Properties

Label 2240.2.g.p.449.7
Level $2240$
Weight $2$
Character 2240.449
Analytic conductor $17.886$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(449,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, 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("2240.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.7
Root \(2.64856i\) of defining polynomial
Character \(\chi\) \(=\) 2240.449
Dual form 2240.2.g.p.449.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.321637i q^{3} +(-2.10909 - 0.742782i) q^{5} +1.00000i q^{7} +2.89655 q^{9} +O(q^{10})\) \(q+0.321637i q^{3} +(-2.10909 - 0.742782i) q^{5} +1.00000i q^{7} +2.89655 q^{9} +4.37985 q^{11} +5.86542i q^{13} +(0.238906 - 0.678363i) q^{15} -4.85766i q^{17} -7.75195 q^{19} -0.321637 q^{21} +1.35673i q^{23} +(3.89655 + 3.13319i) q^{25} +1.89655i q^{27} -0.539824 q^{29} +2.97113 q^{31} +1.40872i q^{33} +(0.742782 - 2.10909i) q^{35} -6.26639i q^{37} -1.88654 q^{39} +2.64327 q^{41} +4.64327i q^{43} +(-6.10909 - 2.15151i) q^{45} +10.3329i q^{47} -1.00000 q^{49} +1.56241 q^{51} -0.477813i q^{53} +(-9.23752 - 3.25328i) q^{55} -2.49332i q^{57} +7.75195 q^{59} -7.57491 q^{61} +2.89655i q^{63} +(4.35673 - 12.3707i) q^{65} +3.79310i q^{67} -0.436373 q^{69} +9.23752 q^{71} -0.477813i q^{73} +(-1.00775 + 1.25328i) q^{75} +4.37985i q^{77} +1.88654 q^{79} +8.07965 q^{81} +15.2978i q^{83} +(-3.60819 + 10.2453i) q^{85} -0.173627i q^{87} -6.00000 q^{89} -5.86542 q^{91} +0.955625i q^{93} +(16.3496 + 5.75801i) q^{95} +13.6174i q^{97} +12.6865 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 0.321637i 0.185697i 0.995680 + 0.0928487i \(0.0295973\pi\)
−0.995680 + 0.0928487i \(0.970403\pi\)
\(4\) 0 0
\(5\) −2.10909 0.742782i −0.943215 0.332182i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.89655 0.965517
\(10\) 0 0
\(11\) 4.37985 1.32057 0.660287 0.751013i \(-0.270434\pi\)
0.660287 + 0.751013i \(0.270434\pi\)
\(12\) 0 0
\(13\) 5.86542i 1.62677i 0.581723 + 0.813387i \(0.302379\pi\)
−0.581723 + 0.813387i \(0.697621\pi\)
\(14\) 0 0
\(15\) 0.238906 0.678363i 0.0616853 0.175153i
\(16\) 0 0
\(17\) 4.85766i 1.17816i −0.808076 0.589078i \(-0.799491\pi\)
0.808076 0.589078i \(-0.200509\pi\)
\(18\) 0 0
\(19\) −7.75195 −1.77842 −0.889210 0.457500i \(-0.848745\pi\)
−0.889210 + 0.457500i \(0.848745\pi\)
\(20\) 0 0
\(21\) −0.321637 −0.0701870
\(22\) 0 0
\(23\) 1.35673i 0.282897i 0.989946 + 0.141448i \(0.0451759\pi\)
−0.989946 + 0.141448i \(0.954824\pi\)
\(24\) 0 0
\(25\) 3.89655 + 3.13319i 0.779310 + 0.626639i
\(26\) 0 0
\(27\) 1.89655i 0.364991i
\(28\) 0 0
\(29\) −0.539824 −0.100243 −0.0501214 0.998743i \(-0.515961\pi\)
−0.0501214 + 0.998743i \(0.515961\pi\)
\(30\) 0 0
\(31\) 2.97113 0.533630 0.266815 0.963748i \(-0.414029\pi\)
0.266815 + 0.963748i \(0.414029\pi\)
\(32\) 0 0
\(33\) 1.40872i 0.245227i
\(34\) 0 0
\(35\) 0.742782 2.10909i 0.125553 0.356502i
\(36\) 0 0
\(37\) 6.26639i 1.03019i −0.857134 0.515094i \(-0.827757\pi\)
0.857134 0.515094i \(-0.172243\pi\)
\(38\) 0 0
\(39\) −1.88654 −0.302087
\(40\) 0 0
\(41\) 2.64327 0.412810 0.206405 0.978467i \(-0.433824\pi\)
0.206405 + 0.978467i \(0.433824\pi\)
\(42\) 0 0
\(43\) 4.64327i 0.708093i 0.935228 + 0.354046i \(0.115194\pi\)
−0.935228 + 0.354046i \(0.884806\pi\)
\(44\) 0 0
\(45\) −6.10909 2.15151i −0.910690 0.320727i
\(46\) 0 0
\(47\) 10.3329i 1.50721i 0.657327 + 0.753606i \(0.271687\pi\)
−0.657327 + 0.753606i \(0.728313\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.56241 0.218781
\(52\) 0 0
\(53\) 0.477813i 0.0656326i −0.999461 0.0328163i \(-0.989552\pi\)
0.999461 0.0328163i \(-0.0104476\pi\)
\(54\) 0 0
\(55\) −9.23752 3.25328i −1.24559 0.438672i
\(56\) 0 0
\(57\) 2.49332i 0.330248i
\(58\) 0 0
\(59\) 7.75195 1.00922 0.504609 0.863348i \(-0.331637\pi\)
0.504609 + 0.863348i \(0.331637\pi\)
\(60\) 0 0
\(61\) −7.57491 −0.969868 −0.484934 0.874551i \(-0.661156\pi\)
−0.484934 + 0.874551i \(0.661156\pi\)
\(62\) 0 0
\(63\) 2.89655i 0.364931i
\(64\) 0 0
\(65\) 4.35673 12.3707i 0.540385 1.53440i
\(66\) 0 0
\(67\) 3.79310i 0.463401i 0.972787 + 0.231700i \(0.0744289\pi\)
−0.972787 + 0.231700i \(0.925571\pi\)
\(68\) 0 0
\(69\) −0.436373 −0.0525332
\(70\) 0 0
\(71\) 9.23752 1.09629 0.548146 0.836383i \(-0.315334\pi\)
0.548146 + 0.836383i \(0.315334\pi\)
\(72\) 0 0
\(73\) 0.477813i 0.0559237i −0.999609 0.0279619i \(-0.991098\pi\)
0.999609 0.0279619i \(-0.00890170\pi\)
\(74\) 0 0
\(75\) −1.00775 + 1.25328i −0.116365 + 0.144716i
\(76\) 0 0
\(77\) 4.37985i 0.499130i
\(78\) 0 0
\(79\) 1.88654 0.212252 0.106126 0.994353i \(-0.466155\pi\)
0.106126 + 0.994353i \(0.466155\pi\)
\(80\) 0 0
\(81\) 8.07965 0.897739
\(82\) 0 0
\(83\) 15.2978i 1.67916i 0.543240 + 0.839578i \(0.317198\pi\)
−0.543240 + 0.839578i \(0.682802\pi\)
\(84\) 0 0
\(85\) −3.60819 + 10.2453i −0.391363 + 1.11126i
\(86\) 0 0
\(87\) 0.173627i 0.0186148i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −5.86542 −0.614863
\(92\) 0 0
\(93\) 0.955625i 0.0990937i
\(94\) 0 0
\(95\) 16.3496 + 5.75801i 1.67743 + 0.590759i
\(96\) 0 0
\(97\) 13.6174i 1.38263i 0.722552 + 0.691317i \(0.242969\pi\)
−0.722552 + 0.691317i \(0.757031\pi\)
\(98\) 0 0
\(99\) 12.6865 1.27504
\(100\) 0 0
\(101\) −6.28836 −0.625716 −0.312858 0.949800i \(-0.601286\pi\)
−0.312858 + 0.949800i \(0.601286\pi\)
\(102\) 0 0
\(103\) 9.25328i 0.911752i 0.890043 + 0.455876i \(0.150674\pi\)
−0.890043 + 0.455876i \(0.849326\pi\)
\(104\) 0 0
\(105\) 0.678363 + 0.238906i 0.0662014 + 0.0233149i
\(106\) 0 0
\(107\) 15.5862i 1.50677i 0.657577 + 0.753387i \(0.271581\pi\)
−0.657577 + 0.753387i \(0.728419\pi\)
\(108\) 0 0
\(109\) 11.2533 1.07787 0.538934 0.842348i \(-0.318827\pi\)
0.538934 + 0.842348i \(0.318827\pi\)
\(110\) 0 0
\(111\) 2.01550 0.191303
\(112\) 0 0
\(113\) 12.2086i 1.14849i 0.818683 + 0.574246i \(0.194705\pi\)
−0.818683 + 0.574246i \(0.805295\pi\)
\(114\) 0 0
\(115\) 1.00775 2.86146i 0.0939733 0.266833i
\(116\) 0 0
\(117\) 16.9895i 1.57068i
\(118\) 0 0
\(119\) 4.85766 0.445301
\(120\) 0 0
\(121\) 8.18310 0.743918
\(122\) 0 0
\(123\) 0.850175i 0.0766577i
\(124\) 0 0
\(125\) −5.89091 9.50248i −0.526899 0.849928i
\(126\) 0 0
\(127\) 9.14982i 0.811916i 0.913892 + 0.405958i \(0.133062\pi\)
−0.913892 + 0.405958i \(0.866938\pi\)
\(128\) 0 0
\(129\) −1.49345 −0.131491
\(130\) 0 0
\(131\) 16.5117 1.44263 0.721315 0.692607i \(-0.243538\pi\)
0.721315 + 0.692607i \(0.243538\pi\)
\(132\) 0 0
\(133\) 7.75195i 0.672179i
\(134\) 0 0
\(135\) 1.40872 4.00000i 0.121244 0.344265i
\(136\) 0 0
\(137\) 8.75970i 0.748392i −0.927350 0.374196i \(-0.877919\pi\)
0.927350 0.374196i \(-0.122081\pi\)
\(138\) 0 0
\(139\) 9.76745 0.828465 0.414232 0.910171i \(-0.364050\pi\)
0.414232 + 0.910171i \(0.364050\pi\)
\(140\) 0 0
\(141\) −3.32345 −0.279885
\(142\) 0 0
\(143\) 25.6896i 2.14828i
\(144\) 0 0
\(145\) 1.13854 + 0.400972i 0.0945505 + 0.0332989i
\(146\) 0 0
\(147\) 0.321637i 0.0265282i
\(148\) 0 0
\(149\) 22.8727 1.87381 0.936904 0.349586i \(-0.113678\pi\)
0.936904 + 0.349586i \(0.113678\pi\)
\(150\) 0 0
\(151\) −16.1107 −1.31107 −0.655534 0.755165i \(-0.727557\pi\)
−0.655534 + 0.755165i \(0.727557\pi\)
\(152\) 0 0
\(153\) 14.0705i 1.13753i
\(154\) 0 0
\(155\) −6.26639 2.20690i −0.503328 0.177263i
\(156\) 0 0
\(157\) 0.529939i 0.0422937i −0.999776 0.0211469i \(-0.993268\pi\)
0.999776 0.0211469i \(-0.00673176\pi\)
\(158\) 0 0
\(159\) 0.153682 0.0121878
\(160\) 0 0
\(161\) −1.35673 −0.106925
\(162\) 0 0
\(163\) 21.5160i 1.68526i 0.538489 + 0.842632i \(0.318995\pi\)
−0.538489 + 0.842632i \(0.681005\pi\)
\(164\) 0 0
\(165\) 1.04637 2.97113i 0.0814601 0.231302i
\(166\) 0 0
\(167\) 10.5398i 0.815596i −0.913072 0.407798i \(-0.866297\pi\)
0.913072 0.407798i \(-0.133703\pi\)
\(168\) 0 0
\(169\) −21.4031 −1.64639
\(170\) 0 0
\(171\) −22.4539 −1.71709
\(172\) 0 0
\(173\) 11.8077i 0.897721i −0.893602 0.448860i \(-0.851830\pi\)
0.893602 0.448860i \(-0.148170\pi\)
\(174\) 0 0
\(175\) −3.13319 + 3.89655i −0.236847 + 0.294551i
\(176\) 0 0
\(177\) 2.49332i 0.187409i
\(178\) 0 0
\(179\) −18.4750 −1.38089 −0.690444 0.723386i \(-0.742585\pi\)
−0.690444 + 0.723386i \(0.742585\pi\)
\(180\) 0 0
\(181\) −4.86146 −0.361350 −0.180675 0.983543i \(-0.557828\pi\)
−0.180675 + 0.983543i \(0.557828\pi\)
\(182\) 0 0
\(183\) 2.43637i 0.180102i
\(184\) 0 0
\(185\) −4.65456 + 13.2164i −0.342210 + 0.971689i
\(186\) 0 0
\(187\) 21.2758i 1.55584i
\(188\) 0 0
\(189\) −1.89655 −0.137954
\(190\) 0 0
\(191\) −15.6329 −1.13115 −0.565577 0.824695i \(-0.691347\pi\)
−0.565577 + 0.824695i \(0.691347\pi\)
\(192\) 0 0
\(193\) 20.9683i 1.50933i −0.656108 0.754667i \(-0.727798\pi\)
0.656108 0.754667i \(-0.272202\pi\)
\(194\) 0 0
\(195\) 3.97888 + 1.40128i 0.284934 + 0.100348i
\(196\) 0 0
\(197\) 15.1798i 1.08151i −0.841179 0.540757i \(-0.818138\pi\)
0.841179 0.540757i \(-0.181862\pi\)
\(198\) 0 0
\(199\) 11.7308 0.831577 0.415788 0.909461i \(-0.363506\pi\)
0.415788 + 0.909461i \(0.363506\pi\)
\(200\) 0 0
\(201\) −1.22000 −0.0860523
\(202\) 0 0
\(203\) 0.539824i 0.0378882i
\(204\) 0 0
\(205\) −5.57491 1.96338i −0.389369 0.137128i
\(206\) 0 0
\(207\) 3.92982i 0.273142i
\(208\) 0 0
\(209\) −33.9524 −2.34854
\(210\) 0 0
\(211\) −1.40872 −0.0969805 −0.0484902 0.998824i \(-0.515441\pi\)
−0.0484902 + 0.998824i \(0.515441\pi\)
\(212\) 0 0
\(213\) 2.97113i 0.203578i
\(214\) 0 0
\(215\) 3.44894 9.79310i 0.235216 0.667884i
\(216\) 0 0
\(217\) 2.97113i 0.201693i
\(218\) 0 0
\(219\) 0.153682 0.0103849
\(220\) 0 0
\(221\) 28.4922 1.91659
\(222\) 0 0
\(223\) 4.61000i 0.308708i −0.988016 0.154354i \(-0.950670\pi\)
0.988016 0.154354i \(-0.0493297\pi\)
\(224\) 0 0
\(225\) 11.2865 + 9.07545i 0.752437 + 0.605030i
\(226\) 0 0
\(227\) 5.74854i 0.381544i −0.981634 0.190772i \(-0.938901\pi\)
0.981634 0.190772i \(-0.0610991\pi\)
\(228\) 0 0
\(229\) 17.5047 1.15675 0.578373 0.815773i \(-0.303688\pi\)
0.578373 + 0.815773i \(0.303688\pi\)
\(230\) 0 0
\(231\) −1.40872 −0.0926872
\(232\) 0 0
\(233\) 18.4750i 1.21034i 0.796096 + 0.605170i \(0.206895\pi\)
−0.796096 + 0.605170i \(0.793105\pi\)
\(234\) 0 0
\(235\) 7.67511 21.7931i 0.500669 1.42163i
\(236\) 0 0
\(237\) 0.606780i 0.0394146i
\(238\) 0 0
\(239\) −7.35098 −0.475495 −0.237748 0.971327i \(-0.576409\pi\)
−0.237748 + 0.971327i \(0.576409\pi\)
\(240\) 0 0
\(241\) −3.51602 −0.226487 −0.113243 0.993567i \(-0.536124\pi\)
−0.113243 + 0.993567i \(0.536124\pi\)
\(242\) 0 0
\(243\) 8.28836i 0.531699i
\(244\) 0 0
\(245\) 2.10909 + 0.742782i 0.134745 + 0.0474546i
\(246\) 0 0
\(247\) 45.4684i 2.89309i
\(248\) 0 0
\(249\) −4.92035 −0.311815
\(250\) 0 0
\(251\) −5.73645 −0.362081 −0.181041 0.983476i \(-0.557947\pi\)
−0.181041 + 0.983476i \(0.557947\pi\)
\(252\) 0 0
\(253\) 5.94226i 0.373586i
\(254\) 0 0
\(255\) −3.29526 1.16053i −0.206357 0.0726750i
\(256\) 0 0
\(257\) 17.9972i 1.12264i 0.827600 + 0.561318i \(0.189705\pi\)
−0.827600 + 0.561318i \(0.810295\pi\)
\(258\) 0 0
\(259\) 6.26639 0.389374
\(260\) 0 0
\(261\) −1.56363 −0.0967861
\(262\) 0 0
\(263\) 16.8727i 1.04042i 0.854039 + 0.520209i \(0.174146\pi\)
−0.854039 + 0.520209i \(0.825854\pi\)
\(264\) 0 0
\(265\) −0.354911 + 1.00775i −0.0218020 + 0.0619057i
\(266\) 0 0
\(267\) 1.92982i 0.118103i
\(268\) 0 0
\(269\) 25.8746 1.57760 0.788800 0.614650i \(-0.210703\pi\)
0.788800 + 0.614650i \(0.210703\pi\)
\(270\) 0 0
\(271\) 27.2347 1.65439 0.827196 0.561913i \(-0.189935\pi\)
0.827196 + 0.561913i \(0.189935\pi\)
\(272\) 0 0
\(273\) 1.88654i 0.114178i
\(274\) 0 0
\(275\) 17.0663 + 13.7229i 1.02914 + 0.827523i
\(276\) 0 0
\(277\) 23.9395i 1.43838i −0.694812 0.719192i \(-0.744512\pi\)
0.694812 0.719192i \(-0.255488\pi\)
\(278\) 0 0
\(279\) 8.60602 0.515229
\(280\) 0 0
\(281\) −19.5493 −1.16621 −0.583107 0.812396i \(-0.698163\pi\)
−0.583107 + 0.812396i \(0.698163\pi\)
\(282\) 0 0
\(283\) 30.9173i 1.83784i −0.394440 0.918922i \(-0.629062\pi\)
0.394440 0.918922i \(-0.370938\pi\)
\(284\) 0 0
\(285\) −1.85199 + 5.25864i −0.109702 + 0.311495i
\(286\) 0 0
\(287\) 2.64327i 0.156028i
\(288\) 0 0
\(289\) −6.59690 −0.388053
\(290\) 0 0
\(291\) −4.37985 −0.256751
\(292\) 0 0
\(293\) 9.63849i 0.563086i 0.959549 + 0.281543i \(0.0908463\pi\)
−0.959549 + 0.281543i \(0.909154\pi\)
\(294\) 0 0
\(295\) −16.3496 5.75801i −0.951910 0.335244i
\(296\) 0 0
\(297\) 8.30661i 0.481998i
\(298\) 0 0
\(299\) −7.95776 −0.460209
\(300\) 0 0
\(301\) −4.64327 −0.267634
\(302\) 0 0
\(303\) 2.02257i 0.116194i
\(304\) 0 0
\(305\) 15.9762 + 5.62651i 0.914794 + 0.322173i
\(306\) 0 0
\(307\) 10.8282i 0.617997i −0.951062 0.308999i \(-0.900006\pi\)
0.951062 0.308999i \(-0.0999939\pi\)
\(308\) 0 0
\(309\) −2.97620 −0.169310
\(310\) 0 0
\(311\) −9.71533 −0.550906 −0.275453 0.961315i \(-0.588828\pi\)
−0.275453 + 0.961315i \(0.588828\pi\)
\(312\) 0 0
\(313\) 4.85766i 0.274571i −0.990531 0.137286i \(-0.956162\pi\)
0.990531 0.137286i \(-0.0438378\pi\)
\(314\) 0 0
\(315\) 2.15151 6.10909i 0.121224 0.344208i
\(316\) 0 0
\(317\) 20.8147i 1.16907i 0.811369 + 0.584534i \(0.198723\pi\)
−0.811369 + 0.584534i \(0.801277\pi\)
\(318\) 0 0
\(319\) −2.36435 −0.132378
\(320\) 0 0
\(321\) −5.01310 −0.279804
\(322\) 0 0
\(323\) 37.6564i 2.09526i
\(324\) 0 0
\(325\) −18.3775 + 22.8549i −1.01940 + 1.26776i
\(326\) 0 0
\(327\) 3.61947i 0.200157i
\(328\) 0 0
\(329\) −10.3329 −0.569672
\(330\) 0 0
\(331\) 16.4595 0.904697 0.452349 0.891841i \(-0.350586\pi\)
0.452349 + 0.891841i \(0.350586\pi\)
\(332\) 0 0
\(333\) 18.1509i 0.994663i
\(334\) 0 0
\(335\) 2.81745 8.00000i 0.153934 0.437087i
\(336\) 0 0
\(337\) 33.5011i 1.82492i −0.409163 0.912461i \(-0.634179\pi\)
0.409163 0.912461i \(-0.365821\pi\)
\(338\) 0 0
\(339\) −3.92675 −0.213272
\(340\) 0 0
\(341\) 13.0131 0.704699
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.920352 + 0.324130i 0.0495501 + 0.0174506i
\(346\) 0 0
\(347\) 26.7360i 1.43526i −0.696422 0.717632i \(-0.745226\pi\)
0.696422 0.717632i \(-0.254774\pi\)
\(348\) 0 0
\(349\) −12.0113 −0.642949 −0.321475 0.946918i \(-0.604179\pi\)
−0.321475 + 0.946918i \(0.604179\pi\)
\(350\) 0 0
\(351\) −11.1241 −0.593758
\(352\) 0 0
\(353\) 4.70398i 0.250368i −0.992134 0.125184i \(-0.960048\pi\)
0.992134 0.125184i \(-0.0399521\pi\)
\(354\) 0 0
\(355\) −19.4828 6.86146i −1.03404 0.364169i
\(356\) 0 0
\(357\) 1.56241i 0.0826913i
\(358\) 0 0
\(359\) −24.4173 −1.28870 −0.644348 0.764733i \(-0.722871\pi\)
−0.644348 + 0.764733i \(0.722871\pi\)
\(360\) 0 0
\(361\) 41.0927 2.16278
\(362\) 0 0
\(363\) 2.63199i 0.138144i
\(364\) 0 0
\(365\) −0.354911 + 1.00775i −0.0185769 + 0.0527481i
\(366\) 0 0
\(367\) 3.18310i 0.166156i 0.996543 + 0.0830782i \(0.0264751\pi\)
−0.996543 + 0.0830782i \(0.973525\pi\)
\(368\) 0 0
\(369\) 7.65638 0.398575
\(370\) 0 0
\(371\) 0.477813 0.0248068
\(372\) 0 0
\(373\) 6.42007i 0.332419i −0.986090 0.166209i \(-0.946847\pi\)
0.986090 0.166209i \(-0.0531527\pi\)
\(374\) 0 0
\(375\) 3.05635 1.89473i 0.157829 0.0978437i
\(376\) 0 0
\(377\) 3.16629i 0.163072i
\(378\) 0 0
\(379\) −27.3884 −1.40685 −0.703424 0.710770i \(-0.748347\pi\)
−0.703424 + 0.710770i \(0.748347\pi\)
\(380\) 0 0
\(381\) −2.94292 −0.150771
\(382\) 0 0
\(383\) 27.0095i 1.38012i 0.723752 + 0.690060i \(0.242416\pi\)
−0.723752 + 0.690060i \(0.757584\pi\)
\(384\) 0 0
\(385\) 3.25328 9.23752i 0.165802 0.470787i
\(386\) 0 0
\(387\) 13.4495i 0.683675i
\(388\) 0 0
\(389\) 15.1129 0.766256 0.383128 0.923695i \(-0.374847\pi\)
0.383128 + 0.923695i \(0.374847\pi\)
\(390\) 0 0
\(391\) 6.59052 0.333297
\(392\) 0 0
\(393\) 5.31076i 0.267892i
\(394\) 0 0
\(395\) −3.97888 1.40128i −0.200199 0.0705063i
\(396\) 0 0
\(397\) 13.5652i 0.680820i −0.940277 0.340410i \(-0.889434\pi\)
0.940277 0.340410i \(-0.110566\pi\)
\(398\) 0 0
\(399\) 2.49332 0.124822
\(400\) 0 0
\(401\) −16.4031 −0.819132 −0.409566 0.912281i \(-0.634320\pi\)
−0.409566 + 0.912281i \(0.634320\pi\)
\(402\) 0 0
\(403\) 17.4269i 0.868096i
\(404\) 0 0
\(405\) −17.0407 6.00142i −0.846761 0.298213i
\(406\) 0 0
\(407\) 27.4458i 1.36044i
\(408\) 0 0
\(409\) −8.07018 −0.399045 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(410\) 0 0
\(411\) 2.81745 0.138974
\(412\) 0 0
\(413\) 7.75195i 0.381449i
\(414\) 0 0
\(415\) 11.3630 32.2646i 0.557786 1.58381i
\(416\) 0 0
\(417\) 3.14158i 0.153844i
\(418\) 0 0
\(419\) −3.82520 −0.186873 −0.0934366 0.995625i \(-0.529785\pi\)
−0.0934366 + 0.995625i \(0.529785\pi\)
\(420\) 0 0
\(421\) −13.5304 −0.659429 −0.329715 0.944081i \(-0.606952\pi\)
−0.329715 + 0.944081i \(0.606952\pi\)
\(422\) 0 0
\(423\) 29.9298i 1.45524i
\(424\) 0 0
\(425\) 15.2200 18.9281i 0.738279 0.918149i
\(426\) 0 0
\(427\) 7.57491i 0.366576i
\(428\) 0 0
\(429\) −8.26275 −0.398929
\(430\) 0 0
\(431\) 22.0529 1.06225 0.531126 0.847293i \(-0.321769\pi\)
0.531126 + 0.847293i \(0.321769\pi\)
\(432\) 0 0
\(433\) 32.5455i 1.56404i 0.623255 + 0.782018i \(0.285810\pi\)
−0.623255 + 0.782018i \(0.714190\pi\)
\(434\) 0 0
\(435\) −0.128967 + 0.366196i −0.00618351 + 0.0175578i
\(436\) 0 0
\(437\) 10.5173i 0.503109i
\(438\) 0 0
\(439\) −20.6442 −0.985295 −0.492647 0.870229i \(-0.663971\pi\)
−0.492647 + 0.870229i \(0.663971\pi\)
\(440\) 0 0
\(441\) −2.89655 −0.137931
\(442\) 0 0
\(443\) 10.5066i 0.499181i 0.968351 + 0.249591i \(0.0802961\pi\)
−0.968351 + 0.249591i \(0.919704\pi\)
\(444\) 0 0
\(445\) 12.6546 + 4.45669i 0.599884 + 0.211267i
\(446\) 0 0
\(447\) 7.35673i 0.347961i
\(448\) 0 0
\(449\) −8.76930 −0.413849 −0.206924 0.978357i \(-0.566345\pi\)
−0.206924 + 0.978357i \(0.566345\pi\)
\(450\) 0 0
\(451\) 11.5771 0.545147
\(452\) 0 0
\(453\) 5.18179i 0.243462i
\(454\) 0 0
\(455\) 12.3707 + 4.35673i 0.579948 + 0.204246i
\(456\) 0 0
\(457\) 18.1509i 0.849063i −0.905413 0.424532i \(-0.860439\pi\)
0.905413 0.424532i \(-0.139561\pi\)
\(458\) 0 0
\(459\) 9.21280 0.430017
\(460\) 0 0
\(461\) −3.22766 −0.150327 −0.0751635 0.997171i \(-0.523948\pi\)
−0.0751635 + 0.997171i \(0.523948\pi\)
\(462\) 0 0
\(463\) 19.7931i 0.919863i −0.887954 0.459932i \(-0.847874\pi\)
0.887954 0.459932i \(-0.152126\pi\)
\(464\) 0 0
\(465\) 0.709821 2.01550i 0.0329172 0.0934667i
\(466\) 0 0
\(467\) 14.3918i 0.665974i −0.942931 0.332987i \(-0.891944\pi\)
0.942931 0.332987i \(-0.108056\pi\)
\(468\) 0 0
\(469\) −3.79310 −0.175149
\(470\) 0 0
\(471\) 0.170448 0.00785383
\(472\) 0 0
\(473\) 20.3369i 0.935089i
\(474\) 0 0
\(475\) −30.2059 24.2884i −1.38594 1.11443i
\(476\) 0 0
\(477\) 1.38401i 0.0633694i
\(478\) 0 0
\(479\) 3.92675 0.179418 0.0897090 0.995968i \(-0.471406\pi\)
0.0897090 + 0.995968i \(0.471406\pi\)
\(480\) 0 0
\(481\) 36.7550 1.67588
\(482\) 0 0
\(483\) 0.436373i 0.0198557i
\(484\) 0 0
\(485\) 10.1147 28.7203i 0.459287 1.30412i
\(486\) 0 0
\(487\) 10.4364i 0.472917i −0.971642 0.236459i \(-0.924013\pi\)
0.971642 0.236459i \(-0.0759868\pi\)
\(488\) 0 0
\(489\) −6.92035 −0.312949
\(490\) 0 0
\(491\) 28.6435 1.29266 0.646331 0.763058i \(-0.276303\pi\)
0.646331 + 0.763058i \(0.276303\pi\)
\(492\) 0 0
\(493\) 2.62228i 0.118102i
\(494\) 0 0
\(495\) −26.7569 9.42327i −1.20263 0.423545i
\(496\) 0 0
\(497\) 9.23752i 0.414359i
\(498\) 0 0
\(499\) 34.5857 1.54827 0.774135 0.633021i \(-0.218185\pi\)
0.774135 + 0.633021i \(0.218185\pi\)
\(500\) 0 0
\(501\) 3.39000 0.151454
\(502\) 0 0
\(503\) 18.5624i 0.827656i −0.910355 0.413828i \(-0.864191\pi\)
0.910355 0.413828i \(-0.135809\pi\)
\(504\) 0 0
\(505\) 13.2627 + 4.67088i 0.590184 + 0.207852i
\(506\) 0 0
\(507\) 6.88403i 0.305731i
\(508\) 0 0
\(509\) 27.5975 1.22324 0.611618 0.791153i \(-0.290519\pi\)
0.611618 + 0.791153i \(0.290519\pi\)
\(510\) 0 0
\(511\) 0.477813 0.0211372
\(512\) 0 0
\(513\) 14.7020i 0.649107i
\(514\) 0 0
\(515\) 6.87317 19.5160i 0.302868 0.859979i
\(516\) 0 0
\(517\) 45.2567i 1.99039i
\(518\) 0 0
\(519\) 3.79779 0.166704
\(520\) 0 0
\(521\) −21.0095 −0.920442 −0.460221 0.887804i \(-0.652230\pi\)
−0.460221 + 0.887804i \(0.652230\pi\)
\(522\) 0 0
\(523\) 32.1706i 1.40672i −0.710833 0.703361i \(-0.751682\pi\)
0.710833 0.703361i \(-0.248318\pi\)
\(524\) 0 0
\(525\) −1.25328 1.00775i −0.0546974 0.0439819i
\(526\) 0 0
\(527\) 14.4327i 0.628700i
\(528\) 0 0
\(529\) 21.1593 0.919969
\(530\) 0 0
\(531\) 22.4539 0.974417
\(532\) 0 0
\(533\) 15.5039i 0.671549i
\(534\) 0 0
\(535\) 11.5771 32.8727i 0.500524 1.42121i
\(536\) 0 0
\(537\) 5.94226i 0.256427i
\(538\) 0 0
\(539\) −4.37985 −0.188654
\(540\) 0 0
\(541\) −35.3424 −1.51949 −0.759744 0.650222i \(-0.774676\pi\)
−0.759744 + 0.650222i \(0.774676\pi\)
\(542\) 0 0
\(543\) 1.56363i 0.0671016i
\(544\) 0 0
\(545\) −23.7342 8.35873i −1.01666 0.358049i
\(546\) 0 0
\(547\) 19.3567i 0.827634i −0.910360 0.413817i \(-0.864195\pi\)
0.910360 0.413817i \(-0.135805\pi\)
\(548\) 0 0
\(549\) −21.9411 −0.936424
\(550\) 0 0
\(551\) 4.18469 0.178274
\(552\) 0 0
\(553\) 1.88654i 0.0802237i
\(554\) 0 0
\(555\) −4.25088 1.49708i −0.180440 0.0635475i
\(556\) 0 0
\(557\) 15.8280i 0.670655i 0.942102 + 0.335328i \(0.108847\pi\)
−0.942102 + 0.335328i \(0.891153\pi\)
\(558\) 0 0
\(559\) −27.2347 −1.15191
\(560\) 0 0
\(561\) 6.84310 0.288916
\(562\) 0 0
\(563\) 25.5749i 1.07785i −0.842352 0.538927i \(-0.818830\pi\)
0.842352 0.538927i \(-0.181170\pi\)
\(564\) 0 0
\(565\) 9.06836 25.7492i 0.381509 1.08328i
\(566\) 0 0
\(567\) 8.07965i 0.339313i
\(568\) 0 0
\(569\) 18.3662 0.769951 0.384976 0.922927i \(-0.374210\pi\)
0.384976 + 0.922927i \(0.374210\pi\)
\(570\) 0 0
\(571\) −44.9078 −1.87933 −0.939667 0.342091i \(-0.888865\pi\)
−0.939667 + 0.342091i \(0.888865\pi\)
\(572\) 0 0
\(573\) 5.02811i 0.210052i
\(574\) 0 0
\(575\) −4.25088 + 5.28655i −0.177274 + 0.220464i
\(576\) 0 0
\(577\) 2.84216i 0.118321i 0.998248 + 0.0591604i \(0.0188423\pi\)
−0.998248 + 0.0591604i \(0.981158\pi\)
\(578\) 0 0
\(579\) 6.74420 0.280279
\(580\) 0 0
\(581\) −15.2978 −0.634661
\(582\) 0 0
\(583\) 2.09275i 0.0866728i
\(584\) 0 0
\(585\) 12.6195 35.8324i 0.521751 1.48149i
\(586\) 0 0
\(587\) 9.16111i 0.378119i 0.981966 + 0.189060i \(0.0605440\pi\)
−0.981966 + 0.189060i \(0.939456\pi\)
\(588\) 0 0
\(589\) −23.0320 −0.949019
\(590\) 0 0
\(591\) 4.88238 0.200834
\(592\) 0 0
\(593\) 42.8676i 1.76036i −0.474639 0.880181i \(-0.657421\pi\)
0.474639 0.880181i \(-0.342579\pi\)
\(594\) 0 0
\(595\) −10.2453 3.60819i −0.420015 0.147921i
\(596\) 0 0
\(597\) 3.77307i 0.154422i
\(598\) 0 0
\(599\) 11.6019 0.474039 0.237020 0.971505i \(-0.423829\pi\)
0.237020 + 0.971505i \(0.423829\pi\)
\(600\) 0 0
\(601\) 40.5066 1.65230 0.826148 0.563453i \(-0.190527\pi\)
0.826148 + 0.563453i \(0.190527\pi\)
\(602\) 0 0
\(603\) 10.9869i 0.447421i
\(604\) 0 0
\(605\) −17.2589 6.07826i −0.701675 0.247116i
\(606\) 0 0
\(607\) 2.10345i 0.0853764i 0.999088 + 0.0426882i \(0.0135922\pi\)
−0.999088 + 0.0426882i \(0.986408\pi\)
\(608\) 0 0
\(609\) 0.173627 0.00703574
\(610\) 0 0
\(611\) −60.6069 −2.45189
\(612\) 0 0
\(613\) 44.1721i 1.78409i 0.451943 + 0.892047i \(0.350731\pi\)
−0.451943 + 0.892047i \(0.649269\pi\)
\(614\) 0 0
\(615\) 0.631495 1.79310i 0.0254643 0.0723047i
\(616\) 0 0
\(617\) 33.5011i 1.34870i 0.738410 + 0.674352i \(0.235577\pi\)
−0.738410 + 0.674352i \(0.764423\pi\)
\(618\) 0 0
\(619\) −30.4117 −1.22235 −0.611174 0.791496i \(-0.709302\pi\)
−0.611174 + 0.791496i \(0.709302\pi\)
\(620\) 0 0
\(621\) −2.57310 −0.103255
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 5.36620 + 24.4173i 0.214648 + 0.976692i
\(626\) 0 0
\(627\) 10.9204i 0.436117i
\(628\) 0 0
\(629\) −30.4400 −1.21372
\(630\) 0 0
\(631\) −9.99798 −0.398013 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(632\) 0 0
\(633\) 0.453098i 0.0180090i
\(634\) 0 0
\(635\) 6.79633 19.2978i 0.269704 0.765811i
\(636\) 0 0
\(637\) 5.86542i 0.232396i
\(638\) 0 0
\(639\) 26.7569 1.05849
\(640\) 0 0
\(641\) 7.79310 0.307809 0.153904 0.988086i \(-0.450815\pi\)
0.153904 + 0.988086i \(0.450815\pi\)
\(642\) 0 0
\(643\) 31.0351i 1.22390i 0.790895 + 0.611952i \(0.209616\pi\)
−0.790895 + 0.611952i \(0.790384\pi\)
\(644\) 0 0
\(645\) 3.14982 + 1.10931i 0.124024 + 0.0436789i
\(646\) 0 0
\(647\) 22.2771i 0.875802i −0.899023 0.437901i \(-0.855722\pi\)
0.899023 0.437901i \(-0.144278\pi\)
\(648\) 0 0
\(649\) 33.9524 1.33275
\(650\) 0 0
\(651\) −0.955625 −0.0374539
\(652\) 0 0
\(653\) 7.47995i 0.292713i 0.989232 + 0.146356i \(0.0467547\pi\)
−0.989232 + 0.146356i \(0.953245\pi\)
\(654\) 0 0
\(655\) −34.8246 12.2646i −1.36071 0.479216i
\(656\) 0 0
\(657\) 1.38401i 0.0539953i
\(658\) 0 0
\(659\) 6.54904 0.255114 0.127557 0.991831i \(-0.459286\pi\)
0.127557 + 0.991831i \(0.459286\pi\)
\(660\) 0 0
\(661\) −37.9637 −1.47662 −0.738308 0.674464i \(-0.764375\pi\)
−0.738308 + 0.674464i \(0.764375\pi\)
\(662\) 0 0
\(663\) 9.16416i 0.355906i
\(664\) 0 0
\(665\) −5.75801 + 16.3496i −0.223286 + 0.634010i
\(666\) 0 0
\(667\) 0.732393i 0.0283584i
\(668\) 0 0
\(669\) 1.48275 0.0573263
\(670\) 0 0
\(671\) −33.1770 −1.28078
\(672\) 0 0
\(673\) 3.12481i 0.120453i 0.998185 + 0.0602263i \(0.0191822\pi\)
−0.998185 + 0.0602263i \(0.980818\pi\)
\(674\) 0 0
\(675\) −5.94226 + 7.39000i −0.228718 + 0.284441i
\(676\) 0 0
\(677\) 20.4137i 0.784562i −0.919845 0.392281i \(-0.871686\pi\)
0.919845 0.392281i \(-0.128314\pi\)
\(678\) 0 0
\(679\) −13.6174 −0.522587
\(680\) 0 0
\(681\) 1.84894 0.0708517
\(682\) 0 0
\(683\) 9.63380i 0.368627i 0.982867 + 0.184314i \(0.0590062\pi\)
−0.982867 + 0.184314i \(0.940994\pi\)
\(684\) 0 0
\(685\) −6.50655 + 18.4750i −0.248603 + 0.705895i
\(686\) 0 0
\(687\) 5.63017i 0.214805i
\(688\) 0 0
\(689\) 2.80257 0.106769
\(690\) 0 0
\(691\) −20.4384 −0.777514 −0.388757 0.921340i \(-0.627095\pi\)
−0.388757 + 0.921340i \(0.627095\pi\)
\(692\) 0 0
\(693\) 12.6865i 0.481919i
\(694\) 0 0
\(695\) −20.6005 7.25509i −0.781420 0.275201i
\(696\) 0 0
\(697\) 12.8401i 0.486355i
\(698\) 0 0
\(699\) −5.94226 −0.224757
\(700\) 0 0
\(701\) −12.4733 −0.471109 −0.235555 0.971861i \(-0.575691\pi\)
−0.235555 + 0.971861i \(0.575691\pi\)
\(702\) 0 0
\(703\) 48.5767i 1.83211i
\(704\) 0 0
\(705\) 7.00947 + 2.46860i 0.263992 + 0.0929729i
\(706\) 0 0
\(707\) 6.28836i 0.236498i
\(708\) 0 0
\(709\) 31.7788 1.19348 0.596738 0.802436i \(-0.296463\pi\)
0.596738 + 0.802436i \(0.296463\pi\)
\(710\) 0 0
\(711\) 5.46444 0.204933
\(712\) 0 0
\(713\) 4.03101i 0.150962i
\(714\) 0 0
\(715\) 19.0818 54.1819i 0.713619 2.02629i
\(716\) 0 0
\(717\) 2.36435i 0.0882982i
\(718\) 0 0
\(719\) 20.4905 0.764168 0.382084 0.924128i \(-0.375207\pi\)
0.382084 + 0.924128i \(0.375207\pi\)
\(720\) 0 0
\(721\) −9.25328 −0.344610
\(722\) 0 0
\(723\) 1.13088i 0.0420580i
\(724\) 0 0
\(725\) −2.10345 1.69137i −0.0781202 0.0628160i
\(726\) 0 0
\(727\) 39.8822i 1.47915i −0.673074 0.739575i \(-0.735027\pi\)
0.673074 0.739575i \(-0.264973\pi\)
\(728\) 0 0
\(729\) 21.5731 0.799004
\(730\) 0 0
\(731\) 22.5555 0.834244
\(732\) 0 0
\(733\) 12.4559i 0.460070i 0.973182 + 0.230035i \(0.0738841\pi\)
−0.973182 + 0.230035i \(0.926116\pi\)
\(734\) 0 0
\(735\) −0.238906 + 0.678363i −0.00881219 + 0.0250218i
\(736\) 0 0
\(737\) 16.6132i 0.611955i
\(738\) 0 0
\(739\) 33.5258 1.23327 0.616634 0.787250i \(-0.288496\pi\)
0.616634 + 0.787250i \(0.288496\pi\)
\(740\) 0 0
\(741\) 14.6243 0.537238
\(742\) 0 0
\(743\) 3.72292i 0.136581i 0.997665 + 0.0682904i \(0.0217544\pi\)
−0.997665 + 0.0682904i \(0.978246\pi\)
\(744\) 0 0
\(745\) −48.2408 16.9895i −1.76740 0.622446i
\(746\) 0 0
\(747\) 44.3109i 1.62125i
\(748\) 0 0
\(749\) −15.5862 −0.569507
\(750\) 0 0
\(751\) −29.5991 −1.08009 −0.540043 0.841637i \(-0.681592\pi\)
−0.540043 + 0.841637i \(0.681592\pi\)
\(752\) 0 0
\(753\) 1.84506i 0.0672375i
\(754\) 0 0
\(755\) 33.9789 + 11.9667i 1.23662 + 0.435514i
\(756\) 0 0
\(757\) 4.66250i 0.169461i 0.996404 + 0.0847307i \(0.0270030\pi\)
−0.996404 + 0.0847307i \(0.972997\pi\)
\(758\) 0 0
\(759\) −1.91125 −0.0693740
\(760\) 0 0
\(761\) −27.5160 −0.997455 −0.498727 0.866759i \(-0.666199\pi\)
−0.498727 + 0.866759i \(0.666199\pi\)
\(762\) 0 0
\(763\) 11.2533i 0.407396i
\(764\) 0 0
\(765\) −10.4513 + 29.6759i −0.377867 + 1.07294i
\(766\) 0 0
\(767\) 45.4684i 1.64177i
\(768\) 0 0
\(769\) −16.8026 −0.605916 −0.302958 0.953004i \(-0.597974\pi\)
−0.302958 + 0.953004i \(0.597974\pi\)
\(770\) 0 0
\(771\) −5.78857 −0.208470
\(772\) 0 0
\(773\) 13.6695i 0.491657i −0.969313 0.245829i \(-0.920940\pi\)
0.969313 0.245829i \(-0.0790601\pi\)
\(774\) 0 0
\(775\) 11.5771 + 9.30912i 0.415863 + 0.334393i
\(776\) 0 0
\(777\) 2.01550i 0.0723058i
\(778\) 0 0
\(779\) −20.4905 −0.734150
\(780\) 0 0
\(781\) 40.4589 1.44773
\(782\) 0 0
\(783\) 1.02380i 0.0365877i
\(784\) 0 0
\(785\) −0.393629 + 1.11769i −0.0140492 + 0.0398921i
\(786\) 0 0
\(787\) 5.81509i 0.207285i 0.994615 + 0.103643i \(0.0330499\pi\)
−0.994615 + 0.103643i \(0.966950\pi\)
\(788\) 0 0
\(789\) −5.42690 −0.193203
\(790\) 0 0
\(791\) −12.2086 −0.434089
\(792\) 0 0
\(793\) 44.4300i 1.57776i
\(794\) 0 0
\(795\) −0.324130 0.114152i −0.0114957 0.00404857i
\(796\) 0 0
\(797\) 39.6907i 1.40592i −0.711232 0.702958i \(-0.751862\pi\)
0.711232 0.702958i \(-0.248138\pi\)
\(798\) 0 0
\(799\) 50.1939 1.77573
\(800\) 0 0
\(801\) −17.3793 −0.614067
\(802\) 0 0
\(803\) 2.09275i 0.0738515i
\(804\) 0 0
\(805\) 2.86146 + 1.00775i 0.100853 + 0.0355186i
\(806\) 0 0
\(807\) 8.32222i 0.292956i
\(808\) 0 0
\(809\) 16.6765 0.586316 0.293158 0.956064i \(-0.405294\pi\)
0.293158 + 0.956064i \(0.405294\pi\)
\(810\) 0 0
\(811\) 15.2981 0.537189 0.268594 0.963253i \(-0.413441\pi\)
0.268594 + 0.963253i \(0.413441\pi\)
\(812\) 0 0
\(813\) 8.75970i 0.307216i
\(814\) 0 0
\(815\) 15.9817 45.3793i 0.559815 1.58957i
\(816\) 0 0
\(817\) 35.9944i 1.25929i
\(818\) 0 0
\(819\) −16.9895 −0.593660
\(820\) 0 0
\(821\) 20.9097 0.729752 0.364876 0.931056i \(-0.381111\pi\)
0.364876 + 0.931056i \(0.381111\pi\)
\(822\) 0 0
\(823\) 51.9048i 1.80929i −0.426169 0.904644i \(-0.640137\pi\)
0.426169 0.904644i \(-0.359863\pi\)
\(824\) 0 0
\(825\) −4.41380 + 5.48916i −0.153669 + 0.191108i
\(826\) 0 0
\(827\) 5.63380i 0.195907i −0.995191 0.0979533i \(-0.968770\pi\)
0.995191 0.0979533i \(-0.0312296\pi\)
\(828\) 0 0
\(829\) −27.8044 −0.965686 −0.482843 0.875707i \(-0.660396\pi\)
−0.482843 + 0.875707i \(0.660396\pi\)
\(830\) 0 0
\(831\) 7.69983 0.267104
\(832\) 0 0
\(833\) 4.85766i 0.168308i
\(834\) 0 0
\(835\) −7.82879 + 22.2295i −0.270927 + 0.769283i
\(836\) 0 0
\(837\) 5.63489i 0.194770i
\(838\) 0 0
\(839\) −3.61939 −0.124955 −0.0624776 0.998046i \(-0.519900\pi\)
−0.0624776 + 0.998046i \(0.519900\pi\)
\(840\) 0 0
\(841\) −28.7086 −0.989951
\(842\) 0 0
\(843\) 6.28778i 0.216563i
\(844\) 0 0
\(845\) 45.1411 + 15.8978i 1.55290 + 0.546902i
\(846\) 0 0
\(847\) 8.18310i 0.281175i
\(848\) 0 0
\(849\) 9.94415 0.341283
\(850\) 0 0
\(851\) 8.50177 0.291437
\(852\) 0 0
\(853\) 57.9705i 1.98487i −0.122758 0.992437i \(-0.539174\pi\)
0.122758 0.992437i \(-0.460826\pi\)
\(854\) 0 0
\(855\) 47.3574 + 16.6784i 1.61959 + 0.570388i
\(856\) 0 0
\(857\) 4.09720i 0.139958i −0.997548 0.0699789i \(-0.977707\pi\)
0.997548 0.0699789i \(-0.0222932\pi\)
\(858\) 0 0
\(859\) 8.96551 0.305899 0.152950 0.988234i \(-0.451123\pi\)
0.152950 + 0.988234i \(0.451123\pi\)
\(860\) 0 0
\(861\) −0.850175 −0.0289739
\(862\) 0 0
\(863\) 48.4589i 1.64956i 0.565453 + 0.824781i \(0.308702\pi\)
−0.565453 + 0.824781i \(0.691298\pi\)
\(864\) 0 0
\(865\) −8.77053 + 24.9035i −0.298207 + 0.846744i
\(866\) 0 0
\(867\) 2.12181i 0.0720604i
\(868\) 0 0
\(869\) 8.26275 0.280294
\(870\) 0 0
\(871\) −22.2481 −0.753848
\(872\) 0 0
\(873\) 39.4434i 1.33496i
\(874\) 0 0
\(875\) 9.50248 5.89091i 0.321243 0.199149i
\(876\) 0 0
\(877\) 27.5589i 0.930597i −0.885154 0.465298i \(-0.845947\pi\)
0.885154 0.465298i \(-0.154053\pi\)
\(878\) 0 0
\(879\) −3.10010 −0.104564
\(880\) 0 0
\(881\) −21.2389 −0.715558 −0.357779 0.933806i \(-0.616466\pi\)
−0.357779 + 0.933806i \(0.616466\pi\)
\(882\) 0 0
\(883\) 10.9204i 0.367499i 0.982973 + 0.183750i \(0.0588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(884\) 0 0
\(885\) 1.85199 5.25864i 0.0622540 0.176767i
\(886\) 0 0
\(887\) 41.5898i 1.39645i −0.715878 0.698225i \(-0.753974\pi\)
0.715878 0.698225i \(-0.246026\pi\)
\(888\) 0 0
\(889\) −9.14982 −0.306875
\(890\) 0 0
\(891\) 35.3877 1.18553
\(892\) 0 0
\(893\) 80.1003i 2.68045i
\(894\) 0 0
\(895\) 38.9656 + 13.7229i 1.30248 + 0.458707i
\(896\) 0 0
\(897\) 2.55951i 0.0854596i
\(898\) 0 0
\(899\) −1.60389 −0.0534926
\(900\) 0 0
\(901\) −2.32105 −0.0773255
\(902\) 0 0
\(903\) 1.49345i 0.0496989i
\(904\) 0 0
\(905\) 10.2533 + 3.61101i 0.340830 + 0.120034i
\(906\) 0 0
\(907\) 11.8596i 0.393793i 0.980424 + 0.196897i \(0.0630863\pi\)
−0.980424 + 0.196897i \(0.936914\pi\)
\(908\) 0 0
\(909\) −18.2146 −0.604139
\(910\) 0 0
\(911\) 30.0522 0.995673 0.497837 0.867271i \(-0.334128\pi\)
0.497837 + 0.867271i \(0.334128\pi\)
\(912\) 0 0
\(913\) 67.0022i 2.21745i
\(914\) 0 0
\(915\) −1.80969 + 5.13854i −0.0598267 + 0.169875i
\(916\) 0 0
\(917\) 16.5117i 0.545263i
\(918\) 0 0
\(919\) −21.5751 −0.711698 −0.355849 0.934544i \(-0.615808\pi\)
−0.355849 + 0.934544i \(0.615808\pi\)
\(920\) 0 0
\(921\) 3.48275 0.114760
\(922\) 0 0
\(923\) 54.1819i 1.78342i
\(924\) 0 0
\(925\) 19.6338 24.4173i 0.645555 0.802835i
\(926\) 0 0
\(927\) 26.8026i 0.880312i
\(928\) 0 0
\(929\) 23.7229 0.778324 0.389162 0.921169i \(-0.372765\pi\)
0.389162 + 0.921169i \(0.372765\pi\)
\(930\) 0 0
\(931\) 7.75195 0.254060
\(932\) 0 0
\(933\) 3.12481i 0.102302i
\(934\) 0 0
\(935\) −15.8033 + 44.8727i −0.516824 + 1.46750i
\(936\) 0 0
\(937\) 12.8154i 0.418662i −0.977845 0.209331i \(-0.932871\pi\)
0.977845 0.209331i \(-0.0671286\pi\)
\(938\) 0 0
\(939\) 1.56241 0.0509872
\(940\) 0 0
\(941\) 40.0339 1.30507 0.652533 0.757760i \(-0.273706\pi\)
0.652533 + 0.757760i \(0.273706\pi\)
\(942\) 0 0
\(943\) 3.58620i 0.116783i
\(944\) 0 0
\(945\) 4.00000 + 1.40872i 0.130120 + 0.0458258i
\(946\) 0 0
\(947\) 31.8822i 1.03603i −0.855371 0.518016i \(-0.826671\pi\)
0.855371 0.518016i \(-0.173329\pi\)
\(948\) 0 0
\(949\) 2.80257 0.0909753
\(950\) 0 0
\(951\) −6.69477 −0.217093
\(952\) 0 0
\(953\) 53.5138i 1.73348i −0.498757 0.866742i \(-0.666210\pi\)
0.498757 0.866742i \(-0.333790\pi\)
\(954\) 0 0
\(955\) 32.9712 + 11.6118i 1.06692 + 0.375750i
\(956\) 0 0
\(957\) 0.760462i 0.0245823i
\(958\) 0 0
\(959\) 8.75970 0.282866
\(960\) 0 0
\(961\) −22.1724 −0.715239
\(962\) 0 0
\(963\) 45.1462i 1.45482i
\(964\) 0 0
\(965\) −15.5749 + 44.2242i −0.501374 + 1.42363i
\(966\) 0 0
\(967\) 22.2295i 0.714852i 0.933942 + 0.357426i \(0.116346\pi\)
−0.933942 + 0.357426i \(0.883654\pi\)
\(968\) 0 0
\(969\) −12.1117 −0.389084
\(970\) 0 0
\(971\) −28.0888 −0.901413 −0.450706 0.892672i \(-0.648828\pi\)
−0.450706 + 0.892672i \(0.648828\pi\)
\(972\) 0 0
\(973\) 9.76745i 0.313130i
\(974\) 0 0
\(975\) −7.35098 5.91088i −0.235420 0.189300i
\(976\) 0 0
\(977\) 8.75970i 0.280248i −0.990134 0.140124i \(-0.955250\pi\)
0.990134 0.140124i \(-0.0447501\pi\)
\(978\) 0 0
\(979\) −26.2791 −0.839884
\(980\) 0 0
\(981\) 32.5957 1.04070
\(982\) 0 0
\(983\) 34.7467i 1.10825i 0.832434 + 0.554124i \(0.186947\pi\)
−0.832434 + 0.554124i \(0.813053\pi\)
\(984\) 0 0
\(985\) −11.2753 + 32.0156i −0.359260 + 1.02010i
\(986\) 0 0
\(987\) 3.32345i 0.105787i
\(988\) 0 0
\(989\) −6.29965 −0.200317
\(990\) 0 0
\(991\) −15.1798 −0.482201 −0.241101 0.970500i \(-0.577508\pi\)
−0.241101 + 0.970500i \(0.577508\pi\)
\(992\) 0 0
\(993\) 5.29400i 0.168000i
\(994\) 0 0
\(995\) −24.7414 8.71345i −0.784356 0.276235i
\(996\) 0 0
\(997\) 55.3483i 1.75290i 0.481496 + 0.876448i \(0.340094\pi\)
−0.481496 + 0.876448i \(0.659906\pi\)
\(998\) 0 0
\(999\) 11.8845 0.376009
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 2240.2.g.p.449.7 12
4.3 odd 2 inner 2240.2.g.p.449.5 12
5.4 even 2 inner 2240.2.g.p.449.6 12
8.3 odd 2 1120.2.g.d.449.8 yes 12
8.5 even 2 1120.2.g.d.449.6 yes 12
20.19 odd 2 inner 2240.2.g.p.449.8 12
40.3 even 4 5600.2.a.by.1.4 6
40.13 odd 4 5600.2.a.bz.1.3 6
40.19 odd 2 1120.2.g.d.449.5 12
40.27 even 4 5600.2.a.bz.1.4 6
40.29 even 2 1120.2.g.d.449.7 yes 12
40.37 odd 4 5600.2.a.by.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.g.d.449.5 12 40.19 odd 2
1120.2.g.d.449.6 yes 12 8.5 even 2
1120.2.g.d.449.7 yes 12 40.29 even 2
1120.2.g.d.449.8 yes 12 8.3 odd 2
2240.2.g.p.449.5 12 4.3 odd 2 inner
2240.2.g.p.449.6 12 5.4 even 2 inner
2240.2.g.p.449.7 12 1.1 even 1 trivial
2240.2.g.p.449.8 12 20.19 odd 2 inner
5600.2.a.by.1.3 6 40.37 odd 4
5600.2.a.by.1.4 6 40.3 even 4
5600.2.a.bz.1.3 6 40.13 odd 4
5600.2.a.bz.1.4 6 40.27 even 4