) −4.52377 + 4.07724i −0.431324 + 0.388749i
\(111\) 0 0
\(112\) 0 0
\(113\) 12.6060 1.18587 0.592937 0.805249i \(-0.297968\pi\)
0.592937 + 0.805249i \(0.297968\pi\)
\(114\) 0 0
\(115\) 10.3963 0.969461
\(116\) 0.975845 9.37293i 0.0906049 0.870254i
\(117\) 0 0
\(118\) −9.50445 + 8.56628i −0.874955 + 0.788590i
\(119\) 0 0
\(120\) 0 0
\(121\) 3.82843 0.348039
\(122\) −3.19879 3.54911i −0.289604 0.321321i
\(123\) 0 0
\(124\) 2.15315 20.6808i 0.193358 1.85719i
\(125\) 11.9223i 1.06637i
\(126\) 0 0
\(127\) 11.2529i 0.998532i 0.866449 + 0.499266i \(0.166397\pi\)
−0.866449 + 0.499266i \(0.833603\pi\)
\(128\) 4.63714 + 10.3197i 0.409869 + 0.912144i
\(129\) 0 0
\(130\) −5.70711 + 5.14377i −0.500546 + 0.451138i
\(131\) −12.7951 −1.11792 −0.558958 0.829196i \(-0.688799\pi\)
−0.558958 + 0.829196i \(0.688799\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.6543 11.8212i −0.920394 1.02119i
\(135\) 0 0
\(136\) 3.67565 2.67878i 0.315184 0.229704i
\(137\) 19.0583 1.62826 0.814132 0.580680i \(-0.197213\pi\)
0.814132 + 0.580680i \(0.197213\pi\)
\(138\) 0 0
\(139\) 6.09002 0.516549 0.258274 0.966072i \(-0.416846\pi\)
0.258274 + 0.966072i \(0.416846\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.60660 + 8.43966i 0.638332 + 0.708241i
\(143\) −9.04753 −0.756593
\(144\) 0 0
\(145\) 7.57675i 0.629214i
\(146\) −5.84875 6.48929i −0.484046 0.537058i
\(147\) 0 0
\(148\) 0.0502525 0.482672i 0.00413073 0.0396754i
\(149\) −23.7701 −1.94733 −0.973663 0.227993i \(-0.926784\pi\)
−0.973663 + 0.227993i \(0.926784\pi\)
\(150\) 0 0
\(151\) 7.95699i 0.647531i −0.946137 0.323765i \(-0.895051\pi\)
0.946137 0.323765i \(-0.104949\pi\)
\(152\) −7.17373 9.84332i −0.581866 0.798398i
\(153\) 0 0
\(154\) 0 0
\(155\) 16.7177i 1.34279i
\(156\) 0 0
\(157\) 10.9552i 0.874323i 0.899383 + 0.437162i \(0.144016\pi\)
−0.899383 + 0.437162i \(0.855984\pi\)
\(158\) −3.12058 3.46234i −0.248260 0.275449i
\(159\) 0 0
\(160\) −4.56751 7.86657i −0.361094 0.621907i
\(161\) 0 0
\(162\) 0 0
\(163\) 11.2529i 0.881394i −0.897656 0.440697i \(-0.854731\pi\)
0.897656 0.440697i \(-0.145269\pi\)
\(164\) 23.1677 + 2.41206i 1.80909 + 0.188350i
\(165\) 0 0
\(166\) −13.4413 + 12.1146i −1.04325 + 0.940272i
\(167\) −21.8427 −1.69024 −0.845119 0.534579i \(-0.820470\pi\)
−0.845119 + 0.534579i \(0.820470\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) −2.71637 + 2.44824i −0.208336 + 0.187772i
\(171\) 0 0
\(172\) 15.8284 + 1.64795i 1.20691 + 0.125655i
\(173\) 11.2563i 0.855798i −0.903827 0.427899i \(-0.859254\pi\)
0.903827 0.427899i \(-0.140746\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.20658 10.4822i 0.166328 0.790124i
\(177\) 0 0
\(178\) 0.261220 + 0.289829i 0.0195793 + 0.0217236i
\(179\) 12.4710i 0.932123i −0.884752 0.466062i \(-0.845672\pi\)
0.884752 0.466062i \(-0.154328\pi\)
\(180\) 0 0
\(181\) 14.9134i 1.10850i 0.832349 + 0.554252i \(0.186996\pi\)
−0.832349 + 0.554252i \(0.813004\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −14.7782 + 10.7702i −1.08946 + 0.793991i
\(185\) 0.390175i 0.0286863i
\(186\) 0 0
\(187\) −4.30629 −0.314907
\(188\) 1.87381 17.9978i 0.136661 1.31262i
\(189\) 0 0
\(190\) 6.55635 + 7.27439i 0.475648 + 0.527740i
\(191\) 10.2524i 0.741841i −0.928665 0.370921i \(-0.879042\pi\)
0.928665 0.370921i \(-0.120958\pi\)
\(192\) 0 0
\(193\) −7.65685 −0.551152 −0.275576 0.961279i \(-0.588869\pi\)
−0.275576 + 0.961279i \(0.588869\pi\)
\(194\) −12.2463 13.5875i −0.879235 0.975527i
\(195\) 0 0
\(196\) 0 0
\(197\) 10.1445 0.722769 0.361384 0.932417i \(-0.382304\pi\)
0.361384 + 0.932417i \(0.382304\pi\)
\(198\) 0 0
\(199\) −14.7026 −1.04224 −0.521120 0.853483i \(-0.674486\pi\)
−0.521120 + 0.853483i \(0.674486\pi\)
\(200\) −4.02177 5.51841i −0.284382 0.390210i
\(201\) 0 0
\(202\) 15.3332 + 17.0125i 1.07884 + 1.19700i
\(203\) 0 0
\(204\) 0 0
\(205\) −18.7279 −1.30801
\(206\) 10.9213 9.84332i 0.760926 0.685816i
\(207\) 0 0
\(208\) 2.78379 13.2241i 0.193021 0.916929i
\(209\) 11.5322i 0.797696i
\(210\) 0 0
\(211\) 15.9140i 1.09556i −0.836621 0.547782i \(-0.815472\pi\)
0.836621 0.547782i \(-0.184528\pi\)
\(212\) −0.509789 + 4.89649i −0.0350124 + 0.336292i
\(213\) 0 0
\(214\) −6.12132 6.79172i −0.418445 0.464272i
\(215\) −12.7951 −0.872622
\(216\) 0 0
\(217\) 0 0
\(218\) −7.78864 + 7.01984i −0.527513 + 0.475444i
\(219\) 0 0
\(220\) −0.891862 + 8.56628i −0.0601294 + 0.577539i
\(221\) −5.43275 −0.365446
\(222\) 0 0
\(223\) 8.61259 0.576741 0.288371 0.957519i \(-0.406886\pi\)
0.288371 + 0.957519i \(0.406886\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 13.2426 11.9355i 0.880887 0.793937i
\(227\) 3.74761 0.248738 0.124369 0.992236i \(-0.460309\pi\)
0.124369 + 0.992236i \(0.460309\pi\)
\(228\) 0 0
\(229\) 4.19825i 0.277428i −0.990332 0.138714i \(-0.955703\pi\)
0.990332 0.138714i \(-0.0442969\pi\)
\(230\) 10.9213 9.84332i 0.720132 0.649049i
\(231\) 0 0
\(232\) −7.84924 10.7702i −0.515328 0.707099i
\(233\) −3.69222 −0.241885 −0.120943 0.992660i \(-0.538592\pi\)
−0.120943 + 0.992660i \(0.538592\pi\)
\(234\) 0 0
\(235\) 14.5488i 0.949058i
\(236\) −1.87381 + 17.9978i −0.121974 + 1.17156i
\(237\) 0 0
\(238\) 0 0
\(239\) 1.10926i 0.0717518i −0.999356 0.0358759i \(-0.988578\pi\)
0.999356 0.0358759i \(-0.0114221\pi\)
\(240\) 0 0
\(241\) 1.39942i 0.0901444i −0.998984 0.0450722i \(-0.985648\pi\)
0.998984 0.0450722i \(-0.0143518\pi\)
\(242\) 4.02177 3.62479i 0.258529 0.233010i
\(243\) 0 0
\(244\) −6.72066 0.699709i −0.430246 0.0447943i
\(245\) 0 0
\(246\) 0 0
\(247\) 14.5488i 0.925717i
\(248\) −17.3189 23.7639i −1.09975 1.50901i
\(249\) 0 0
\(250\) 11.2882 + 12.5244i 0.713927 + 0.792115i
\(251\) −21.8427 −1.37870 −0.689349 0.724430i \(-0.742103\pi\)
−0.689349 + 0.724430i \(0.742103\pi\)
\(252\) 0 0
\(253\) 17.3137 1.08850
\(254\) 10.6543 + 11.8212i 0.668512 + 0.741726i
\(255\) 0 0
\(256\) 14.6421 + 6.45042i 0.915133 + 0.403151i
\(257\) 14.3107i 0.892679i −0.894864 0.446339i \(-0.852727\pi\)
0.894864 0.446339i \(-0.147273\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.12516 + 10.8071i −0.0697794 + 0.670226i
\(261\) 0 0
\(262\) −13.4413 + 12.1146i −0.830407 + 0.748440i
\(263\) 2.67798i 0.165131i 0.996586 + 0.0825656i \(0.0263114\pi\)
−0.996586 + 0.0825656i \(0.973689\pi\)
\(264\) 0 0
\(265\) 3.95815i 0.243147i
\(266\) 0 0
\(267\) 0 0
\(268\) −22.3848 2.33055i −1.36737 0.142361i
\(269\) 16.1947i 0.987406i 0.869631 + 0.493703i \(0.164357\pi\)
−0.869631 + 0.493703i \(0.835643\pi\)
\(270\) 0 0
\(271\) −16.4863 −1.00147 −0.500737 0.865600i \(-0.666937\pi\)
−0.500737 + 0.865600i \(0.666937\pi\)
\(272\) 1.32498 6.29420i 0.0803388 0.381642i
\(273\) 0 0
\(274\) 20.0208 18.0446i 1.20950 1.09011i
\(275\) 6.46521i 0.389867i
\(276\) 0 0
\(277\) 23.7990 1.42994 0.714971 0.699154i \(-0.246440\pi\)
0.714971 + 0.699154i \(0.246440\pi\)
\(278\) 6.39757 5.76608i 0.383701 0.345827i
\(279\) 0 0
\(280\) 0 0
\(281\) −8.19285 −0.488744 −0.244372 0.969681i \(-0.578582\pi\)
−0.244372 + 0.969681i \(0.578582\pi\)
\(282\) 0 0
\(283\) −10.3963 −0.617996 −0.308998 0.951063i \(-0.599994\pi\)
−0.308998 + 0.951063i \(0.599994\pi\)
\(284\) 15.9815 + 1.66388i 0.948327 + 0.0987333i
\(285\) 0 0
\(286\) −9.50445 + 8.56628i −0.562010 + 0.506535i
\(287\) 0 0
\(288\) 0 0
\(289\) 14.4142 0.847895
\(290\) 7.17373 + 7.95938i 0.421256 + 0.467391i
\(291\) 0 0
\(292\) −12.2882 1.27937i −0.719115 0.0748693i
\(293\) 1.21786i 0.0711483i −0.999367 0.0355741i \(-0.988674\pi\)
0.999367 0.0355741i \(-0.0113260\pi\)
\(294\) 0 0
\(295\) 14.5488i 0.847063i
\(296\) −0.404208 0.554628i −0.0234941 0.0322371i
\(297\) 0 0
\(298\) −24.9706 + 22.5058i −1.44651 + 1.30372i
\(299\) 21.8427 1.26319
\(300\) 0 0
\(301\) 0 0
\(302\) −7.53375 8.35883i −0.433518 0.480997i
\(303\) 0 0
\(304\) −16.8557 3.54827i −0.966743 0.203507i
\(305\) 5.43275 0.311078
\(306\) 0 0
\(307\) −22.5763 −1.28850 −0.644250 0.764815i \(-0.722830\pi\)
−0.644250 + 0.764815i \(0.722830\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 15.8284 + 17.5619i 0.898994 + 0.997451i
\(311\) −21.8427 −1.23858 −0.619292 0.785161i \(-0.712580\pi\)
−0.619292 + 0.785161i \(0.712580\pi\)
\(312\) 0 0
\(313\) 6.17733i 0.349163i −0.984643 0.174582i \(-0.944143\pi\)
0.984643 0.174582i \(-0.0558573\pi\)
\(314\) 10.3725 + 11.5085i 0.585355 + 0.649462i
\(315\) 0 0
\(316\) −6.55635 0.682602i −0.368823 0.0383994i
\(317\) −10.1445 −0.569774 −0.284887 0.958561i \(-0.591956\pi\)
−0.284887 + 0.958561i \(0.591956\pi\)
\(318\) 0 0
\(319\) 12.6181i 0.706477i
\(320\) −12.2463 3.93929i −0.684590 0.220213i
\(321\) 0 0
\(322\) 0 0
\(323\) 6.92468i 0.385300i
\(324\) 0 0
\(325\) 8.15640i 0.452436i
\(326\) −10.6543 11.8212i −0.590089 0.654714i
\(327\) 0 0
\(328\) 26.6214 19.4015i 1.46992 1.07127i
\(329\) 0 0
\(330\) 0 0
\(331\) 22.5058i 1.23703i 0.785773 + 0.618514i \(0.212265\pi\)
−0.785773 + 0.618514i \(0.787735\pi\)
\(332\) −2.64996 + 25.4527i −0.145436 + 1.39690i
\(333\) 0 0
\(334\) −22.9458 + 20.6808i −1.25554 + 1.13161i
\(335\) 18.0951 0.988639
\(336\) 0 0
\(337\) −34.3848 −1.87306 −0.936529 0.350590i \(-0.885981\pi\)
−0.936529 + 0.350590i \(0.885981\pi\)
\(338\) 1.66587 1.50144i 0.0906114 0.0816674i
\(339\) 0 0
\(340\) −0.535534 + 5.14377i −0.0290434 + 0.278960i
\(341\) 27.8411i 1.50768i
\(342\) 0 0
\(343\) 0 0
\(344\) 18.1881 13.2553i 0.980635 0.714679i
\(345\) 0 0
\(346\) −10.6575 11.8247i −0.572952 0.635701i
\(347\) 14.0397i 0.753690i 0.926276 + 0.376845i \(0.122991\pi\)
−0.926276 + 0.376845i \(0.877009\pi\)
\(348\) 0 0
\(349\) 8.15640i 0.436602i 0.975881 + 0.218301i \(0.0700515\pi\)
−0.975881 + 0.218301i \(0.929949\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7.60660 13.1008i −0.405433 0.698273i
\(353\) 1.21786i 0.0648203i 0.999475 + 0.0324101i \(0.0103183\pi\)
−0.999475 + 0.0324101i \(0.989682\pi\)
\(354\) 0 0
\(355\) −12.9189 −0.685663
\(356\) 0.548825 + 0.0571399i 0.0290877 + 0.00302841i
\(357\) 0 0
\(358\) −11.8076 13.1008i −0.624052 0.692397i
\(359\) 6.46521i 0.341221i 0.985339 + 0.170610i \(0.0545740\pi\)
−0.985339 + 0.170610i \(0.945426\pi\)
\(360\) 0 0
\(361\) −0.455844 −0.0239918
\(362\) 14.1201 + 15.6665i 0.742137 + 0.823415i
\(363\) 0 0
\(364\) 0 0
\(365\) 9.93338 0.519937
\(366\) 0 0
\(367\) −18.2701 −0.953689 −0.476844 0.878988i \(-0.658220\pi\)
−0.476844 + 0.878988i \(0.658220\pi\)
\(368\) −5.32716 + 25.3062i −0.277698 + 1.31918i
\(369\) 0 0
\(370\) 0.369422 + 0.409880i 0.0192053 + 0.0213086i
\(371\) 0 0
\(372\) 0 0
\(373\) 2.82843 0.146450 0.0732252 0.997315i \(-0.476671\pi\)
0.0732252 + 0.997315i \(0.476671\pi\)
\(374\) −4.52377 + 4.07724i −0.233918 + 0.210829i
\(375\) 0 0
\(376\) −15.0720 20.6808i −0.777280 1.06653i
\(377\) 15.9188i 0.819858i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 13.7749 + 1.43415i 0.706638 + 0.0735703i
\(381\) 0 0
\(382\) −9.70711 10.7702i −0.496659 0.551052i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.04354 + 7.24958i −0.409405 + 0.368994i
\(387\) 0 0
\(388\) −25.7296 2.67878i −1.30622 0.135995i
\(389\) −0.211161 −0.0107063 −0.00535315 0.999986i \(-0.501704\pi\)
−0.00535315 + 0.999986i \(0.501704\pi\)
\(390\) 0 0
\(391\) 10.3963 0.525764
\(392\) 0 0
\(393\) 0 0
\(394\) 10.6569 9.60494i 0.536885 0.483890i
\(395\) 5.29992 0.266668
\(396\) 0 0
\(397\) 7.33664i 0.368216i 0.982906 + 0.184108i \(0.0589395\pi\)
−0.982906 + 0.184108i \(0.941060\pi\)
\(398\) −15.4451 + 13.9206i −0.774193 + 0.697774i
\(399\) 0 0
\(400\) −9.44975 1.98925i −0.472487 0.0994624i
\(401\) 29.9238 1.49432 0.747162 0.664642i \(-0.231416\pi\)
0.747162 + 0.664642i \(0.231416\pi\)
\(402\) 0 0
\(403\) 35.1239i 1.74964i
\(404\) 32.2152 + 3.35402i 1.60277 + 0.166869i
\(405\) 0 0
\(406\) 0 0
\(407\) 0.649787i 0.0322087i
\(408\) 0 0
\(409\) 33.2053i 1.64189i −0.571004 0.820947i \(-0.693446\pi\)
0.571004 0.820947i \(-0.306554\pi\)
\(410\) −19.6737 + 17.7318i −0.971615 + 0.875709i
\(411\) 0 0
\(412\) 2.15315 20.6808i 0.106078 1.01887i
\(413\) 0 0
\(414\) 0 0
\(415\) 20.5751i 1.00999i
\(416\) −9.59636 16.5277i −0.470500 0.810337i
\(417\) 0 0
\(418\) 10.9188 + 12.1146i 0.534054 + 0.592542i
\(419\) 34.6378 1.69217 0.846084 0.533049i \(-0.178954\pi\)
0.846084 + 0.533049i \(0.178954\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) −15.0675 16.7177i −0.733474 0.813803i
\(423\) 0 0
\(424\) 4.10051 + 5.62644i 0.199138 + 0.273244i
\(425\) 3.88215i 0.188312i
\(426\) 0 0
\(427\) 0 0
\(428\) −12.8609 1.33899i −0.621656 0.0647225i
\(429\) 0 0
\(430\) −13.4413 + 12.1146i −0.648198 + 0.584216i
\(431\) 2.67798i 0.128994i −0.997918 0.0644969i \(-0.979456\pi\)
0.997918 0.0644969i \(-0.0205442\pi\)
\(432\) 0 0
\(433\) 25.6285i 1.23163i −0.787891 0.615814i \(-0.788827\pi\)
0.787891 0.615814i \(-0.211173\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.53553 + 14.7487i −0.0735387 + 0.706335i
\(437\) 27.8411i 1.33182i
\(438\) 0 0
\(439\) −14.7026 −0.701717 −0.350858 0.936429i \(-0.614110\pi\)
−0.350858 + 0.936429i \(0.614110\pi\)
\(440\) 7.17373 + 9.84332i 0.341994 + 0.469262i
\(441\) 0 0
\(442\) −5.70711 + 5.14377i −0.271459 + 0.244664i
\(443\) 33.8948i 1.61039i −0.593010 0.805195i \(-0.702061\pi\)
0.593010 0.805195i \(-0.297939\pi\)
\(444\) 0 0
\(445\) −0.443651 −0.0210311
\(446\) 9.04753 8.15447i 0.428413 0.386125i
\(447\) 0 0
\(448\) 0 0
\(449\) 8.70264 0.410703 0.205351 0.978688i \(-0.434166\pi\)
0.205351 + 0.978688i \(0.434166\pi\)
\(450\) 0 0
\(451\) −31.1889 −1.46863
\(452\) 2.61079 25.0765i 0.122801 1.17950i
\(453\) 0 0
\(454\) 3.93687 3.54827i 0.184767 0.166529i
\(455\) 0 0
\(456\) 0 0
\(457\) 32.4853 1.51960 0.759799 0.650158i \(-0.225297\pi\)
0.759799 + 0.650158i \(0.225297\pi\)
\(458\) −3.97494 4.41027i −0.185737 0.206078i
\(459\) 0 0
\(460\) 2.15315 20.6808i 0.100391 0.964249i
\(461\) 29.4491i 1.37158i −0.727798 0.685792i \(-0.759456\pi\)
0.727798 0.685792i \(-0.240544\pi\)
\(462\) 0 0
\(463\) 38.4198i 1.78552i −0.450535 0.892759i \(-0.648767\pi\)
0.450535 0.892759i \(-0.351233\pi\)
\(464\) −18.4430 3.88239i −0.856193 0.180236i
\(465\) 0 0
\(466\) −3.87868 + 3.49582i −0.179676 + 0.161941i
\(467\) 9.04753 0.418670 0.209335 0.977844i \(-0.432870\pi\)
0.209335 + 0.977844i \(0.432870\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 13.7749 + 15.2835i 0.635389 + 0.704976i
\(471\) 0 0
\(472\) 15.0720 + 20.6808i 0.693746 + 0.951913i
\(473\) −21.3087 −0.979773
\(474\) 0 0
\(475\) 10.3963 0.477015
\(476\) 0 0
\(477\) 0 0
\(478\) −1.05025 1.16527i −0.0480374 0.0532984i
\(479\) 34.6378 1.58264 0.791321 0.611401i \(-0.209394\pi\)
0.791321 + 0.611401i \(0.209394\pi\)
\(480\) 0 0
\(481\) 0.819760i 0.0373778i
\(482\) −1.32498 1.47009i −0.0603512 0.0669608i
\(483\) 0 0
\(484\) 0.792893 7.61569i 0.0360406 0.346168i
\(485\) 20.7989 0.944428
\(486\) 0 0
\(487\) 14.5488i 0.659268i 0.944109 + 0.329634i \(0.106925\pi\)
−0.944109 + 0.329634i \(0.893075\pi\)
\(488\) −7.72255 + 5.62813i −0.349584 + 0.254774i
\(489\) 0 0
\(490\) 0 0
\(491\) 32.3261i 1.45886i −0.684058 0.729428i \(-0.739787\pi\)
0.684058 0.729428i \(-0.260213\pi\)
\(492\) 0 0
\(493\) 7.57675i 0.341239i
\(494\) 13.7749 + 15.2835i 0.619762 + 0.687638i
\(495\) 0 0
\(496\) −40.6934 8.56628i −1.82719 0.384637i
\(497\) 0 0
\(498\) 0 0
\(499\) 19.2099i 0.859952i 0.902840 + 0.429976i \(0.141478\pi\)
−0.902840 + 0.429976i \(0.858522\pi\)
\(500\) 23.7165 + 2.46920i 1.06063 + 0.110426i
\(501\) 0 0
\(502\) −22.9458 + 20.6808i −1.02412 + 0.923031i
\(503\) −5.29992 −0.236312 −0.118156 0.992995i \(-0.537698\pi\)
−0.118156 + 0.992995i \(0.537698\pi\)
\(504\) 0 0
\(505\) −26.0416 −1.15884
\(506\) 18.1881 16.3928i 0.808559 0.728747i
\(507\) 0 0
\(508\) 22.3848 + 2.33055i 0.993164 + 0.103401i
\(509\) 1.21786i 0.0539808i 0.999636 + 0.0269904i \(0.00859236\pi\)
−0.999636 + 0.0269904i \(0.991408\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 21.4889 7.08713i 0.949684 0.313210i
\(513\) 0 0
\(514\) −13.5495 15.0334i −0.597644 0.663097i
\(515\) 16.7177i 0.736668i
\(516\) 0 0
\(517\) 24.2291i 1.06559i
\(518\) 0 0
\(519\) 0 0
\(520\) 9.05025 + 12.4182i 0.396880 + 0.544572i
\(521\) 15.6429i 0.685327i −0.939458 0.342663i \(-0.888671\pi\)
0.939458 0.342663i \(-0.111329\pi\)
\(522\) 0 0
\(523\) 14.7026 0.642900 0.321450 0.946927i \(-0.395830\pi\)
0.321450 + 0.946927i \(0.395830\pi\)
\(524\) −2.64996 + 25.4527i −0.115764 + 1.11191i
\(525\) 0 0
\(526\) 2.53553 + 2.81322i 0.110555 + 0.122662i
\(527\) 16.7177i 0.728233i
\(528\) 0 0
\(529\) −18.7990 −0.817347
\(530\) −3.74761 4.15804i −0.162786 0.180614i
\(531\) 0 0
\(532\) 0 0
\(533\) −39.3474 −1.70433
\(534\) 0 0
\(535\) 10.3963 0.449472
\(536\) −25.7218 + 18.7459i −1.11101 + 0.809698i
\(537\) 0 0
\(538\) 15.3332 + 17.0125i 0.661063 + 0.733462i
\(539\) 0 0
\(540\) 0 0
\(541\) −33.1127 −1.42363 −0.711813 0.702369i \(-0.752126\pi\)
−0.711813 + 0.702369i \(0.752126\pi\)
\(542\) −17.3189 + 15.6094i −0.743911 + 0.670481i
\(543\) 0 0
\(544\) −4.56751 7.86657i −0.195831 0.337277i
\(545\) 11.9223i 0.510697i
\(546\) 0 0
\(547\) 3.29589i 0.140922i 0.997515 + 0.0704611i \(0.0224471\pi\)
−0.997515 + 0.0704611i \(0.977553\pi\)
\(548\) 3.94711 37.9118i 0.168612 1.61951i
\(549\) 0 0
\(550\) 6.12132 + 6.79172i 0.261014 + 0.289600i
\(551\) 20.2904 0.864399
\(552\) 0 0
\(553\) 0 0
\(554\) 25.0009 22.5331i 1.06219 0.957339i
\(555\) 0 0
\(556\) 1.26128 12.1146i 0.0534904 0.513772i
\(557\) −31.4532 −1.33271 −0.666357 0.745633i \(-0.732147\pi\)
−0.666357 + 0.745633i \(0.732147\pi\)
\(558\) 0 0
\(559\) −26.8826 −1.13701
\(560\) 0 0
\(561\) 0 0
\(562\) −8.60660 + 7.75706i −0.363048 + 0.327212i
\(563\) −3.74761 −0.157943 −0.0789715 0.996877i \(-0.525164\pi\)
−0.0789715 + 0.996877i \(0.525164\pi\)
\(564\) 0 0
\(565\) 20.2710i 0.852806i
\(566\) −10.9213 + 9.84332i −0.459058 + 0.413745i
\(567\) 0 0
\(568\) 18.3640 13.3835i 0.770535 0.561559i
\(569\) −12.0962 −0.507100 −0.253550 0.967322i \(-0.581598\pi\)
−0.253550 + 0.967322i \(0.581598\pi\)
\(570\) 0 0
\(571\) 30.4628i 1.27483i −0.770522 0.637413i \(-0.780004\pi\)
0.770522 0.637413i \(-0.219996\pi\)
\(572\) −1.87381 + 17.9978i −0.0783477 + 0.752525i
\(573\) 0 0
\(574\) 0 0
\(575\) 15.6084i 0.650916i
\(576\) 0 0
\(577\) 18.8715i 0.785632i 0.919617 + 0.392816i \(0.128499\pi\)
−0.919617 + 0.392816i \(0.871501\pi\)
\(578\) 15.1422 13.6475i 0.629831 0.567661i
\(579\) 0 0
\(580\) 15.0720 + 1.56920i 0.625832 + 0.0651573i
\(581\) 0 0
\(582\) 0 0
\(583\) 6.59179i 0.273004i
\(584\) −14.1201 + 10.2906i −0.584295 + 0.425829i
\(585\) 0 0
\(586\) −1.15308 1.27937i −0.0476334 0.0528501i
\(587\) 34.6378 1.42966 0.714828 0.699300i \(-0.246505\pi\)
0.714828 + 0.699300i \(0.246505\pi\)
\(588\) 0 0
\(589\) 44.7696 1.84470
\(590\) −13.7749 15.2835i −0.567104 0.629212i
\(591\) 0 0
\(592\) −0.949747 0.199929i −0.0390344 0.00821705i
\(593\) 18.4688i 0.758421i 0.925310 + 0.379211i \(0.123804\pi\)
−0.925310 + 0.379211i \(0.876196\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.92296 + 47.2847i −0.201652 + 1.93686i
\(597\) 0 0
\(598\) 22.9458 20.6808i 0.938322 0.845702i
\(599\) 16.5273i 0.675289i −0.941274 0.337644i \(-0.890370\pi\)
0.941274 0.337644i \(-0.109630\pi\)
\(600\) 0 0
\(601\) 16.8925i 0.689058i −0.938776 0.344529i \(-0.888039\pi\)
0.938776 0.344529i \(-0.111961\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −15.8284 1.64795i −0.644050 0.0670540i
\(605\) 6.15626i 0.250287i
\(606\) 0 0
\(607\) −2.52257 −0.102388 −0.0511939 0.998689i \(-0.516303\pi\)
−0.0511939 + 0.998689i \(0.516303\pi\)
\(608\) −21.0665 + 12.2317i −0.854360 + 0.496061i
\(609\) 0 0
\(610\) 5.70711 5.14377i 0.231074 0.208265i
\(611\) 30.5670i 1.23661i
\(612\) 0 0
\(613\) 22.0416 0.890253 0.445127 0.895468i \(-0.353159\pi\)
0.445127 + 0.895468i \(0.353159\pi\)
\(614\) −23.7165 + 21.3755i −0.957119 + 0.862644i
\(615\) 0 0
\(616\) 0 0
\(617\) −16.5969 −0.668165 −0.334082 0.942544i \(-0.608426\pi\)
−0.334082 + 0.942544i \(0.608426\pi\)
\(618\) 0 0
\(619\) 6.09002 0.244778 0.122389 0.992482i \(-0.460944\pi\)
0.122389 + 0.992482i \(0.460944\pi\)
\(620\) 33.2556 + 3.46234i 1.33558 + 0.139051i
\(621\) 0 0
\(622\) −22.9458 + 20.6808i −0.920041 + 0.829226i
\(623\) 0 0
\(624\) 0 0
\(625\) −7.10051 −0.284020
\(626\) −5.84875 6.48929i −0.233763 0.259364i
\(627\) 0 0
\(628\) 21.7927 + 2.26890i 0.869623 + 0.0905391i
\(629\) 0.390175i 0.0155573i
\(630\) 0 0
\(631\) 45.0115i 1.79188i 0.444174 + 0.895941i \(0.353497\pi\)
−0.444174 + 0.895941i \(0.646503\pi\)
\(632\) −7.53375 + 5.49053i −0.299676 + 0.218402i
\(633\) 0 0
\(634\) −10.6569 + 9.60494i −0.423238 + 0.381461i
\(635\) −18.0951 −0.718081
\(636\) 0 0
\(637\) 0 0
\(638\) 11.9469 + 13.2553i 0.472983 + 0.524783i
\(639\) 0 0
\(640\) −16.5945 + 7.45669i −0.655956 + 0.294752i
\(641\) 28.9043 1.14165 0.570825 0.821072i \(-0.306624\pi\)
0.570825 + 0.821072i \(0.306624\pi\)
\(642\) 0 0
\(643\) 45.8915 1.80979 0.904893 0.425640i \(-0.139951\pi\)
0.904893 + 0.425640i \(0.139951\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.55635 + 7.27439i 0.257956 + 0.286207i
\(647\) 27.1426 1.06709 0.533543 0.845773i \(-0.320860\pi\)
0.533543 + 0.845773i \(0.320860\pi\)
\(648\) 0 0
\(649\) 24.2291i 0.951076i
\(650\) 7.72255 + 8.56831i 0.302903 + 0.336077i
\(651\) 0 0
\(652\) −22.3848 2.33055i −0.876655 0.0912713i
\(653\) 37.3083 1.45999 0.729993 0.683455i \(-0.239523\pi\)
0.729993 + 0.683455i \(0.239523\pi\)
\(654\) 0 0
\(655\) 20.5751i 0.803935i
\(656\) 9.59636 45.5867i 0.374675 1.77986i
\(657\) 0 0
\(658\) 0 0
\(659\) 31.6763i 1.23393i −0.786990 0.616966i \(-0.788361\pi\)
0.786990 0.616966i \(-0.211639\pi\)
\(660\) 0 0
\(661\) 43.5809i 1.69510i 0.530717 + 0.847549i \(0.321923\pi\)
−0.530717 + 0.847549i \(0.678077\pi\)
\(662\) 21.3087 + 23.6423i 0.828184 + 0.918886i
\(663\) 0 0
\(664\) 21.3151 + 29.2471i 0.827185 + 1.13501i
\(665\) 0 0
\(666\) 0 0
\(667\) 30.4628i 1.17952i
\(668\) −4.52377 + 43.4505i −0.175030 + 1.68115i
\(669\) 0 0
\(670\) 19.0089 17.1326i 0.734378 0.661889i
\(671\) 9.04753 0.349276
\(672\) 0 0
\(673\) −16.1005 −0.620629 −0.310314 0.950634i \(-0.600434\pi\)
−0.310314 + 0.950634i \(0.600434\pi\)
\(674\) −36.1213 + 32.5558i −1.39134 + 1.25400i
\(675\) 0 0
\(676\) 0.328427 3.15452i 0.0126318 0.121328i
\(677\) 31.8849i 1.22543i −0.790302 0.612717i \(-0.790076\pi\)
0.790302 0.612717i \(-0.209924\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.30759 + 5.91059i 0.165188 + 0.226661i
\(681\) 0 0
\(682\) 26.3602 + 29.2471i 1.00938 + 1.11993i
\(683\) 16.5273i 0.632401i −0.948692 0.316201i \(-0.897593\pi\)
0.948692 0.316201i \(-0.102407\pi\)
\(684\) 0 0
\(685\) 30.6465i 1.17094i
\(686\) 0 0
\(687\) 0 0
\(688\) 6.55635 31.1454i 0.249958 1.18741i
\(689\) 8.31609i 0.316818i
\(690\) 0 0
\(691\) −12.1800 −0.463350 −0.231675 0.972793i \(-0.574421\pi\)
−0.231675 + 0.972793i \(0.574421\pi\)
\(692\) −22.3915 2.33125i −0.851197 0.0886208i
\(693\) 0 0
\(694\) 13.2929 + 14.7487i 0.504591 + 0.559853i
\(695\) 9.79298i 0.371469i
\(696\) 0 0
\(697\) −18.7279 −0.709371
\(698\) 7.72255 + 8.56831i 0.292303 + 0.324315i
\(699\) 0 0
\(700\) 0 0
\(701\) −19.0583 −0.719824 −0.359912 0.932986i \(-0.617193\pi\)
−0.359912 + 0.932986i \(0.617193\pi\)
\(702\) 0 0
\(703\) 1.04488 0.0394085
\(704\) −20.3947 6.56037i −0.768653 0.247253i
\(705\) 0 0
\(706\) 1.15308 + 1.27937i 0.0433968 + 0.0481496i
\(707\) 0 0
\(708\) 0 0
\(709\) −31.4142 −1.17979 −0.589893 0.807482i \(-0.700830\pi\)
−0.589893 + 0.807482i \(0.700830\pi\)
\(710\) −13.5713 + 12.2317i −0.509322 + 0.459048i
\(711\) 0 0
\(712\) 0.630642 0.459607i 0.0236343 0.0172245i
\(713\) 67.2144i 2.51720i
\(714\) 0 0
\(715\) 14.5488i 0.544093i
\(716\) −24.8078 2.58282i −0.927112 0.0965245i
\(717\) 0 0
\(718\) 6.12132 + 6.79172i 0.228446 + 0.253465i
\(719\) −43.6854 −1.62919 −0.814594 0.580031i \(-0.803040\pi\)
−0.814594 + 0.580031i \(0.803040\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.478865 + 0.431597i −0.0178215 + 0.0160624i
\(723\) 0 0
\(724\) 29.6664 + 3.08866i 1.10254 + 0.114789i
\(725\) 11.3753 0.422467
\(726\) 0 0
\(727\) 1.78372 0.0661547 0.0330773 0.999453i \(-0.489469\pi\)
0.0330773 + 0.999453i \(0.489469\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10.4350 9.40501i 0.386218 0.348095i
\(731\) −12.7951 −0.473246
\(732\) 0 0
\(733\) 23.3099i 0.860971i −0.902598 0.430485i \(-0.858342\pi\)
0.902598 0.430485i \(-0.141658\pi\)
\(734\) −19.1927 + 17.2982i −0.708416 + 0.638490i
\(735\) 0 0
\(736\) 18.3640 + 31.6280i 0.676905 + 1.16582i
\(737\) 30.1350 1.11004
\(738\) 0 0
\(739\) 31.8280i 1.17081i −0.810741 0.585405i \(-0.800935\pi\)
0.810741 0.585405i \(-0.199065\pi\)
\(740\) 0.776156 + 0.0808080i 0.0285321 + 0.00297056i
\(741\) 0 0
\(742\) 0 0
\(743\) 17.8269i 0.654006i 0.945023 + 0.327003i \(0.106039\pi\)
−0.945023 + 0.327003i \(0.893961\pi\)
\(744\) 0 0
\(745\) 38.2233i 1.40039i
\(746\) 2.97127 2.67798i 0.108786 0.0980478i
\(747\) 0 0
\(748\) −0.891862 + 8.56628i −0.0326097 + 0.313214i
\(749\) 0 0
\(750\) 0 0
\(751\) 36.4891i 1.33150i 0.746173 + 0.665752i \(0.231889\pi\)
−0.746173 + 0.665752i \(0.768111\pi\)
\(752\) −35.4140 7.45493i −1.29141 0.271853i
\(753\) 0 0
\(754\) 15.0720 + 16.7227i 0.548891 + 0.609004i
\(755\) 12.7951 0.465663
\(756\) 0 0
\(757\) 11.8995 0.432494 0.216247 0.976339i \(-0.430618\pi\)
0.216247 + 0.976339i \(0.430618\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 15.8284 11.5356i 0.574157 0.418441i
\(761\) 8.26875i 0.299742i 0.988706 + 0.149871i \(0.0478858\pi\)
−0.988706 + 0.149871i \(0.952114\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.3947 2.12335i −0.737853 0.0768202i
\(765\) 0 0
\(766\) 0 0
\(767\) 30.5670i 1.10371i
\(768\) 0 0
\(769\) 43.5809i 1.57157i −0.618502 0.785783i \(-0.712260\pi\)
0.618502 0.785783i \(-0.287740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.58579 + 15.2314i −0.0570737 + 0.548189i
\(773\) 41.7617i 1.50206i −0.660267 0.751031i \(-0.729557\pi\)
0.660267 0.751031i \(-0.270443\pi\)
\(774\) 0 0
\(775\) 25.0989 0.901580
\(776\) −29.5652 + 21.5469i −1.06133 + 0.773489i
\(777\) 0 0
\(778\) −0.221825 + 0.199929i −0.00795283 + 0.00716782i
\(779\) 50.1530i 1.79692i
\(780\) 0 0
\(781\) −21.5147 −0.769857
\(782\) 10.9213 9.84332i 0.390546 0.351996i
\(783\) 0 0
\(784\) 0 0
\(785\) −17.6164 −0.628758
\(786\) 0 0
\(787\) 44.1078 1.57227 0.786137 0.618053i \(-0.212078\pi\)
0.786137 + 0.618053i \(0.212078\pi\)
\(788\) 2.10100 20.1800i 0.0748451 0.718883i
\(789\) 0 0
\(790\) 5.56758 5.01801i 0.198085 0.178533i
\(791\) 0 0
\(792\) 0 0
\(793\) 11.4142 0.405331
\(794\) 6.94640 + 7.70715i 0.246518 + 0.273517i
\(795\) 0 0
\(796\) −3.04501 + 29.2471i −0.107927 + 1.03664i
\(797\) 28.5072i 1.00978i −0.863185 0.504888i \(-0.831534\pi\)
0.863185 0.504888i \(-0.168466\pi\)
\(798\) 0 0
\(799\) 14.5488i 0.514699i
\(800\) −11.8104 + 6.85739i −0.417561 + 0.242446i
\(801\) 0 0
\(802\) 31.4350 28.3321i 1.11001 1.00044i
\(803\) 16.5428 0.583781
\(804\) 0 0
\(805\) 0 0
\(806\) 33.2556 + 36.8977i 1.17138 + 1.29966i
\(807\) 0 0
\(808\) 37.0177 26.9782i 1.30228 0.949090i
\(809\) −29.1154 −1.02364 −0.511822 0.859092i \(-0.671029\pi\)
−0.511822 + 0.859092i \(0.671029\pi\)
\(810\) 0 0
\(811\) −50.1978 −1.76268 −0.881342 0.472479i \(-0.843359\pi\)
−0.881342 + 0.472479i \(0.843359\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.615224 + 0.682602i 0.0215636 + 0.0239252i
\(815\) 18.0951 0.633843
\(816\) 0 0
\(817\) 34.2651i 1.19879i
\(818\) −31.4390 34.8822i −1.09924 1.21963i
\(819\) 0 0
\(820\) −3.87868 + 37.2545i −0.135449 + 1.30098i
\(821\) −19.9905 −0.697672 −0.348836 0.937184i \(-0.613423\pi\)
−0.348836 + 0.937184i \(0.613423\pi\)
\(822\) 0 0
\(823\) 1.36520i 0.0475880i 0.999717 + 0.0237940i \(0.00757458\pi\)
−0.999717 + 0.0237940i \(0.992425\pi\)
\(824\) −17.3189 23.7639i −0.603333 0.827854i
\(825\) 0 0
\(826\) 0 0
\(827\) 22.5331i 0.783552i 0.920061 + 0.391776i \(0.128139\pi\)
−0.920061 + 0.391776i \(0.871861\pi\)
\(828\) 0 0
\(829\) 0.579658i 0.0201323i −0.999949 0.0100662i \(-0.996796\pi\)
0.999949 0.0100662i \(-0.00320422\pi\)
\(830\) −19.4807 21.6142i −0.676184 0.750238i
\(831\) 0 0
\(832\) −25.7296 8.27645i −0.892012 0.286934i
\(833\) 0 0
\(834\) 0 0
\(835\) 35.1239i 1.21551i
\(836\) 22.9403 + 2.38839i 0.793408 + 0.0826042i
\(837\) 0 0
\(838\) 36.3871 32.7954i 1.25697 1.13290i
\(839\) 39.9377 1.37880 0.689402 0.724379i \(-0.257873\pi\)
0.689402 + 0.724379i \(0.257873\pi\)
\(840\) 0 0
\(841\) −6.79899 −0.234448
\(842\) 16.8080 15.1489i 0.579243 0.522067i
\(843\) 0 0
\(844\) −31.6569 3.29589i −1.08967 0.113449i
\(845\) 2.55000i 0.0877228i
\(846\) 0 0
\(847\) 0 0
\(848\) 9.63475 + 2.02819i 0.330859 + 0.0696484i
\(849\) 0 0
\(850\) 3.67565 + 4.07820i 0.126074 + 0.139881i
\(851\) 1.56872i 0.0537752i
\(852\) 0 0
\(853\) 39.1425i 1.34021i 0.742265 + 0.670107i \(0.233752\pi\)
−0.742265 + 0.670107i \(0.766248\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.7782 + 10.7702i −0.505108 + 0.368118i
\(857\) 31.8849i 1.08917i −0.838707 0.544583i \(-0.816688\pi\)
0.838707 0.544583i \(-0.183312\pi\)
\(858\) 0 0
\(859\) 12.9189 0.440786 0.220393 0.975411i \(-0.429266\pi\)
0.220393 + 0.975411i \(0.429266\pi\)
\(860\) −2.64996 + 25.4527i −0.0903629 + 0.867930i
\(861\) 0 0
\(862\) −2.53553 2.81322i −0.0863606 0.0958187i
\(863\) 50.6125i 1.72287i 0.507870 + 0.861434i \(0.330433\pi\)
−0.507870 + 0.861434i \(0.669567\pi\)
\(864\) 0 0
\(865\) 18.1005 0.615436
\(866\) −24.2653 26.9228i −0.824569 0.914874i
\(867\) 0 0
\(868\) 0 0
\(869\) 8.82633 0.299413
\(870\) 0 0
\(871\) 38.0178 1.28818
\(872\) 12.3511 + 16.9474i 0.418262 + 0.573911i
\(873\) 0 0
\(874\) −26.3602 29.2471i −0.891647 0.989299i
\(875\) 0 0
\(876\) 0 0
\(877\) −1.75736 −0.0593418 −0.0296709 0.999560i \(-0.509446\pi\)
−0.0296709 + 0.999560i \(0.509446\pi\)
\(878\) −15.4451 + 13.9206i −0.521247 + 0.469796i
\(879\) 0 0
\(880\) 16.8557 + 3.54827i 0.568207 + 0.119612i
\(881\) 33.9974i 1.14540i −0.819765 0.572700i \(-0.805896\pi\)
0.819765 0.572700i \(-0.194104\pi\)
\(882\) 0 0
\(883\) 44.4461i 1.49573i 0.663852 + 0.747864i \(0.268921\pi\)
−0.663852 + 0.747864i \(0.731079\pi\)
\(884\) −1.12516 + 10.8071i −0.0378432 + 0.363481i
\(885\) 0 0
\(886\) −32.0919 35.6065i −1.07815 1.19622i
\(887\) −9.04753 −0.303786 −0.151893 0.988397i \(-0.548537\pi\)
−0.151893 + 0.988397i \(0.548537\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.466056 + 0.420052i −0.0156222 + 0.0140802i
\(891\) 0 0
\(892\) 1.78372 17.1326i 0.0597235 0.573641i
\(893\) 38.9613 1.30379
\(894\) 0 0
\(895\) 20.0538 0.670324
\(896\) 0 0
\(897\) 0 0
\(898\) 9.14214 8.23973i 0.305077 0.274963i
\(899\) 48.9853 1.63375
\(900\) 0 0
\(901\) 3.95815i 0.131865i
\(902\) −32.7640 + 29.5299i −1.09092 + 0.983240i
\(903\) 0 0
\(904\) −21.0000 28.8148i −0.698450 0.958366i
\(905\) −23.9813 −0.797165
\(906\) 0 0
\(907\) 21.1406i 0.701961i −0.936383 0.350980i \(-0.885848\pi\)
0.936383 0.350980i \(-0.114152\pi\)
\(908\) 0.776156 7.45493i 0.0257576 0.247400i
\(909\) 0 0
\(910\) 0 0
\(911\) 29.4578i 0.975980i −0.872849 0.487990i \(-0.837730\pi\)
0.872849 0.487990i \(-0.162270\pi\)
\(912\) 0 0
\(913\) 34.2651i 1.13401i
\(914\) 34.1258 30.7573i 1.12878 1.01736i
\(915\) 0 0
\(916\) −8.35136 0.869487i −0.275937 0.0287286i
\(917\) 0 0
\(918\) 0 0
\(919\) 14.5488i 0.479920i −0.970783 0.239960i \(-0.922866\pi\)
0.970783 0.239960i \(-0.0771343\pi\)
\(920\) −17.3189 23.7639i −0.570988 0.783472i
\(921\) 0 0
\(922\) −27.8827 30.9364i −0.918268 1.01883i
\(923\) −27.1426 −0.893410
\(924\) 0 0
\(925\) 0.585786 0.0192605
\(926\) −36.3762 40.3600i −1.19539 1.32631i
\(927\) 0 0
\(928\) −23.0503 + 13.3835i −0.756662 + 0.439335i
\(929\) 50.0777i 1.64300i −0.570210 0.821499i \(-0.693138\pi\)
0.570210 0.821499i \(-0.306862\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.764683 + 7.34473i −0.0250480 + 0.240585i
\(933\) 0 0
\(934\) 9.50445 8.56628i 0.310995 0.280297i
\(935\) 6.92468i 0.226461i
\(936\) 0 0
\(937\) 49.5181i 1.61769i −0.588025 0.808843i \(-0.700094\pi\)
0.588025 0.808843i \(-0.299906\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 28.9411 + 3.01315i 0.943955 + 0.0982781i
\(941\) 14.0822i 0.459066i 0.973301 + 0.229533i \(0.0737198\pi\)
−0.973301 + 0.229533i \(0.926280\pi\)
\(942\) 0 0
\(943\) 75.2967 2.45200
\(944\) 35.4140 + 7.45493i 1.15263 + 0.242637i
\(945\) 0 0
\(946\) −22.3848 + 20.1752i −0.727792 + 0.655953i
\(947\) 20.3146i 0.660135i −0.943957 0.330068i \(-0.892928\pi\)
0.943957 0.330068i \(-0.107072\pi\)
\(948\) 0 0
\(949\) 20.8701 0.677471
\(950\) 10.9213 9.84332i 0.354335 0.319359i
\(951\) 0 0
\(952\) 0 0
\(953\) −31.1546 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(954\) 0 0
\(955\) 16.4863 0.533485
\(956\) −2.20658 0.229734i −0.0713660 0.00743014i
\(957\) 0 0
\(958\) 36.3871 32.7954i 1.17561 1.05957i
\(959\) 0 0
\(960\) 0 0
\(961\) 77.0833 2.48656
\(962\) 0.776156 + 0.861159i 0.0250243 + 0.0277649i
\(963\) 0 0
\(964\) −2.78379 0.289829i −0.0896598 0.00933476i
\(965\) 12.3125i 0.396354i
\(966\) 0 0
\(967\) 15.9140i 0.511759i 0.966709 + 0.255880i \(0.0823651\pi\)
−0.966709 + 0.255880i \(0.917635\pi\)
\(968\) −6.37767 8.75101i −0.204986 0.281268i
\(969\) 0 0
\(970\) 21.8492 19.6925i 0.701537 0.632290i
\(971\) −48.9853 −1.57201 −0.786006 0.618219i \(-0.787855\pi\)
−0.786006 + 0.618219i \(0.787855\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 13.7749 + 15.2835i 0.441377 + 0.489715i
\(975\) 0 0
\(976\) −2.78379 + 13.2241i −0.0891069 + 0.423294i
\(977\) −42.8285 −1.37020 −0.685102 0.728447i \(-0.740242\pi\)
−0.685102 + 0.728447i \(0.740242\pi\)
\(978\) 0 0
\(979\) −0.738843 −0.0236135
\(980\) 0 0
\(981\) 0 0
\(982\) −30.6066 33.9586i −0.976696 1.08366i
\(983\) −18.0951 −0.577143 −0.288572 0.957458i \(-0.593180\pi\)
−0.288572 + 0.957458i \(0.593180\pi\)
\(984\) 0 0
\(985\) 16.3128i 0.519769i
\(986\) 7.17373 + 7.95938i 0.228458 + 0.253478i
\(987\) 0 0
\(988\) 28.9411 + 3.01315i 0.920740 + 0.0958611i
\(989\) 51.4436 1.63581
\(990\) 0 0
\(991\) 3.29589i 0.104698i −0.998629 0.0523488i \(-0.983329\pi\)
0.998629 0.0523488i \(-0.0166707\pi\)
\(992\) −50.8591 + 29.5299i −1.61478 + 0.937577i
\(993\) 0 0
\(994\) 0 0
\(995\) 23.6423i 0.749513i
\(996\) 0 0
\(997\) 30.0669i 0.952228i 0.879384 + 0.476114i \(0.157955\pi\)
−0.879384 + 0.476114i \(0.842045\pi\)
\(998\) 18.1881 + 20.1800i 0.575734 + 0.638787i
\(999\) 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 1764.2.b.n.1567.14 yes 16
3.2 odd 2 inner 1764.2.b.n.1567.3 yes 16
4.3 odd 2 inner 1764.2.b.n.1567.16 yes 16
7.6 odd 2 inner 1764.2.b.n.1567.13 yes 16
12.11 even 2 inner 1764.2.b.n.1567.1 16
21.20 even 2 inner 1764.2.b.n.1567.4 yes 16
28.27 even 2 inner 1764.2.b.n.1567.15 yes 16
84.83 odd 2 inner 1764.2.b.n.1567.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.b.n.1567.1 16 12.11 even 2 inner
1764.2.b.n.1567.2 yes 16 84.83 odd 2 inner
1764.2.b.n.1567.3 yes 16 3.2 odd 2 inner
1764.2.b.n.1567.4 yes 16 21.20 even 2 inner
1764.2.b.n.1567.13 yes 16 7.6 odd 2 inner
1764.2.b.n.1567.14 yes 16 1.1 even 1 trivial
1764.2.b.n.1567.15 yes 16 28.27 even 2 inner
1764.2.b.n.1567.16 yes 16 4.3 odd 2 inner