Properties

Label 1040.2.d.c.209.5
Level $1040$
Weight $2$
Character 1040.209
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(209,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.d (of order \(2\), degree \(1\), 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.5
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 1040.209
Dual form 1040.2.d.c.209.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.31111i q^{3} +(-2.21432 + 0.311108i) q^{5} -2.90321i q^{7} +1.28100 q^{9} +O(q^{10})\) \(q+1.31111i q^{3} +(-2.21432 + 0.311108i) q^{5} -2.90321i q^{7} +1.28100 q^{9} -0.214320 q^{11} +1.00000i q^{13} +(-0.407896 - 2.90321i) q^{15} +6.42864i q^{17} +2.21432 q^{19} +3.80642 q^{21} +4.68889i q^{23} +(4.80642 - 1.37778i) q^{25} +5.61285i q^{27} -8.70964 q^{29} +5.59210 q^{31} -0.280996i q^{33} +(0.903212 + 6.42864i) q^{35} +2.28100i q^{37} -1.31111 q^{39} +3.05086 q^{41} +6.36196i q^{43} +(-2.83654 + 0.398528i) q^{45} -1.09679i q^{47} -1.42864 q^{49} -8.42864 q^{51} +6.23506i q^{53} +(0.474572 - 0.0666765i) q^{55} +2.90321i q^{57} -9.26517 q^{59} -0.280996 q^{61} -3.71900i q^{63} +(-0.311108 - 2.21432i) q^{65} +7.76049i q^{67} -6.14764 q^{69} +6.08097 q^{71} +10.2810i q^{73} +(1.80642 + 6.30174i) q^{75} +0.622216i q^{77} -14.2351 q^{79} -3.51606 q^{81} +9.52543i q^{83} +(-2.00000 - 14.2351i) q^{85} -11.4193i q^{87} +5.61285 q^{89} +2.90321 q^{91} +7.33185i q^{93} +(-4.90321 + 0.688892i) q^{95} -18.0415i q^{97} -0.274543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 12 q^{11} - 16 q^{15} - 4 q^{21} + 2 q^{25} - 12 q^{29} + 20 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} + 18 q^{49} - 24 q^{51} + 16 q^{55} - 16 q^{59} + 12 q^{61} - 2 q^{65} - 24 q^{69} + 24 q^{71} - 16 q^{75} - 32 q^{79} + 46 q^{81} - 12 q^{85} - 20 q^{89} + 4 q^{91} - 16 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.31111i 0.756968i 0.925608 + 0.378484i \(0.123555\pi\)
−0.925608 + 0.378484i \(0.876445\pi\)
\(4\) 0 0
\(5\) −2.21432 + 0.311108i −0.990274 + 0.139132i
\(6\) 0 0
\(7\) 2.90321i 1.09731i −0.836049 0.548655i \(-0.815140\pi\)
0.836049 0.548655i \(-0.184860\pi\)
\(8\) 0 0
\(9\) 1.28100 0.426999
\(10\) 0 0
\(11\) −0.214320 −0.0646198 −0.0323099 0.999478i \(-0.510286\pi\)
−0.0323099 + 0.999478i \(0.510286\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −0.407896 2.90321i −0.105318 0.749606i
\(16\) 0 0
\(17\) 6.42864i 1.55917i 0.626294 + 0.779587i \(0.284571\pi\)
−0.626294 + 0.779587i \(0.715429\pi\)
\(18\) 0 0
\(19\) 2.21432 0.508000 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(20\) 0 0
\(21\) 3.80642 0.830630
\(22\) 0 0
\(23\) 4.68889i 0.977702i 0.872367 + 0.488851i \(0.162584\pi\)
−0.872367 + 0.488851i \(0.837416\pi\)
\(24\) 0 0
\(25\) 4.80642 1.37778i 0.961285 0.275557i
\(26\) 0 0
\(27\) 5.61285i 1.08019i
\(28\) 0 0
\(29\) −8.70964 −1.61734 −0.808669 0.588263i \(-0.799812\pi\)
−0.808669 + 0.588263i \(0.799812\pi\)
\(30\) 0 0
\(31\) 5.59210 1.00437 0.502186 0.864760i \(-0.332529\pi\)
0.502186 + 0.864760i \(0.332529\pi\)
\(32\) 0 0
\(33\) 0.280996i 0.0489152i
\(34\) 0 0
\(35\) 0.903212 + 6.42864i 0.152671 + 1.08664i
\(36\) 0 0
\(37\) 2.28100i 0.374993i 0.982265 + 0.187497i \(0.0600374\pi\)
−0.982265 + 0.187497i \(0.939963\pi\)
\(38\) 0 0
\(39\) −1.31111 −0.209945
\(40\) 0 0
\(41\) 3.05086 0.476464 0.238232 0.971208i \(-0.423432\pi\)
0.238232 + 0.971208i \(0.423432\pi\)
\(42\) 0 0
\(43\) 6.36196i 0.970190i 0.874461 + 0.485095i \(0.161215\pi\)
−0.874461 + 0.485095i \(0.838785\pi\)
\(44\) 0 0
\(45\) −2.83654 + 0.398528i −0.422846 + 0.0594090i
\(46\) 0 0
\(47\) 1.09679i 0.159983i −0.996796 0.0799915i \(-0.974511\pi\)
0.996796 0.0799915i \(-0.0254893\pi\)
\(48\) 0 0
\(49\) −1.42864 −0.204091
\(50\) 0 0
\(51\) −8.42864 −1.18025
\(52\) 0 0
\(53\) 6.23506i 0.856452i 0.903672 + 0.428226i \(0.140861\pi\)
−0.903672 + 0.428226i \(0.859139\pi\)
\(54\) 0 0
\(55\) 0.474572 0.0666765i 0.0639913 0.00899066i
\(56\) 0 0
\(57\) 2.90321i 0.384540i
\(58\) 0 0
\(59\) −9.26517 −1.20622 −0.603112 0.797657i \(-0.706073\pi\)
−0.603112 + 0.797657i \(0.706073\pi\)
\(60\) 0 0
\(61\) −0.280996 −0.0359779 −0.0179889 0.999838i \(-0.505726\pi\)
−0.0179889 + 0.999838i \(0.505726\pi\)
\(62\) 0 0
\(63\) 3.71900i 0.468550i
\(64\) 0 0
\(65\) −0.311108 2.21432i −0.0385882 0.274653i
\(66\) 0 0
\(67\) 7.76049i 0.948095i 0.880499 + 0.474047i \(0.157207\pi\)
−0.880499 + 0.474047i \(0.842793\pi\)
\(68\) 0 0
\(69\) −6.14764 −0.740089
\(70\) 0 0
\(71\) 6.08097 0.721678 0.360839 0.932628i \(-0.382490\pi\)
0.360839 + 0.932628i \(0.382490\pi\)
\(72\) 0 0
\(73\) 10.2810i 1.20330i 0.798760 + 0.601650i \(0.205490\pi\)
−0.798760 + 0.601650i \(0.794510\pi\)
\(74\) 0 0
\(75\) 1.80642 + 6.30174i 0.208588 + 0.727662i
\(76\) 0 0
\(77\) 0.622216i 0.0709081i
\(78\) 0 0
\(79\) −14.2351 −1.60157 −0.800785 0.598952i \(-0.795584\pi\)
−0.800785 + 0.598952i \(0.795584\pi\)
\(80\) 0 0
\(81\) −3.51606 −0.390673
\(82\) 0 0
\(83\) 9.52543i 1.04555i 0.852470 + 0.522776i \(0.175103\pi\)
−0.852470 + 0.522776i \(0.824897\pi\)
\(84\) 0 0
\(85\) −2.00000 14.2351i −0.216930 1.54401i
\(86\) 0 0
\(87\) 11.4193i 1.22427i
\(88\) 0 0
\(89\) 5.61285 0.594961 0.297480 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) 2.90321 0.304339
\(92\) 0 0
\(93\) 7.33185i 0.760278i
\(94\) 0 0
\(95\) −4.90321 + 0.688892i −0.503059 + 0.0706788i
\(96\) 0 0
\(97\) 18.0415i 1.83184i −0.401366 0.915918i \(-0.631464\pi\)
0.401366 0.915918i \(-0.368536\pi\)
\(98\) 0 0
\(99\) −0.274543 −0.0275926
\(100\) 0 0
\(101\) 3.93978 0.392022 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(102\) 0 0
\(103\) 2.82225i 0.278084i 0.990286 + 0.139042i \(0.0444023\pi\)
−0.990286 + 0.139042i \(0.955598\pi\)
\(104\) 0 0
\(105\) −8.42864 + 1.18421i −0.822551 + 0.115567i
\(106\) 0 0
\(107\) 17.1175i 1.65481i −0.561603 0.827407i \(-0.689815\pi\)
0.561603 0.827407i \(-0.310185\pi\)
\(108\) 0 0
\(109\) 16.7239 1.60186 0.800931 0.598757i \(-0.204338\pi\)
0.800931 + 0.598757i \(0.204338\pi\)
\(110\) 0 0
\(111\) −2.99063 −0.283858
\(112\) 0 0
\(113\) 1.18421i 0.111401i −0.998448 0.0557005i \(-0.982261\pi\)
0.998448 0.0557005i \(-0.0177392\pi\)
\(114\) 0 0
\(115\) −1.45875 10.3827i −0.136029 0.968192i
\(116\) 0 0
\(117\) 1.28100i 0.118428i
\(118\) 0 0
\(119\) 18.6637 1.71090
\(120\) 0 0
\(121\) −10.9541 −0.995824
\(122\) 0 0
\(123\) 4.00000i 0.360668i
\(124\) 0 0
\(125\) −10.2143 + 4.54617i −0.913597 + 0.406622i
\(126\) 0 0
\(127\) 2.30174i 0.204246i −0.994772 0.102123i \(-0.967436\pi\)
0.994772 0.102123i \(-0.0325636\pi\)
\(128\) 0 0
\(129\) −8.34122 −0.734403
\(130\) 0 0
\(131\) 13.4193 1.17245 0.586224 0.810149i \(-0.300614\pi\)
0.586224 + 0.810149i \(0.300614\pi\)
\(132\) 0 0
\(133\) 6.42864i 0.557434i
\(134\) 0 0
\(135\) −1.74620 12.4286i −0.150289 1.06969i
\(136\) 0 0
\(137\) 19.1526i 1.63631i 0.574995 + 0.818157i \(0.305004\pi\)
−0.574995 + 0.818157i \(0.694996\pi\)
\(138\) 0 0
\(139\) 19.0923 1.61939 0.809696 0.586850i \(-0.199632\pi\)
0.809696 + 0.586850i \(0.199632\pi\)
\(140\) 0 0
\(141\) 1.43801 0.121102
\(142\) 0 0
\(143\) 0.214320i 0.0179223i
\(144\) 0 0
\(145\) 19.2859 2.70964i 1.60161 0.225023i
\(146\) 0 0
\(147\) 1.87310i 0.154491i
\(148\) 0 0
\(149\) 3.57136 0.292577 0.146289 0.989242i \(-0.453267\pi\)
0.146289 + 0.989242i \(0.453267\pi\)
\(150\) 0 0
\(151\) 1.26517 0.102958 0.0514792 0.998674i \(-0.483606\pi\)
0.0514792 + 0.998674i \(0.483606\pi\)
\(152\) 0 0
\(153\) 8.23506i 0.665765i
\(154\) 0 0
\(155\) −12.3827 + 1.73975i −0.994603 + 0.139740i
\(156\) 0 0
\(157\) 5.61285i 0.447954i −0.974594 0.223977i \(-0.928096\pi\)
0.974594 0.223977i \(-0.0719041\pi\)
\(158\) 0 0
\(159\) −8.17484 −0.648307
\(160\) 0 0
\(161\) 13.6128 1.07284
\(162\) 0 0
\(163\) 3.71900i 0.291295i 0.989337 + 0.145647i \(0.0465265\pi\)
−0.989337 + 0.145647i \(0.953473\pi\)
\(164\) 0 0
\(165\) 0.0874201 + 0.622216i 0.00680565 + 0.0484394i
\(166\) 0 0
\(167\) 7.03657i 0.544506i 0.962226 + 0.272253i \(0.0877687\pi\)
−0.962226 + 0.272253i \(0.912231\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.83654 0.216915
\(172\) 0 0
\(173\) 0.723926i 0.0550391i −0.999621 0.0275195i \(-0.991239\pi\)
0.999621 0.0275195i \(-0.00876085\pi\)
\(174\) 0 0
\(175\) −4.00000 13.9541i −0.302372 1.05483i
\(176\) 0 0
\(177\) 12.1476i 0.913073i
\(178\) 0 0
\(179\) −4.04149 −0.302075 −0.151037 0.988528i \(-0.548261\pi\)
−0.151037 + 0.988528i \(0.548261\pi\)
\(180\) 0 0
\(181\) 2.34122 0.174021 0.0870107 0.996207i \(-0.472269\pi\)
0.0870107 + 0.996207i \(0.472269\pi\)
\(182\) 0 0
\(183\) 0.368416i 0.0272341i
\(184\) 0 0
\(185\) −0.709636 5.05086i −0.0521735 0.371346i
\(186\) 0 0
\(187\) 1.37778i 0.100754i
\(188\) 0 0
\(189\) 16.2953 1.18531
\(190\) 0 0
\(191\) 2.10171 0.152074 0.0760372 0.997105i \(-0.475773\pi\)
0.0760372 + 0.997105i \(0.475773\pi\)
\(192\) 0 0
\(193\) 13.5210i 0.973262i −0.873608 0.486631i \(-0.838226\pi\)
0.873608 0.486631i \(-0.161774\pi\)
\(194\) 0 0
\(195\) 2.90321 0.407896i 0.207903 0.0292100i
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 22.1432 1.56969 0.784845 0.619692i \(-0.212743\pi\)
0.784845 + 0.619692i \(0.212743\pi\)
\(200\) 0 0
\(201\) −10.1748 −0.717678
\(202\) 0 0
\(203\) 25.2859i 1.77472i
\(204\) 0 0
\(205\) −6.75557 + 0.949145i −0.471829 + 0.0662912i
\(206\) 0 0
\(207\) 6.00645i 0.417477i
\(208\) 0 0
\(209\) −0.474572 −0.0328269
\(210\) 0 0
\(211\) −19.6543 −1.35306 −0.676530 0.736415i \(-0.736517\pi\)
−0.676530 + 0.736415i \(0.736517\pi\)
\(212\) 0 0
\(213\) 7.97280i 0.546287i
\(214\) 0 0
\(215\) −1.97926 14.0874i −0.134984 0.960754i
\(216\) 0 0
\(217\) 16.2351i 1.10211i
\(218\) 0 0
\(219\) −13.4795 −0.910860
\(220\) 0 0
\(221\) −6.42864 −0.432437
\(222\) 0 0
\(223\) 19.6686i 1.31711i −0.752533 0.658554i \(-0.771168\pi\)
0.752533 0.658554i \(-0.228832\pi\)
\(224\) 0 0
\(225\) 6.15701 1.76494i 0.410467 0.117662i
\(226\) 0 0
\(227\) 13.2716i 0.880869i −0.897785 0.440434i \(-0.854824\pi\)
0.897785 0.440434i \(-0.145176\pi\)
\(228\) 0 0
\(229\) 2.42864 0.160489 0.0802445 0.996775i \(-0.474430\pi\)
0.0802445 + 0.996775i \(0.474430\pi\)
\(230\) 0 0
\(231\) −0.815792 −0.0536752
\(232\) 0 0
\(233\) 16.1748i 1.05965i 0.848107 + 0.529825i \(0.177742\pi\)
−0.848107 + 0.529825i \(0.822258\pi\)
\(234\) 0 0
\(235\) 0.341219 + 2.42864i 0.0222587 + 0.158427i
\(236\) 0 0
\(237\) 18.6637i 1.21234i
\(238\) 0 0
\(239\) −12.7763 −0.826431 −0.413215 0.910633i \(-0.635594\pi\)
−0.413215 + 0.910633i \(0.635594\pi\)
\(240\) 0 0
\(241\) −5.89829 −0.379942 −0.189971 0.981790i \(-0.560839\pi\)
−0.189971 + 0.981790i \(0.560839\pi\)
\(242\) 0 0
\(243\) 12.2286i 0.784466i
\(244\) 0 0
\(245\) 3.16346 0.444461i 0.202106 0.0283956i
\(246\) 0 0
\(247\) 2.21432i 0.140894i
\(248\) 0 0
\(249\) −12.4889 −0.791450
\(250\) 0 0
\(251\) 2.07313 0.130855 0.0654274 0.997857i \(-0.479159\pi\)
0.0654274 + 0.997857i \(0.479159\pi\)
\(252\) 0 0
\(253\) 1.00492i 0.0631789i
\(254\) 0 0
\(255\) 18.6637 2.62222i 1.16877 0.164210i
\(256\) 0 0
\(257\) 18.3970i 1.14757i −0.819005 0.573787i \(-0.805474\pi\)
0.819005 0.573787i \(-0.194526\pi\)
\(258\) 0 0
\(259\) 6.62222 0.411484
\(260\) 0 0
\(261\) −11.1570 −0.690602
\(262\) 0 0
\(263\) 11.0257i 0.679872i −0.940449 0.339936i \(-0.889595\pi\)
0.940449 0.339936i \(-0.110405\pi\)
\(264\) 0 0
\(265\) −1.93978 13.8064i −0.119160 0.848122i
\(266\) 0 0
\(267\) 7.35905i 0.450366i
\(268\) 0 0
\(269\) 16.1432 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(270\) 0 0
\(271\) −13.0114 −0.790385 −0.395192 0.918598i \(-0.629322\pi\)
−0.395192 + 0.918598i \(0.629322\pi\)
\(272\) 0 0
\(273\) 3.80642i 0.230375i
\(274\) 0 0
\(275\) −1.03011 + 0.295286i −0.0621181 + 0.0178064i
\(276\) 0 0
\(277\) 7.57136i 0.454919i −0.973788 0.227459i \(-0.926958\pi\)
0.973788 0.227459i \(-0.0730419\pi\)
\(278\) 0 0
\(279\) 7.16346 0.428865
\(280\) 0 0
\(281\) 6.75557 0.403003 0.201502 0.979488i \(-0.435418\pi\)
0.201502 + 0.979488i \(0.435418\pi\)
\(282\) 0 0
\(283\) 19.0859i 1.13454i −0.823532 0.567269i \(-0.808000\pi\)
0.823532 0.567269i \(-0.192000\pi\)
\(284\) 0 0
\(285\) −0.903212 6.42864i −0.0535017 0.380800i
\(286\) 0 0
\(287\) 8.85728i 0.522829i
\(288\) 0 0
\(289\) −24.3274 −1.43102
\(290\) 0 0
\(291\) 23.6543 1.38664
\(292\) 0 0
\(293\) 8.08742i 0.472472i −0.971696 0.236236i \(-0.924086\pi\)
0.971696 0.236236i \(-0.0759139\pi\)
\(294\) 0 0
\(295\) 20.5161 2.88247i 1.19449 0.167824i
\(296\) 0 0
\(297\) 1.20294i 0.0698019i
\(298\) 0 0
\(299\) −4.68889 −0.271166
\(300\) 0 0
\(301\) 18.4701 1.06460
\(302\) 0 0
\(303\) 5.16547i 0.296749i
\(304\) 0 0
\(305\) 0.622216 0.0874201i 0.0356280 0.00500566i
\(306\) 0 0
\(307\) 13.4336i 0.766694i −0.923604 0.383347i \(-0.874771\pi\)
0.923604 0.383347i \(-0.125229\pi\)
\(308\) 0 0
\(309\) −3.70027 −0.210501
\(310\) 0 0
\(311\) −20.2034 −1.14563 −0.572815 0.819684i \(-0.694149\pi\)
−0.572815 + 0.819684i \(0.694149\pi\)
\(312\) 0 0
\(313\) 15.1111i 0.854129i 0.904221 + 0.427064i \(0.140452\pi\)
−0.904221 + 0.427064i \(0.859548\pi\)
\(314\) 0 0
\(315\) 1.15701 + 8.23506i 0.0651902 + 0.463993i
\(316\) 0 0
\(317\) 22.2810i 1.25143i −0.780054 0.625713i \(-0.784808\pi\)
0.780054 0.625713i \(-0.215192\pi\)
\(318\) 0 0
\(319\) 1.86665 0.104512
\(320\) 0 0
\(321\) 22.4429 1.25264
\(322\) 0 0
\(323\) 14.2351i 0.792060i
\(324\) 0 0
\(325\) 1.37778 + 4.80642i 0.0764257 + 0.266612i
\(326\) 0 0
\(327\) 21.9269i 1.21256i
\(328\) 0 0
\(329\) −3.18421 −0.175551
\(330\) 0 0
\(331\) −8.25581 −0.453780 −0.226890 0.973920i \(-0.572856\pi\)
−0.226890 + 0.973920i \(0.572856\pi\)
\(332\) 0 0
\(333\) 2.92195i 0.160122i
\(334\) 0 0
\(335\) −2.41435 17.1842i −0.131910 0.938874i
\(336\) 0 0
\(337\) 13.7462i 0.748803i 0.927267 + 0.374402i \(0.122152\pi\)
−0.927267 + 0.374402i \(0.877848\pi\)
\(338\) 0 0
\(339\) 1.55262 0.0843270
\(340\) 0 0
\(341\) −1.19850 −0.0649023
\(342\) 0 0
\(343\) 16.1748i 0.873359i
\(344\) 0 0
\(345\) 13.6128 1.91258i 0.732891 0.102970i
\(346\) 0 0
\(347\) 1.21924i 0.0654523i 0.999464 + 0.0327262i \(0.0104189\pi\)
−0.999464 + 0.0327262i \(0.989581\pi\)
\(348\) 0 0
\(349\) −22.5116 −1.20502 −0.602510 0.798112i \(-0.705832\pi\)
−0.602510 + 0.798112i \(0.705832\pi\)
\(350\) 0 0
\(351\) −5.61285 −0.299592
\(352\) 0 0
\(353\) 14.2810i 0.760101i −0.924966 0.380050i \(-0.875907\pi\)
0.924966 0.380050i \(-0.124093\pi\)
\(354\) 0 0
\(355\) −13.4652 + 1.89184i −0.714659 + 0.100408i
\(356\) 0 0
\(357\) 24.4701i 1.29510i
\(358\) 0 0
\(359\) −12.1541 −0.641469 −0.320734 0.947169i \(-0.603930\pi\)
−0.320734 + 0.947169i \(0.603930\pi\)
\(360\) 0 0
\(361\) −14.0968 −0.741936
\(362\) 0 0
\(363\) 14.3620i 0.753808i
\(364\) 0 0
\(365\) −3.19850 22.7654i −0.167417 1.19160i
\(366\) 0 0
\(367\) 4.65725i 0.243106i −0.992585 0.121553i \(-0.961212\pi\)
0.992585 0.121553i \(-0.0387875\pi\)
\(368\) 0 0
\(369\) 3.90813 0.203449
\(370\) 0 0
\(371\) 18.1017 0.939794
\(372\) 0 0
\(373\) 34.9403i 1.80914i 0.426328 + 0.904569i \(0.359807\pi\)
−0.426328 + 0.904569i \(0.640193\pi\)
\(374\) 0 0
\(375\) −5.96052 13.3921i −0.307800 0.691564i
\(376\) 0 0
\(377\) 8.70964i 0.448569i
\(378\) 0 0
\(379\) −17.4717 −0.897459 −0.448729 0.893668i \(-0.648123\pi\)
−0.448729 + 0.893668i \(0.648123\pi\)
\(380\) 0 0
\(381\) 3.01783 0.154608
\(382\) 0 0
\(383\) 18.6780i 0.954401i 0.878794 + 0.477200i \(0.158348\pi\)
−0.878794 + 0.477200i \(0.841652\pi\)
\(384\) 0 0
\(385\) −0.193576 1.37778i −0.00986555 0.0702184i
\(386\) 0 0
\(387\) 8.14965i 0.414270i
\(388\) 0 0
\(389\) −1.61285 −0.0817746 −0.0408873 0.999164i \(-0.513018\pi\)
−0.0408873 + 0.999164i \(0.513018\pi\)
\(390\) 0 0
\(391\) −30.1432 −1.52441
\(392\) 0 0
\(393\) 17.5941i 0.887506i
\(394\) 0 0
\(395\) 31.5210 4.42864i 1.58599 0.222829i
\(396\) 0 0
\(397\) 6.57628i 0.330054i −0.986289 0.165027i \(-0.947229\pi\)
0.986289 0.165027i \(-0.0527712\pi\)
\(398\) 0 0
\(399\) 8.42864 0.421960
\(400\) 0 0
\(401\) −21.9081 −1.09404 −0.547020 0.837120i \(-0.684238\pi\)
−0.547020 + 0.837120i \(0.684238\pi\)
\(402\) 0 0
\(403\) 5.59210i 0.278563i
\(404\) 0 0
\(405\) 7.78568 1.09387i 0.386874 0.0543550i
\(406\) 0 0
\(407\) 0.488863i 0.0242320i
\(408\) 0 0
\(409\) 10.1936 0.504040 0.252020 0.967722i \(-0.418905\pi\)
0.252020 + 0.967722i \(0.418905\pi\)
\(410\) 0 0
\(411\) −25.1111 −1.23864
\(412\) 0 0
\(413\) 26.8988i 1.32360i
\(414\) 0 0
\(415\) −2.96343 21.0923i −0.145469 1.03538i
\(416\) 0 0
\(417\) 25.0321i 1.22583i
\(418\) 0 0
\(419\) 7.31756 0.357486 0.178743 0.983896i \(-0.442797\pi\)
0.178743 + 0.983896i \(0.442797\pi\)
\(420\) 0 0
\(421\) −7.86665 −0.383397 −0.191698 0.981454i \(-0.561400\pi\)
−0.191698 + 0.981454i \(0.561400\pi\)
\(422\) 0 0
\(423\) 1.40498i 0.0683125i
\(424\) 0 0
\(425\) 8.85728 + 30.8988i 0.429641 + 1.49881i
\(426\) 0 0
\(427\) 0.815792i 0.0394789i
\(428\) 0 0
\(429\) 0.280996 0.0135666
\(430\) 0 0
\(431\) 38.9195 1.87469 0.937343 0.348407i \(-0.113277\pi\)
0.937343 + 0.348407i \(0.113277\pi\)
\(432\) 0 0
\(433\) 20.2034i 0.970914i −0.874260 0.485457i \(-0.838653\pi\)
0.874260 0.485457i \(-0.161347\pi\)
\(434\) 0 0
\(435\) 3.55262 + 25.2859i 0.170335 + 1.21237i
\(436\) 0 0
\(437\) 10.3827i 0.496672i
\(438\) 0 0
\(439\) 10.8889 0.519700 0.259850 0.965649i \(-0.416327\pi\)
0.259850 + 0.965649i \(0.416327\pi\)
\(440\) 0 0
\(441\) −1.83008 −0.0871468
\(442\) 0 0
\(443\) 28.6287i 1.36019i 0.733124 + 0.680095i \(0.238061\pi\)
−0.733124 + 0.680095i \(0.761939\pi\)
\(444\) 0 0
\(445\) −12.4286 + 1.74620i −0.589174 + 0.0827779i
\(446\) 0 0
\(447\) 4.68244i 0.221472i
\(448\) 0 0
\(449\) 10.9304 0.515838 0.257919 0.966167i \(-0.416963\pi\)
0.257919 + 0.966167i \(0.416963\pi\)
\(450\) 0 0
\(451\) −0.653858 −0.0307890
\(452\) 0 0
\(453\) 1.65878i 0.0779363i
\(454\) 0 0
\(455\) −6.42864 + 0.903212i −0.301379 + 0.0423432i
\(456\) 0 0
\(457\) 11.4064i 0.533567i 0.963756 + 0.266784i \(0.0859609\pi\)
−0.963756 + 0.266784i \(0.914039\pi\)
\(458\) 0 0
\(459\) −36.0830 −1.68421
\(460\) 0 0
\(461\) −26.1334 −1.21715 −0.608576 0.793496i \(-0.708259\pi\)
−0.608576 + 0.793496i \(0.708259\pi\)
\(462\) 0 0
\(463\) 7.92242i 0.368186i 0.982909 + 0.184093i \(0.0589348\pi\)
−0.982909 + 0.184093i \(0.941065\pi\)
\(464\) 0 0
\(465\) −2.28100 16.2351i −0.105779 0.752883i
\(466\) 0 0
\(467\) 10.8923i 0.504036i 0.967723 + 0.252018i \(0.0810942\pi\)
−0.967723 + 0.252018i \(0.918906\pi\)
\(468\) 0 0
\(469\) 22.5303 1.04035
\(470\) 0 0
\(471\) 7.35905 0.339087
\(472\) 0 0
\(473\) 1.36349i 0.0626935i
\(474\) 0 0
\(475\) 10.6430 3.05086i 0.488332 0.139983i
\(476\) 0 0
\(477\) 7.98709i 0.365704i
\(478\) 0 0
\(479\) −9.13182 −0.417244 −0.208622 0.977996i \(-0.566898\pi\)
−0.208622 + 0.977996i \(0.566898\pi\)
\(480\) 0 0
\(481\) −2.28100 −0.104004
\(482\) 0 0
\(483\) 17.8479i 0.812108i
\(484\) 0 0
\(485\) 5.61285 + 39.9496i 0.254866 + 1.81402i
\(486\) 0 0
\(487\) 16.1891i 0.733600i −0.930300 0.366800i \(-0.880453\pi\)
0.930300 0.366800i \(-0.119547\pi\)
\(488\) 0 0
\(489\) −4.87601 −0.220501
\(490\) 0 0
\(491\) −26.2636 −1.18526 −0.592631 0.805474i \(-0.701911\pi\)
−0.592631 + 0.805474i \(0.701911\pi\)
\(492\) 0 0
\(493\) 55.9911i 2.52171i
\(494\) 0 0
\(495\) 0.607926 0.0854124i 0.0273242 0.00383900i
\(496\) 0 0
\(497\) 17.6543i 0.791905i
\(498\) 0 0
\(499\) 30.0306 1.34435 0.672177 0.740391i \(-0.265359\pi\)
0.672177 + 0.740391i \(0.265359\pi\)
\(500\) 0 0
\(501\) −9.22570 −0.412174
\(502\) 0 0
\(503\) 16.7304i 0.745971i 0.927837 + 0.372985i \(0.121666\pi\)
−0.927837 + 0.372985i \(0.878334\pi\)
\(504\) 0 0
\(505\) −8.72393 + 1.22570i −0.388210 + 0.0545427i
\(506\) 0 0
\(507\) 1.31111i 0.0582283i
\(508\) 0 0
\(509\) 11.9684 0.530488 0.265244 0.964181i \(-0.414547\pi\)
0.265244 + 0.964181i \(0.414547\pi\)
\(510\) 0 0
\(511\) 29.8479 1.32039
\(512\) 0 0
\(513\) 12.4286i 0.548738i
\(514\) 0 0
\(515\) −0.878023 6.24935i −0.0386903 0.275379i
\(516\) 0 0
\(517\) 0.235063i 0.0103381i
\(518\) 0 0
\(519\) 0.949145 0.0416628
\(520\) 0 0
\(521\) 5.75065 0.251940 0.125970 0.992034i \(-0.459796\pi\)
0.125970 + 0.992034i \(0.459796\pi\)
\(522\) 0 0
\(523\) 20.8035i 0.909674i 0.890575 + 0.454837i \(0.150302\pi\)
−0.890575 + 0.454837i \(0.849698\pi\)
\(524\) 0 0
\(525\) 18.2953 5.24443i 0.798472 0.228886i
\(526\) 0 0
\(527\) 35.9496i 1.56599i
\(528\) 0 0
\(529\) 1.01429 0.0440996
\(530\) 0 0
\(531\) −11.8687 −0.515056
\(532\) 0 0
\(533\) 3.05086i 0.132147i
\(534\) 0 0
\(535\) 5.32540 + 37.9037i 0.230237 + 1.63872i
\(536\) 0 0
\(537\) 5.29883i 0.228661i
\(538\) 0 0
\(539\) 0.306186 0.0131883
\(540\) 0 0
\(541\) 16.6222 0.714645 0.357322 0.933981i \(-0.383690\pi\)
0.357322 + 0.933981i \(0.383690\pi\)
\(542\) 0 0
\(543\) 3.06959i 0.131729i
\(544\) 0 0
\(545\) −37.0321 + 5.20294i −1.58628 + 0.222870i
\(546\) 0 0
\(547\) 29.9748i 1.28163i −0.767695 0.640815i \(-0.778596\pi\)
0.767695 0.640815i \(-0.221404\pi\)
\(548\) 0 0
\(549\) −0.359955 −0.0153625
\(550\) 0 0
\(551\) −19.2859 −0.821608
\(552\) 0 0
\(553\) 41.3274i 1.75742i
\(554\) 0 0
\(555\) 6.62222 0.930409i 0.281097 0.0394937i
\(556\) 0 0
\(557\) 5.03657i 0.213406i 0.994291 + 0.106703i \(0.0340294\pi\)
−0.994291 + 0.106703i \(0.965971\pi\)
\(558\) 0 0
\(559\) −6.36196 −0.269082
\(560\) 0 0
\(561\) 1.80642 0.0762673
\(562\) 0 0
\(563\) 2.88247i 0.121482i −0.998154 0.0607408i \(-0.980654\pi\)
0.998154 0.0607408i \(-0.0193463\pi\)
\(564\) 0 0
\(565\) 0.368416 + 2.62222i 0.0154994 + 0.110317i
\(566\) 0 0
\(567\) 10.2079i 0.428690i
\(568\) 0 0
\(569\) 4.37286 0.183320 0.0916600 0.995790i \(-0.470783\pi\)
0.0916600 + 0.995790i \(0.470783\pi\)
\(570\) 0 0
\(571\) 1.58120 0.0661714 0.0330857 0.999453i \(-0.489467\pi\)
0.0330857 + 0.999453i \(0.489467\pi\)
\(572\) 0 0
\(573\) 2.75557i 0.115116i
\(574\) 0 0
\(575\) 6.46028 + 22.5368i 0.269412 + 0.939850i
\(576\) 0 0
\(577\) 7.61729i 0.317112i 0.987350 + 0.158556i \(0.0506839\pi\)
−0.987350 + 0.158556i \(0.949316\pi\)
\(578\) 0 0
\(579\) 17.7275 0.736728
\(580\) 0 0
\(581\) 27.6543 1.14730
\(582\) 0 0
\(583\) 1.33630i 0.0553438i
\(584\) 0 0
\(585\) −0.398528 2.83654i −0.0164771 0.117276i
\(586\) 0 0
\(587\) 46.8243i 1.93264i −0.257336 0.966322i \(-0.582845\pi\)
0.257336 0.966322i \(-0.417155\pi\)
\(588\) 0 0
\(589\) 12.3827 0.510221
\(590\) 0 0
\(591\) −2.62222 −0.107864
\(592\) 0 0
\(593\) 15.9398i 0.654568i −0.944926 0.327284i \(-0.893867\pi\)
0.944926 0.327284i \(-0.106133\pi\)
\(594\) 0 0
\(595\) −41.3274 + 5.80642i −1.69426 + 0.238040i
\(596\) 0 0
\(597\) 29.0321i 1.18821i
\(598\) 0 0
\(599\) 18.4889 0.755434 0.377717 0.925921i \(-0.376709\pi\)
0.377717 + 0.925921i \(0.376709\pi\)
\(600\) 0 0
\(601\) 20.7556 0.846637 0.423319 0.905981i \(-0.360865\pi\)
0.423319 + 0.905981i \(0.360865\pi\)
\(602\) 0 0
\(603\) 9.94116i 0.404835i
\(604\) 0 0
\(605\) 24.2558 3.40790i 0.986139 0.138551i
\(606\) 0 0
\(607\) 36.0765i 1.46430i 0.681143 + 0.732150i \(0.261483\pi\)
−0.681143 + 0.732150i \(0.738517\pi\)
\(608\) 0 0
\(609\) −33.1526 −1.34341
\(610\) 0 0
\(611\) 1.09679 0.0443713
\(612\) 0 0
\(613\) 9.94962i 0.401861i 0.979605 + 0.200931i \(0.0643966\pi\)
−0.979605 + 0.200931i \(0.935603\pi\)
\(614\) 0 0
\(615\) −1.24443 8.85728i −0.0501803 0.357160i
\(616\) 0 0
\(617\) 2.09187i 0.0842154i −0.999113 0.0421077i \(-0.986593\pi\)
0.999113 0.0421077i \(-0.0134073\pi\)
\(618\) 0 0
\(619\) 18.4681 0.742296 0.371148 0.928574i \(-0.378964\pi\)
0.371148 + 0.928574i \(0.378964\pi\)
\(620\) 0 0
\(621\) −26.3180 −1.05611
\(622\) 0 0
\(623\) 16.2953i 0.652857i
\(624\) 0 0
\(625\) 21.2034 13.2444i 0.848137 0.529777i
\(626\) 0 0
\(627\) 0.622216i 0.0248489i
\(628\) 0 0
\(629\) −14.6637 −0.584680
\(630\) 0 0
\(631\) 38.6657 1.53926 0.769629 0.638492i \(-0.220441\pi\)
0.769629 + 0.638492i \(0.220441\pi\)
\(632\) 0 0
\(633\) 25.7690i 1.02422i
\(634\) 0 0
\(635\) 0.716089 + 5.09679i 0.0284171 + 0.202260i
\(636\) 0 0
\(637\) 1.42864i 0.0566048i
\(638\) 0 0
\(639\) 7.78970 0.308156
\(640\) 0 0
\(641\) 24.5718 0.970529 0.485265 0.874367i \(-0.338723\pi\)
0.485265 + 0.874367i \(0.338723\pi\)
\(642\) 0 0
\(643\) 27.4938i 1.08425i −0.840298 0.542125i \(-0.817620\pi\)
0.840298 0.542125i \(-0.182380\pi\)
\(644\) 0 0
\(645\) 18.4701 2.59502i 0.727261 0.102179i
\(646\) 0 0
\(647\) 13.7812i 0.541796i 0.962608 + 0.270898i \(0.0873207\pi\)
−0.962608 + 0.270898i \(0.912679\pi\)
\(648\) 0 0
\(649\) 1.98571 0.0779459
\(650\) 0 0
\(651\) 21.2859 0.834261
\(652\) 0 0
\(653\) 2.12045i 0.0829795i −0.999139 0.0414897i \(-0.986790\pi\)
0.999139 0.0414897i \(-0.0132104\pi\)
\(654\) 0 0
\(655\) −29.7146 + 4.17484i −1.16104 + 0.163125i
\(656\) 0 0
\(657\) 13.1699i 0.513807i
\(658\) 0 0
\(659\) 33.8894 1.32014 0.660072 0.751203i \(-0.270526\pi\)
0.660072 + 0.751203i \(0.270526\pi\)
\(660\) 0 0
\(661\) −37.3689 −1.45348 −0.726741 0.686912i \(-0.758966\pi\)
−0.726741 + 0.686912i \(0.758966\pi\)
\(662\) 0 0
\(663\) 8.42864i 0.327341i
\(664\) 0 0
\(665\) 2.00000 + 14.2351i 0.0775567 + 0.552012i
\(666\) 0 0
\(667\) 40.8385i 1.58127i
\(668\) 0 0
\(669\) 25.7877 0.997010
\(670\) 0 0
\(671\) 0.0602231 0.00232489
\(672\) 0 0
\(673\) 35.4608i 1.36691i −0.729992 0.683456i \(-0.760476\pi\)
0.729992 0.683456i \(-0.239524\pi\)
\(674\) 0 0
\(675\) 7.73329 + 26.9777i 0.297655 + 1.03837i
\(676\) 0 0
\(677\) 15.3047i 0.588206i −0.955774 0.294103i \(-0.904979\pi\)
0.955774 0.294103i \(-0.0950208\pi\)
\(678\) 0 0
\(679\) −52.3783 −2.01009
\(680\) 0 0
\(681\) 17.4005 0.666790
\(682\) 0 0
\(683\) 13.0968i 0.501135i 0.968099 + 0.250567i \(0.0806171\pi\)
−0.968099 + 0.250567i \(0.919383\pi\)
\(684\) 0 0
\(685\) −5.95851 42.4099i −0.227663 1.62040i
\(686\) 0 0
\(687\) 3.18421i 0.121485i
\(688\) 0 0
\(689\) −6.23506 −0.237537
\(690\) 0 0
\(691\) 18.4079 0.700269 0.350135 0.936699i \(-0.386136\pi\)
0.350135 + 0.936699i \(0.386136\pi\)
\(692\) 0 0
\(693\) 0.797056i 0.0302777i
\(694\) 0 0
\(695\) −42.2766 + 5.93978i −1.60364 + 0.225309i
\(696\) 0 0
\(697\) 19.6128i 0.742890i
\(698\) 0 0
\(699\) −21.2070 −0.802121
\(700\) 0 0
\(701\) −31.3689 −1.18479 −0.592393 0.805649i \(-0.701817\pi\)
−0.592393 + 0.805649i \(0.701817\pi\)
\(702\) 0 0
\(703\) 5.05086i 0.190497i
\(704\) 0 0
\(705\) −3.18421 + 0.447375i −0.119924 + 0.0168491i
\(706\) 0 0
\(707\) 11.4380i 0.430171i
\(708\) 0 0
\(709\) −9.47949 −0.356010 −0.178005 0.984030i \(-0.556964\pi\)
−0.178005 + 0.984030i \(0.556964\pi\)
\(710\) 0 0
\(711\) −18.2351 −0.683868
\(712\) 0 0
\(713\) 26.2208i 0.981976i
\(714\) 0 0
\(715\) 0.0666765 + 0.474572i 0.00249356 + 0.0177480i
\(716\) 0 0
\(717\) 16.7511i 0.625582i
\(718\) 0 0
\(719\) −29.6227 −1.10474 −0.552370 0.833599i \(-0.686276\pi\)
−0.552370 + 0.833599i \(0.686276\pi\)
\(720\) 0 0
\(721\) 8.19358 0.305145
\(722\) 0 0
\(723\) 7.73329i 0.287604i
\(724\) 0 0
\(725\) −41.8622 + 12.0000i −1.55472 + 0.445669i
\(726\) 0 0
\(727\) 42.6702i 1.58255i 0.611461 + 0.791274i \(0.290582\pi\)
−0.611461 + 0.791274i \(0.709418\pi\)
\(728\) 0 0
\(729\) −26.5812 −0.984489
\(730\) 0 0
\(731\) −40.8988 −1.51270
\(732\) 0 0
\(733\) 26.0830i 0.963397i 0.876337 + 0.481698i \(0.159980\pi\)
−0.876337 + 0.481698i \(0.840020\pi\)
\(734\) 0 0
\(735\) 0.582736 + 4.14764i 0.0214945 + 0.152988i
\(736\) 0 0
\(737\) 1.66323i 0.0612657i
\(738\) 0 0
\(739\) −28.2687 −1.03988 −0.519941 0.854202i \(-0.674046\pi\)
−0.519941 + 0.854202i \(0.674046\pi\)
\(740\) 0 0
\(741\) −2.90321 −0.106652
\(742\) 0 0
\(743\) 20.6681i 0.758241i −0.925347 0.379120i \(-0.876227\pi\)
0.925347 0.379120i \(-0.123773\pi\)
\(744\) 0 0
\(745\) −7.90813 + 1.11108i −0.289732 + 0.0407068i
\(746\) 0 0
\(747\) 12.2020i 0.446449i
\(748\) 0 0
\(749\) −49.6958 −1.81585
\(750\) 0 0
\(751\) −2.46028 −0.0897770 −0.0448885 0.998992i \(-0.514293\pi\)
−0.0448885 + 0.998992i \(0.514293\pi\)
\(752\) 0 0
\(753\) 2.71810i 0.0990530i
\(754\) 0 0
\(755\) −2.80150 + 0.393606i −0.101957 + 0.0143248i
\(756\) 0 0
\(757\) 48.6035i 1.76652i 0.468880 + 0.883262i \(0.344658\pi\)
−0.468880 + 0.883262i \(0.655342\pi\)
\(758\) 0 0
\(759\) 1.31756 0.0478244
\(760\) 0 0
\(761\) −13.8252 −0.501162 −0.250581 0.968096i \(-0.580622\pi\)
−0.250581 + 0.968096i \(0.580622\pi\)
\(762\) 0 0
\(763\) 48.5531i 1.75774i
\(764\) 0 0
\(765\) −2.56199 18.2351i −0.0926290 0.659290i
\(766\) 0 0
\(767\) 9.26517i 0.334546i
\(768\) 0 0
\(769\) 38.9688 1.40525 0.702626 0.711559i \(-0.252011\pi\)
0.702626 + 0.711559i \(0.252011\pi\)
\(770\) 0 0
\(771\) 24.1204 0.868677
\(772\) 0 0
\(773\) 0.445992i 0.0160412i −0.999968 0.00802061i \(-0.997447\pi\)
0.999968 0.00802061i \(-0.00255307\pi\)
\(774\) 0 0
\(775\) 26.8780 7.70471i 0.965487 0.276761i
\(776\) 0 0
\(777\) 8.68244i 0.311481i
\(778\) 0 0
\(779\) 6.75557 0.242043
\(780\) 0 0
\(781\) −1.30327 −0.0466347
\(782\) 0 0
\(783\) 48.8859i 1.74704i
\(784\) 0 0
\(785\) 1.74620 + 12.4286i 0.0623246 + 0.443597i
\(786\) 0 0
\(787\) 33.9037i 1.20854i −0.796781 0.604268i \(-0.793466\pi\)
0.796781 0.604268i \(-0.206534\pi\)
\(788\) 0 0
\(789\) 14.4558 0.514641
\(790\) 0 0
\(791\) −3.43801 −0.122241
\(792\) 0 0
\(793\) 0.280996i 0.00997847i
\(794\) 0 0
\(795\) 18.1017 2.54326i 0.642002 0.0902000i
\(796\) 0 0
\(797\) 10.2953i 0.364678i −0.983236 0.182339i \(-0.941633\pi\)
0.983236 0.182339i \(-0.0583668\pi\)
\(798\) 0 0
\(799\) 7.05086 0.249441
\(800\) 0 0
\(801\) 7.19004 0.254047
\(802\) 0 0
\(803\) 2.20342i 0.0777570i
\(804\) 0 0
\(805\) −30.1432 + 4.23506i −1.06241 + 0.149266i
\(806\) 0 0
\(807\) 21.1655i 0.745060i
\(808\) 0 0
\(809\) −7.94422 −0.279304 −0.139652 0.990201i \(-0.544598\pi\)
−0.139652 + 0.990201i \(0.544598\pi\)
\(810\) 0 0
\(811\) 8.12245 0.285218 0.142609 0.989779i \(-0.454451\pi\)
0.142609 + 0.989779i \(0.454451\pi\)
\(812\) 0 0
\(813\) 17.0593i 0.598296i
\(814\) 0 0
\(815\) −1.15701 8.23506i −0.0405283 0.288462i
\(816\) 0 0
\(817\) 14.0874i 0.492856i
\(818\) 0 0
\(819\) 3.71900 0.129953
\(820\) 0 0
\(821\) −22.2065 −0.775012 −0.387506 0.921867i \(-0.626663\pi\)
−0.387506 + 0.921867i \(0.626663\pi\)
\(822\) 0 0
\(823\) 11.1175i 0.387533i 0.981048 + 0.193766i \(0.0620704\pi\)
−0.981048 + 0.193766i \(0.937930\pi\)
\(824\) 0 0
\(825\) −0.387152 1.35059i −0.0134789 0.0470214i
\(826\) 0 0
\(827\) 23.1570i 0.805248i −0.915365 0.402624i \(-0.868098\pi\)
0.915365 0.402624i \(-0.131902\pi\)
\(828\) 0 0
\(829\) 27.1195 0.941901 0.470950 0.882160i \(-0.343911\pi\)
0.470950 + 0.882160i \(0.343911\pi\)
\(830\) 0 0
\(831\) 9.92687 0.344359
\(832\) 0 0
\(833\) 9.18421i 0.318214i
\(834\) 0 0
\(835\) −2.18913 15.5812i −0.0757580 0.539210i
\(836\) 0 0
\(837\) 31.3876i 1.08492i
\(838\) 0 0
\(839\) 25.3955 0.876749 0.438374 0.898792i \(-0.355554\pi\)
0.438374 + 0.898792i \(0.355554\pi\)
\(840\) 0 0
\(841\) 46.8578 1.61578
\(842\) 0 0
\(843\) 8.85728i 0.305061i
\(844\) 0 0
\(845\) 2.21432 0.311108i 0.0761749 0.0107024i
\(846\) 0 0
\(847\) 31.8020i 1.09273i
\(848\) 0 0
\(849\) 25.0237 0.858810
\(850\) 0 0
\(851\) −10.6953 −0.366632
\(852\) 0 0
\(853\) 25.0651i 0.858214i −0.903254 0.429107i \(-0.858828\pi\)
0.903254 0.429107i \(-0.141172\pi\)
\(854\) 0 0
\(855\) −6.28100 + 0.882468i −0.214806 + 0.0301798i
\(856\) 0 0
\(857\) 7.61285i 0.260050i 0.991511 + 0.130025i \(0.0415057\pi\)
−0.991511 + 0.130025i \(0.958494\pi\)
\(858\) 0 0
\(859\) 42.1432 1.43791 0.718954 0.695058i \(-0.244621\pi\)
0.718954 + 0.695058i \(0.244621\pi\)
\(860\) 0 0
\(861\) 11.6128 0.395765
\(862\) 0 0
\(863\) 51.5768i 1.75569i 0.478942 + 0.877847i \(0.341020\pi\)
−0.478942 + 0.877847i \(0.658980\pi\)
\(864\) 0 0
\(865\) 0.225219 + 1.60300i 0.00765768 + 0.0545037i
\(866\) 0 0
\(867\) 31.8959i 1.08324i
\(868\) 0 0
\(869\) 3.05086 0.103493
\(870\) 0 0
\(871\) −7.76049 −0.262954
\(872\) 0 0
\(873\) 23.1111i 0.782191i
\(874\) 0 0
\(875\) 13.1985 + 29.6543i 0.446191 + 1.00250i
\(876\) 0 0
\(877\) 34.0701i 1.15046i −0.817990 0.575232i \(-0.804912\pi\)
0.817990 0.575232i \(-0.195088\pi\)
\(878\) 0 0
\(879\) 10.6035 0.357646
\(880\) 0 0
\(881\) 3.71900 0.125296 0.0626482 0.998036i \(-0.480045\pi\)
0.0626482 + 0.998036i \(0.480045\pi\)
\(882\) 0 0
\(883\) 42.0163i 1.41396i −0.707233 0.706981i \(-0.750057\pi\)
0.707233 0.706981i \(-0.249943\pi\)
\(884\) 0 0
\(885\) 3.77923 + 26.8988i 0.127037 + 0.904192i
\(886\) 0 0
\(887\) 40.3116i 1.35353i 0.736199 + 0.676765i \(0.236619\pi\)
−0.736199 + 0.676765i \(0.763381\pi\)
\(888\) 0 0
\(889\) −6.68244 −0.224122
\(890\) 0 0
\(891\) 0.753561 0.0252452
\(892\) 0 0
\(893\) 2.42864i 0.0812713i
\(894\) 0 0
\(895\) 8.94914 1.25734i 0.299137 0.0420282i
\(896\) 0 0
\(897\) 6.14764i 0.205264i
\(898\) 0 0
\(899\) −48.7052 −1.62441
\(900\) 0 0
\(901\) −40.0830 −1.33536
\(902\) 0 0
\(903\) 24.2163i 0.805869i
\(904\) 0 0
\(905\) −5.18421 + 0.728372i −0.172329 + 0.0242119i
\(906\) 0 0
\(907\) 34.8419i 1.15691i −0.815715 0.578454i \(-0.803656\pi\)
0.815715 0.578454i \(-0.196344\pi\)
\(908\) 0 0
\(909\) 5.04684 0.167393
\(910\) 0 0
\(911\) −23.2672 −0.770876 −0.385438 0.922734i \(-0.625950\pi\)
−0.385438 + 0.922734i \(0.625950\pi\)
\(912\) 0 0
\(913\) 2.04149i 0.0675634i
\(914\) 0 0
\(915\) 0.114617 + 0.815792i 0.00378913 + 0.0269692i
\(916\) 0 0
\(917\) 38.9590i 1.28654i
\(918\) 0 0
\(919\) −3.22570 −0.106406 −0.0532029 0.998584i \(-0.516943\pi\)
−0.0532029 + 0.998584i \(0.516943\pi\)
\(920\) 0 0
\(921\) 17.6128 0.580363
\(922\) 0 0
\(923\) 6.08097i 0.200157i
\(924\) 0 0
\(925\) 3.14272 + 10.9634i 0.103332 + 0.360476i
\(926\) 0 0
\(927\) 3.61529i 0.118742i
\(928\) 0 0
\(929\) 39.3461 1.29091 0.645453 0.763800i \(-0.276669\pi\)
0.645453 + 0.763800i \(0.276669\pi\)
\(930\) 0 0
\(931\) −3.16346 −0.103678
\(932\) 0 0
\(933\) 26.4889i 0.867206i
\(934\) 0 0
\(935\) 0.428639 + 3.05086i 0.0140180 + 0.0997736i
\(936\) 0 0
\(937\) 51.6040i 1.68583i −0.538048 0.842914i \(-0.680838\pi\)
0.538048 0.842914i \(-0.319162\pi\)
\(938\) 0 0
\(939\) −19.8123 −0.646548
\(940\) 0 0
\(941\) 37.5081 1.22273 0.611364 0.791349i \(-0.290621\pi\)
0.611364 + 0.791349i \(0.290621\pi\)
\(942\) 0 0
\(943\) 14.3051i 0.465839i
\(944\) 0 0
\(945\) −36.0830 + 5.06959i −1.17378 + 0.164914i
\(946\) 0 0
\(947\) 38.1160i 1.23860i −0.785153 0.619302i \(-0.787416\pi\)
0.785153 0.619302i \(-0.212584\pi\)
\(948\) 0 0
\(949\) −10.2810 −0.333735
\(950\) 0 0
\(951\) 29.2128 0.947290
\(952\) 0 0
\(953\) 28.7368i 0.930877i −0.885080 0.465439i \(-0.845897\pi\)
0.885080 0.465439i \(-0.154103\pi\)
\(954\) 0 0
\(955\) −4.65386 + 0.653858i −0.150595 + 0.0211584i
\(956\) 0 0
\(957\) 2.44738i 0.0791124i
\(958\) 0 0
\(959\) 55.6040 1.79555
\(960\) 0 0
\(961\) 0.271628 0.00876221
\(962\) 0 0
\(963\) 21.9275i 0.706604i
\(964\) 0 0
\(965\) 4.20648 + 29.9398i 0.135411 + 0.963796i
\(966\) 0 0
\(967\) 29.0593i 0.934485i 0.884129 + 0.467242i \(0.154752\pi\)
−0.884129 + 0.467242i \(0.845248\pi\)
\(968\) 0 0
\(969\) −18.6637 −0.599565
\(970\) 0 0
\(971\) 39.8578 1.27910 0.639548 0.768751i \(-0.279121\pi\)
0.639548 + 0.768751i \(0.279121\pi\)
\(972\) 0 0
\(973\) 55.4291i 1.77698i
\(974\) 0 0
\(975\) −6.30174 + 1.80642i −0.201817 + 0.0578519i
\(976\) 0 0
\(977\) 12.8617i 0.411483i 0.978606 + 0.205742i \(0.0659606\pi\)
−0.978606 + 0.205742i \(0.934039\pi\)
\(978\) 0 0
\(979\) −1.20294 −0.0384463
\(980\) 0 0
\(981\) 21.4233 0.683993
\(982\) 0 0
\(983\) 45.4880i 1.45084i −0.688306 0.725420i \(-0.741645\pi\)
0.688306 0.725420i \(-0.258355\pi\)
\(984\) 0 0
\(985\) −0.622216 4.42864i −0.0198254 0.141108i
\(986\) 0 0
\(987\) 4.17484i 0.132887i
\(988\) 0 0
\(989\) −29.8306 −0.948557
\(990\) 0 0
\(991\) −8.07007 −0.256354 −0.128177 0.991751i \(-0.540913\pi\)
−0.128177 + 0.991751i \(0.540913\pi\)
\(992\) 0 0
\(993\) 10.8243i 0.343497i
\(994\) 0 0
\(995\) −49.0321 + 6.88892i −1.55442 + 0.218394i
\(996\) 0 0
\(997\) 32.8158i 1.03929i 0.854383 + 0.519643i \(0.173935\pi\)
−0.854383 + 0.519643i \(0.826065\pi\)
\(998\) 0 0
\(999\) −12.8029 −0.405065
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 1040.2.d.c.209.5 6
4.3 odd 2 65.2.b.a.14.3 6
5.2 odd 4 5200.2.a.cj.1.2 3
5.3 odd 4 5200.2.a.cb.1.2 3
5.4 even 2 inner 1040.2.d.c.209.2 6
12.11 even 2 585.2.c.b.469.4 6
20.3 even 4 325.2.a.k.1.1 3
20.7 even 4 325.2.a.j.1.3 3
20.19 odd 2 65.2.b.a.14.4 yes 6
52.3 odd 6 845.2.n.f.529.4 12
52.7 even 12 845.2.l.e.699.1 12
52.11 even 12 845.2.l.e.654.2 12
52.15 even 12 845.2.l.d.654.6 12
52.19 even 12 845.2.l.d.699.5 12
52.23 odd 6 845.2.n.g.529.3 12
52.31 even 4 845.2.d.b.844.1 6
52.35 odd 6 845.2.n.f.484.3 12
52.43 odd 6 845.2.n.g.484.4 12
52.47 even 4 845.2.d.a.844.5 6
52.51 odd 2 845.2.b.c.339.4 6
60.23 odd 4 2925.2.a.bf.1.3 3
60.47 odd 4 2925.2.a.bj.1.1 3
60.59 even 2 585.2.c.b.469.3 6
260.19 even 12 845.2.l.e.699.2 12
260.59 even 12 845.2.l.d.699.6 12
260.99 even 4 845.2.d.b.844.2 6
260.103 even 4 4225.2.a.ba.1.3 3
260.119 even 12 845.2.l.e.654.1 12
260.139 odd 6 845.2.n.f.484.4 12
260.159 odd 6 845.2.n.f.529.3 12
260.179 odd 6 845.2.n.g.529.4 12
260.199 odd 6 845.2.n.g.484.3 12
260.207 even 4 4225.2.a.bh.1.1 3
260.219 even 12 845.2.l.d.654.5 12
260.239 even 4 845.2.d.a.844.6 6
260.259 odd 2 845.2.b.c.339.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.b.a.14.3 6 4.3 odd 2
65.2.b.a.14.4 yes 6 20.19 odd 2
325.2.a.j.1.3 3 20.7 even 4
325.2.a.k.1.1 3 20.3 even 4
585.2.c.b.469.3 6 60.59 even 2
585.2.c.b.469.4 6 12.11 even 2
845.2.b.c.339.3 6 260.259 odd 2
845.2.b.c.339.4 6 52.51 odd 2
845.2.d.a.844.5 6 52.47 even 4
845.2.d.a.844.6 6 260.239 even 4
845.2.d.b.844.1 6 52.31 even 4
845.2.d.b.844.2 6 260.99 even 4
845.2.l.d.654.5 12 260.219 even 12
845.2.l.d.654.6 12 52.15 even 12
845.2.l.d.699.5 12 52.19 even 12
845.2.l.d.699.6 12 260.59 even 12
845.2.l.e.654.1 12 260.119 even 12
845.2.l.e.654.2 12 52.11 even 12
845.2.l.e.699.1 12 52.7 even 12
845.2.l.e.699.2 12 260.19 even 12
845.2.n.f.484.3 12 52.35 odd 6
845.2.n.f.484.4 12 260.139 odd 6
845.2.n.f.529.3 12 260.159 odd 6
845.2.n.f.529.4 12 52.3 odd 6
845.2.n.g.484.3 12 260.199 odd 6
845.2.n.g.484.4 12 52.43 odd 6
845.2.n.g.529.3 12 52.23 odd 6
845.2.n.g.529.4 12 260.179 odd 6
1040.2.d.c.209.2 6 5.4 even 2 inner
1040.2.d.c.209.5 6 1.1 even 1 trivial
2925.2.a.bf.1.3 3 60.23 odd 4
2925.2.a.bj.1.1 3 60.47 odd 4
4225.2.a.ba.1.3 3 260.103 even 4
4225.2.a.bh.1.1 3 260.207 even 4
5200.2.a.cb.1.2 3 5.3 odd 4
5200.2.a.cj.1.2 3 5.2 odd 4