Properties

Label 160.6.c.c.129.11
Level $160$
Weight $6$
Character 160.129
Analytic conductor $25.661$
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] = [160,6,Mod(129,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.c (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: \(25.6614111701\)
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: \( x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{44}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.11
Root \(4.23250i\) of defining polynomial
Character \(\chi\) \(=\) 160.129
Dual form 160.6.c.c.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+26.2718i q^{3} +(48.8634 - 27.1545i) q^{5} -23.9306i q^{7} -447.206 q^{9} +O(q^{10})\) \(q+26.2718i q^{3} +(48.8634 - 27.1545i) q^{5} -23.9306i q^{7} -447.206 q^{9} +704.032 q^{11} +1037.56i q^{13} +(713.397 + 1283.73i) q^{15} -217.433i q^{17} +1290.13 q^{19} +628.700 q^{21} +2806.24i q^{23} +(1650.27 - 2653.72i) q^{25} -5364.86i q^{27} -4905.42 q^{29} +415.555 q^{31} +18496.2i q^{33} +(-649.824 - 1169.33i) q^{35} +6450.71i q^{37} -27258.5 q^{39} -16402.5 q^{41} -13772.3i q^{43} +(-21852.0 + 12143.7i) q^{45} +12942.4i q^{47} +16234.3 q^{49} +5712.36 q^{51} -5239.24i q^{53} +(34401.4 - 19117.6i) q^{55} +33894.0i q^{57} -24042.4 q^{59} -1037.24 q^{61} +10701.9i q^{63} +(28174.3 + 50698.6i) q^{65} +47902.5i q^{67} -73725.0 q^{69} +74761.8 q^{71} -22746.5i q^{73} +(69718.0 + 43355.5i) q^{75} -16847.9i q^{77} -64213.8 q^{79} +32273.2 q^{81} +22523.0i q^{83} +(-5904.29 - 10624.5i) q^{85} -128874. i q^{87} +37886.4 q^{89} +24829.4 q^{91} +10917.4i q^{93} +(63040.1 - 35032.8i) q^{95} -84119.7i q^{97} -314847. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 60 q^{5} - 1676 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 60 q^{5} - 1676 q^{9} + 688 q^{21} + 1500 q^{25} - 21304 q^{29} - 109672 q^{41} - 75140 q^{45} - 32348 q^{49} + 64552 q^{61} - 160 q^{65} - 166352 q^{69} - 173764 q^{81} + 151360 q^{85} - 3720 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 26.2718i 1.68534i 0.538434 + 0.842668i \(0.319016\pi\)
−0.538434 + 0.842668i \(0.680984\pi\)
\(4\) 0 0
\(5\) 48.8634 27.1545i 0.874095 0.485754i
\(6\) 0 0
\(7\) 23.9306i 0.184590i −0.995732 0.0922952i \(-0.970580\pi\)
0.995732 0.0922952i \(-0.0294203\pi\)
\(8\) 0 0
\(9\) −447.206 −1.84035
\(10\) 0 0
\(11\) 704.032 1.75433 0.877164 0.480191i \(-0.159433\pi\)
0.877164 + 0.480191i \(0.159433\pi\)
\(12\) 0 0
\(13\) 1037.56i 1.70276i 0.524549 + 0.851380i \(0.324234\pi\)
−0.524549 + 0.851380i \(0.675766\pi\)
\(14\) 0 0
\(15\) 713.397 + 1283.73i 0.818659 + 1.47314i
\(16\) 0 0
\(17\) 217.433i 0.182475i −0.995829 0.0912375i \(-0.970918\pi\)
0.995829 0.0912375i \(-0.0290823\pi\)
\(18\) 0 0
\(19\) 1290.13 0.819878 0.409939 0.912113i \(-0.365550\pi\)
0.409939 + 0.912113i \(0.365550\pi\)
\(20\) 0 0
\(21\) 628.700 0.311097
\(22\) 0 0
\(23\) 2806.24i 1.10613i 0.833138 + 0.553065i \(0.186542\pi\)
−0.833138 + 0.553065i \(0.813458\pi\)
\(24\) 0 0
\(25\) 1650.27 2653.72i 0.528086 0.849191i
\(26\) 0 0
\(27\) 5364.86i 1.41628i
\(28\) 0 0
\(29\) −4905.42 −1.08313 −0.541566 0.840658i \(-0.682168\pi\)
−0.541566 + 0.840658i \(0.682168\pi\)
\(30\) 0 0
\(31\) 415.555 0.0776648 0.0388324 0.999246i \(-0.487636\pi\)
0.0388324 + 0.999246i \(0.487636\pi\)
\(32\) 0 0
\(33\) 18496.2i 2.95663i
\(34\) 0 0
\(35\) −649.824 1169.33i −0.0896656 0.161350i
\(36\) 0 0
\(37\) 6450.71i 0.774646i 0.921944 + 0.387323i \(0.126600\pi\)
−0.921944 + 0.387323i \(0.873400\pi\)
\(38\) 0 0
\(39\) −27258.5 −2.86972
\(40\) 0 0
\(41\) −16402.5 −1.52388 −0.761940 0.647648i \(-0.775753\pi\)
−0.761940 + 0.647648i \(0.775753\pi\)
\(42\) 0 0
\(43\) 13772.3i 1.13589i −0.823067 0.567944i \(-0.807739\pi\)
0.823067 0.567944i \(-0.192261\pi\)
\(44\) 0 0
\(45\) −21852.0 + 12143.7i −1.60865 + 0.893960i
\(46\) 0 0
\(47\) 12942.4i 0.854612i 0.904107 + 0.427306i \(0.140537\pi\)
−0.904107 + 0.427306i \(0.859463\pi\)
\(48\) 0 0
\(49\) 16234.3 0.965926
\(50\) 0 0
\(51\) 5712.36 0.307532
\(52\) 0 0
\(53\) 5239.24i 0.256200i −0.991761 0.128100i \(-0.959112\pi\)
0.991761 0.128100i \(-0.0408878\pi\)
\(54\) 0 0
\(55\) 34401.4 19117.6i 1.53345 0.852172i
\(56\) 0 0
\(57\) 33894.0i 1.38177i
\(58\) 0 0
\(59\) −24042.4 −0.899182 −0.449591 0.893234i \(-0.648430\pi\)
−0.449591 + 0.893234i \(0.648430\pi\)
\(60\) 0 0
\(61\) −1037.24 −0.0356908 −0.0178454 0.999841i \(-0.505681\pi\)
−0.0178454 + 0.999841i \(0.505681\pi\)
\(62\) 0 0
\(63\) 10701.9i 0.339712i
\(64\) 0 0
\(65\) 28174.3 + 50698.6i 0.827123 + 1.48838i
\(66\) 0 0
\(67\) 47902.5i 1.30368i 0.758357 + 0.651840i \(0.226002\pi\)
−0.758357 + 0.651840i \(0.773998\pi\)
\(68\) 0 0
\(69\) −73725.0 −1.86420
\(70\) 0 0
\(71\) 74761.8 1.76009 0.880043 0.474893i \(-0.157513\pi\)
0.880043 + 0.474893i \(0.157513\pi\)
\(72\) 0 0
\(73\) 22746.5i 0.499582i −0.968300 0.249791i \(-0.919638\pi\)
0.968300 0.249791i \(-0.0803619\pi\)
\(74\) 0 0
\(75\) 69718.0 + 43355.5i 1.43117 + 0.890001i
\(76\) 0 0
\(77\) 16847.9i 0.323832i
\(78\) 0 0
\(79\) −64213.8 −1.15761 −0.578803 0.815468i \(-0.696480\pi\)
−0.578803 + 0.815468i \(0.696480\pi\)
\(80\) 0 0
\(81\) 32273.2 0.546550
\(82\) 0 0
\(83\) 22523.0i 0.358865i 0.983770 + 0.179432i \(0.0574261\pi\)
−0.983770 + 0.179432i \(0.942574\pi\)
\(84\) 0 0
\(85\) −5904.29 10624.5i −0.0886381 0.159501i
\(86\) 0 0
\(87\) 128874.i 1.82544i
\(88\) 0 0
\(89\) 37886.4 0.507000 0.253500 0.967335i \(-0.418418\pi\)
0.253500 + 0.967335i \(0.418418\pi\)
\(90\) 0 0
\(91\) 24829.4 0.314313
\(92\) 0 0
\(93\) 10917.4i 0.130891i
\(94\) 0 0
\(95\) 63040.1 35032.8i 0.716651 0.398259i
\(96\) 0 0
\(97\) 84119.7i 0.907755i −0.891064 0.453877i \(-0.850040\pi\)
0.891064 0.453877i \(-0.149960\pi\)
\(98\) 0 0
\(99\) −314847. −3.22858
\(100\) 0 0
\(101\) −77492.2 −0.755883 −0.377941 0.925830i \(-0.623368\pi\)
−0.377941 + 0.925830i \(0.623368\pi\)
\(102\) 0 0
\(103\) 68641.0i 0.637516i 0.947836 + 0.318758i \(0.103266\pi\)
−0.947836 + 0.318758i \(0.896734\pi\)
\(104\) 0 0
\(105\) 30720.4 17072.0i 0.271928 0.151117i
\(106\) 0 0
\(107\) 84264.9i 0.711520i −0.934577 0.355760i \(-0.884222\pi\)
0.934577 0.355760i \(-0.115778\pi\)
\(108\) 0 0
\(109\) 51640.6 0.416318 0.208159 0.978095i \(-0.433253\pi\)
0.208159 + 0.978095i \(0.433253\pi\)
\(110\) 0 0
\(111\) −169472. −1.30554
\(112\) 0 0
\(113\) 88744.4i 0.653800i 0.945059 + 0.326900i \(0.106004\pi\)
−0.945059 + 0.326900i \(0.893996\pi\)
\(114\) 0 0
\(115\) 76202.1 + 137123.i 0.537307 + 0.966863i
\(116\) 0 0
\(117\) 464002.i 3.13368i
\(118\) 0 0
\(119\) −5203.31 −0.0336832
\(120\) 0 0
\(121\) 334610. 2.07767
\(122\) 0 0
\(123\) 430923.i 2.56825i
\(124\) 0 0
\(125\) 8577.22 174482.i 0.0490989 0.998794i
\(126\) 0 0
\(127\) 178878.i 0.984121i 0.870561 + 0.492061i \(0.163756\pi\)
−0.870561 + 0.492061i \(0.836244\pi\)
\(128\) 0 0
\(129\) 361823. 1.91435
\(130\) 0 0
\(131\) 235194. 1.19742 0.598711 0.800965i \(-0.295680\pi\)
0.598711 + 0.800965i \(0.295680\pi\)
\(132\) 0 0
\(133\) 30873.6i 0.151342i
\(134\) 0 0
\(135\) −145680. 262145.i −0.687963 1.23796i
\(136\) 0 0
\(137\) 216936.i 0.987486i −0.869608 0.493743i \(-0.835628\pi\)
0.869608 0.493743i \(-0.164372\pi\)
\(138\) 0 0
\(139\) 347938. 1.52744 0.763721 0.645546i \(-0.223370\pi\)
0.763721 + 0.645546i \(0.223370\pi\)
\(140\) 0 0
\(141\) −340019. −1.44031
\(142\) 0 0
\(143\) 730473.i 2.98720i
\(144\) 0 0
\(145\) −239696. + 133204.i −0.946760 + 0.526136i
\(146\) 0 0
\(147\) 426505.i 1.62791i
\(148\) 0 0
\(149\) −136495. −0.503677 −0.251839 0.967769i \(-0.581035\pi\)
−0.251839 + 0.967769i \(0.581035\pi\)
\(150\) 0 0
\(151\) 318940. 1.13833 0.569163 0.822225i \(-0.307267\pi\)
0.569163 + 0.822225i \(0.307267\pi\)
\(152\) 0 0
\(153\) 97237.5i 0.335819i
\(154\) 0 0
\(155\) 20305.4 11284.2i 0.0678864 0.0377260i
\(156\) 0 0
\(157\) 54247.3i 0.175642i 0.996136 + 0.0878211i \(0.0279904\pi\)
−0.996136 + 0.0878211i \(0.972010\pi\)
\(158\) 0 0
\(159\) 137644. 0.431783
\(160\) 0 0
\(161\) 67155.2 0.204181
\(162\) 0 0
\(163\) 504961.i 1.48864i −0.667825 0.744318i \(-0.732775\pi\)
0.667825 0.744318i \(-0.267225\pi\)
\(164\) 0 0
\(165\) 502254. + 903786.i 1.43620 + 2.58438i
\(166\) 0 0
\(167\) 264305.i 0.733355i −0.930348 0.366677i \(-0.880495\pi\)
0.930348 0.366677i \(-0.119505\pi\)
\(168\) 0 0
\(169\) −705231. −1.89939
\(170\) 0 0
\(171\) −576953. −1.50887
\(172\) 0 0
\(173\) 449650.i 1.14225i −0.820865 0.571123i \(-0.806508\pi\)
0.820865 0.571123i \(-0.193492\pi\)
\(174\) 0 0
\(175\) −63505.3 39491.9i −0.156753 0.0974795i
\(176\) 0 0
\(177\) 631636.i 1.51542i
\(178\) 0 0
\(179\) 657282. 1.53327 0.766636 0.642082i \(-0.221929\pi\)
0.766636 + 0.642082i \(0.221929\pi\)
\(180\) 0 0
\(181\) 640421. 1.45301 0.726506 0.687160i \(-0.241143\pi\)
0.726506 + 0.687160i \(0.241143\pi\)
\(182\) 0 0
\(183\) 27250.2i 0.0601510i
\(184\) 0 0
\(185\) 175166. + 315204.i 0.376288 + 0.677115i
\(186\) 0 0
\(187\) 153080.i 0.320121i
\(188\) 0 0
\(189\) −128384. −0.261431
\(190\) 0 0
\(191\) −689612. −1.36780 −0.683898 0.729578i \(-0.739717\pi\)
−0.683898 + 0.729578i \(0.739717\pi\)
\(192\) 0 0
\(193\) 398342.i 0.769773i 0.922964 + 0.384887i \(0.125759\pi\)
−0.922964 + 0.384887i \(0.874241\pi\)
\(194\) 0 0
\(195\) −1.33194e6 + 740190.i −2.50841 + 1.39398i
\(196\) 0 0
\(197\) 19330.2i 0.0354870i 0.999843 + 0.0177435i \(0.00564823\pi\)
−0.999843 + 0.0177435i \(0.994352\pi\)
\(198\) 0 0
\(199\) 314706. 0.563342 0.281671 0.959511i \(-0.409111\pi\)
0.281671 + 0.959511i \(0.409111\pi\)
\(200\) 0 0
\(201\) −1.25848e6 −2.19714
\(202\) 0 0
\(203\) 117390.i 0.199936i
\(204\) 0 0
\(205\) −801482. + 445402.i −1.33202 + 0.740231i
\(206\) 0 0
\(207\) 1.25497e6i 2.03567i
\(208\) 0 0
\(209\) 908292. 1.43833
\(210\) 0 0
\(211\) −478488. −0.739885 −0.369943 0.929055i \(-0.620623\pi\)
−0.369943 + 0.929055i \(0.620623\pi\)
\(212\) 0 0
\(213\) 1.96413e6i 2.96634i
\(214\) 0 0
\(215\) −373980. 672962.i −0.551762 0.992874i
\(216\) 0 0
\(217\) 9944.49i 0.0143362i
\(218\) 0 0
\(219\) 597590. 0.841963
\(220\) 0 0
\(221\) 225599. 0.310711
\(222\) 0 0
\(223\) 205163.i 0.276273i 0.990413 + 0.138136i \(0.0441112\pi\)
−0.990413 + 0.138136i \(0.955889\pi\)
\(224\) 0 0
\(225\) −738010. + 1.18676e6i −0.971865 + 1.56281i
\(226\) 0 0
\(227\) 409093.i 0.526936i 0.964668 + 0.263468i \(0.0848663\pi\)
−0.964668 + 0.263468i \(0.915134\pi\)
\(228\) 0 0
\(229\) −405619. −0.511128 −0.255564 0.966792i \(-0.582261\pi\)
−0.255564 + 0.966792i \(0.582261\pi\)
\(230\) 0 0
\(231\) 442625. 0.545766
\(232\) 0 0
\(233\) 64192.9i 0.0774635i 0.999250 + 0.0387318i \(0.0123318\pi\)
−0.999250 + 0.0387318i \(0.987668\pi\)
\(234\) 0 0
\(235\) 351443. + 632408.i 0.415131 + 0.747012i
\(236\) 0 0
\(237\) 1.68701e6i 1.95095i
\(238\) 0 0
\(239\) 1.36063e6 1.54080 0.770401 0.637560i \(-0.220056\pi\)
0.770401 + 0.637560i \(0.220056\pi\)
\(240\) 0 0
\(241\) −1.34482e6 −1.49149 −0.745745 0.666231i \(-0.767906\pi\)
−0.745745 + 0.666231i \(0.767906\pi\)
\(242\) 0 0
\(243\) 455786.i 0.495159i
\(244\) 0 0
\(245\) 793265. 440835.i 0.844312 0.469203i
\(246\) 0 0
\(247\) 1.33858e6i 1.39605i
\(248\) 0 0
\(249\) −591719. −0.604808
\(250\) 0 0
\(251\) 171857. 0.172180 0.0860898 0.996287i \(-0.472563\pi\)
0.0860898 + 0.996287i \(0.472563\pi\)
\(252\) 0 0
\(253\) 1.97569e6i 1.94051i
\(254\) 0 0
\(255\) 279125. 155116.i 0.268812 0.149385i
\(256\) 0 0
\(257\) 1.78486e6i 1.68567i −0.538173 0.842834i \(-0.680885\pi\)
0.538173 0.842834i \(-0.319115\pi\)
\(258\) 0 0
\(259\) 154370. 0.142992
\(260\) 0 0
\(261\) 2.19373e6 1.99335
\(262\) 0 0
\(263\) 1.23207e6i 1.09836i 0.835703 + 0.549182i \(0.185060\pi\)
−0.835703 + 0.549182i \(0.814940\pi\)
\(264\) 0 0
\(265\) −142269. 256007.i −0.124450 0.223943i
\(266\) 0 0
\(267\) 995343.i 0.854466i
\(268\) 0 0
\(269\) 309231. 0.260557 0.130278 0.991477i \(-0.458413\pi\)
0.130278 + 0.991477i \(0.458413\pi\)
\(270\) 0 0
\(271\) −143525. −0.118714 −0.0593572 0.998237i \(-0.518905\pi\)
−0.0593572 + 0.998237i \(0.518905\pi\)
\(272\) 0 0
\(273\) 652312.i 0.529723i
\(274\) 0 0
\(275\) 1.16184e6 1.86831e6i 0.926435 1.48976i
\(276\) 0 0
\(277\) 635705.i 0.497802i −0.968529 0.248901i \(-0.919931\pi\)
0.968529 0.248901i \(-0.0800693\pi\)
\(278\) 0 0
\(279\) −185839. −0.142931
\(280\) 0 0
\(281\) −240440. −0.181652 −0.0908262 0.995867i \(-0.528951\pi\)
−0.0908262 + 0.995867i \(0.528951\pi\)
\(282\) 0 0
\(283\) 442548.i 0.328469i −0.986421 0.164234i \(-0.947485\pi\)
0.986421 0.164234i \(-0.0525153\pi\)
\(284\) 0 0
\(285\) 920373. + 1.65618e6i 0.671200 + 1.20780i
\(286\) 0 0
\(287\) 392522.i 0.281293i
\(288\) 0 0
\(289\) 1.37258e6 0.966703
\(290\) 0 0
\(291\) 2.20997e6 1.52987
\(292\) 0 0
\(293\) 2.02835e6i 1.38030i −0.723667 0.690149i \(-0.757545\pi\)
0.723667 0.690149i \(-0.242455\pi\)
\(294\) 0 0
\(295\) −1.17479e6 + 652859.i −0.785971 + 0.436782i
\(296\) 0 0
\(297\) 3.77703e6i 2.48462i
\(298\) 0 0
\(299\) −2.91164e6 −1.88347
\(300\) 0 0
\(301\) −329580. −0.209674
\(302\) 0 0
\(303\) 2.03586e6i 1.27392i
\(304\) 0 0
\(305\) −50683.3 + 28165.8i −0.0311972 + 0.0173370i
\(306\) 0 0
\(307\) 2.31899e6i 1.40427i −0.712042 0.702137i \(-0.752229\pi\)
0.712042 0.702137i \(-0.247771\pi\)
\(308\) 0 0
\(309\) −1.80332e6 −1.07443
\(310\) 0 0
\(311\) −2.17253e6 −1.27369 −0.636846 0.770991i \(-0.719761\pi\)
−0.636846 + 0.770991i \(0.719761\pi\)
\(312\) 0 0
\(313\) 1.93017e6i 1.11362i −0.830641 0.556808i \(-0.812026\pi\)
0.830641 0.556808i \(-0.187974\pi\)
\(314\) 0 0
\(315\) 290605. + 522933.i 0.165016 + 0.296940i
\(316\) 0 0
\(317\) 2.16542e6i 1.21030i −0.796111 0.605151i \(-0.793113\pi\)
0.796111 0.605151i \(-0.206887\pi\)
\(318\) 0 0
\(319\) −3.45357e6 −1.90017
\(320\) 0 0
\(321\) 2.21379e6 1.19915
\(322\) 0 0
\(323\) 280517.i 0.149607i
\(324\) 0 0
\(325\) 2.75339e6 + 1.71225e6i 1.44597 + 0.899203i
\(326\) 0 0
\(327\) 1.35669e6i 0.701635i
\(328\) 0 0
\(329\) 309719. 0.157753
\(330\) 0 0
\(331\) −1.27143e6 −0.637857 −0.318929 0.947779i \(-0.603323\pi\)
−0.318929 + 0.947779i \(0.603323\pi\)
\(332\) 0 0
\(333\) 2.88480e6i 1.42562i
\(334\) 0 0
\(335\) 1.30077e6 + 2.34068e6i 0.633268 + 1.13954i
\(336\) 0 0
\(337\) 2.23785e6i 1.07338i 0.843778 + 0.536692i \(0.180326\pi\)
−0.843778 + 0.536692i \(0.819674\pi\)
\(338\) 0 0
\(339\) −2.33147e6 −1.10187
\(340\) 0 0
\(341\) 292564. 0.136250
\(342\) 0 0
\(343\) 790700.i 0.362891i
\(344\) 0 0
\(345\) −3.60246e6 + 2.00197e6i −1.62949 + 0.905542i
\(346\) 0 0
\(347\) 4.29048e6i 1.91285i −0.291971 0.956427i \(-0.594311\pi\)
0.291971 0.956427i \(-0.405689\pi\)
\(348\) 0 0
\(349\) −907418. −0.398790 −0.199395 0.979919i \(-0.563898\pi\)
−0.199395 + 0.979919i \(0.563898\pi\)
\(350\) 0 0
\(351\) 5.56634e6 2.41158
\(352\) 0 0
\(353\) 383963.i 0.164003i −0.996632 0.0820017i \(-0.973869\pi\)
0.996632 0.0820017i \(-0.0261313\pi\)
\(354\) 0 0
\(355\) 3.65312e6 2.03012e6i 1.53848 0.854970i
\(356\) 0 0
\(357\) 136700.i 0.0567674i
\(358\) 0 0
\(359\) −2.38837e6 −0.978060 −0.489030 0.872267i \(-0.662649\pi\)
−0.489030 + 0.872267i \(0.662649\pi\)
\(360\) 0 0
\(361\) −811667. −0.327801
\(362\) 0 0
\(363\) 8.79080e6i 3.50156i
\(364\) 0 0
\(365\) −617668. 1.11147e6i −0.242674 0.436682i
\(366\) 0 0
\(367\) 2.80989e6i 1.08899i −0.838764 0.544495i \(-0.816721\pi\)
0.838764 0.544495i \(-0.183279\pi\)
\(368\) 0 0
\(369\) 7.33530e6 2.80448
\(370\) 0 0
\(371\) −125378. −0.0472920
\(372\) 0 0
\(373\) 602108.i 0.224080i 0.993704 + 0.112040i \(0.0357384\pi\)
−0.993704 + 0.112040i \(0.964262\pi\)
\(374\) 0 0
\(375\) 4.58395e6 + 225339.i 1.68330 + 0.0827481i
\(376\) 0 0
\(377\) 5.08965e6i 1.84431i
\(378\) 0 0
\(379\) 153274. 0.0548115 0.0274057 0.999624i \(-0.491275\pi\)
0.0274057 + 0.999624i \(0.491275\pi\)
\(380\) 0 0
\(381\) −4.69945e6 −1.65857
\(382\) 0 0
\(383\) 4.29677e6i 1.49673i −0.663285 0.748367i \(-0.730838\pi\)
0.663285 0.748367i \(-0.269162\pi\)
\(384\) 0 0
\(385\) −457497. 823248.i −0.157303 0.283060i
\(386\) 0 0
\(387\) 6.15906e6i 2.09044i
\(388\) 0 0
\(389\) 1.29468e6 0.433800 0.216900 0.976194i \(-0.430405\pi\)
0.216900 + 0.976194i \(0.430405\pi\)
\(390\) 0 0
\(391\) 610171. 0.201841
\(392\) 0 0
\(393\) 6.17895e6i 2.01806i
\(394\) 0 0
\(395\) −3.13771e6 + 1.74369e6i −1.01186 + 0.562312i
\(396\) 0 0
\(397\) 2.41262e6i 0.768266i −0.923278 0.384133i \(-0.874500\pi\)
0.923278 0.384133i \(-0.125500\pi\)
\(398\) 0 0
\(399\) 811104. 0.255061
\(400\) 0 0
\(401\) 5.88260e6 1.82687 0.913437 0.406980i \(-0.133418\pi\)
0.913437 + 0.406980i \(0.133418\pi\)
\(402\) 0 0
\(403\) 431162.i 0.132245i
\(404\) 0 0
\(405\) 1.57698e6 876363.i 0.477737 0.265489i
\(406\) 0 0
\(407\) 4.54151e6i 1.35898i
\(408\) 0 0
\(409\) −166944. −0.0493473 −0.0246737 0.999696i \(-0.507855\pi\)
−0.0246737 + 0.999696i \(0.507855\pi\)
\(410\) 0 0
\(411\) 5.69930e6 1.66424
\(412\) 0 0
\(413\) 575350.i 0.165980i
\(414\) 0 0
\(415\) 611601. + 1.10055e6i 0.174320 + 0.313682i
\(416\) 0 0
\(417\) 9.14095e6i 2.57425i
\(418\) 0 0
\(419\) 2.19513e6 0.610838 0.305419 0.952218i \(-0.401204\pi\)
0.305419 + 0.952218i \(0.401204\pi\)
\(420\) 0 0
\(421\) −1.54531e6 −0.424924 −0.212462 0.977169i \(-0.568148\pi\)
−0.212462 + 0.977169i \(0.568148\pi\)
\(422\) 0 0
\(423\) 5.78790e6i 1.57279i
\(424\) 0 0
\(425\) −577007. 358823.i −0.154956 0.0963625i
\(426\) 0 0
\(427\) 24821.9i 0.00658818i
\(428\) 0 0
\(429\) −1.91908e7 −5.03443
\(430\) 0 0
\(431\) −3.05590e6 −0.792402 −0.396201 0.918164i \(-0.629672\pi\)
−0.396201 + 0.918164i \(0.629672\pi\)
\(432\) 0 0
\(433\) 5.34883e6i 1.37100i 0.728071 + 0.685502i \(0.240417\pi\)
−0.728071 + 0.685502i \(0.759583\pi\)
\(434\) 0 0
\(435\) −3.49951e6 6.29723e6i −0.886715 1.59561i
\(436\) 0 0
\(437\) 3.62042e6i 0.906891i
\(438\) 0 0
\(439\) −1.65736e6 −0.410446 −0.205223 0.978715i \(-0.565792\pi\)
−0.205223 + 0.978715i \(0.565792\pi\)
\(440\) 0 0
\(441\) −7.26009e6 −1.77765
\(442\) 0 0
\(443\) 1.02478e6i 0.248098i −0.992276 0.124049i \(-0.960412\pi\)
0.992276 0.124049i \(-0.0395880\pi\)
\(444\) 0 0
\(445\) 1.85126e6 1.02879e6i 0.443167 0.246278i
\(446\) 0 0
\(447\) 3.58598e6i 0.848865i
\(448\) 0 0
\(449\) 1.87947e6 0.439966 0.219983 0.975504i \(-0.429400\pi\)
0.219983 + 0.975504i \(0.429400\pi\)
\(450\) 0 0
\(451\) −1.15479e7 −2.67338
\(452\) 0 0
\(453\) 8.37912e6i 1.91846i
\(454\) 0 0
\(455\) 1.21325e6 674230.i 0.274740 0.152679i
\(456\) 0 0
\(457\) 4.51688e6i 1.01169i 0.862624 + 0.505846i \(0.168820\pi\)
−0.862624 + 0.505846i \(0.831180\pi\)
\(458\) 0 0
\(459\) −1.16650e6 −0.258436
\(460\) 0 0
\(461\) 6.74712e6 1.47865 0.739326 0.673348i \(-0.235144\pi\)
0.739326 + 0.673348i \(0.235144\pi\)
\(462\) 0 0
\(463\) 3.43370e6i 0.744407i 0.928151 + 0.372203i \(0.121398\pi\)
−0.928151 + 0.372203i \(0.878602\pi\)
\(464\) 0 0
\(465\) 296456. + 533460.i 0.0635810 + 0.114411i
\(466\) 0 0
\(467\) 791564.i 0.167955i 0.996468 + 0.0839777i \(0.0267625\pi\)
−0.996468 + 0.0839777i \(0.973238\pi\)
\(468\) 0 0
\(469\) 1.14634e6 0.240647
\(470\) 0 0
\(471\) −1.42517e6 −0.296016
\(472\) 0 0
\(473\) 9.69614e6i 1.99272i
\(474\) 0 0
\(475\) 2.12906e6 3.42364e6i 0.432965 0.696233i
\(476\) 0 0
\(477\) 2.34302e6i 0.471499i
\(478\) 0 0
\(479\) 799322. 0.159178 0.0795890 0.996828i \(-0.474639\pi\)
0.0795890 + 0.996828i \(0.474639\pi\)
\(480\) 0 0
\(481\) −6.69298e6 −1.31904
\(482\) 0 0
\(483\) 1.76429e6i 0.344113i
\(484\) 0 0
\(485\) −2.28423e6 4.11038e6i −0.440946 0.793464i
\(486\) 0 0
\(487\) 6.68591e6i 1.27743i 0.769442 + 0.638717i \(0.220534\pi\)
−0.769442 + 0.638717i \(0.779466\pi\)
\(488\) 0 0
\(489\) 1.32662e7 2.50885
\(490\) 0 0
\(491\) 235143. 0.0440179 0.0220089 0.999758i \(-0.492994\pi\)
0.0220089 + 0.999758i \(0.492994\pi\)
\(492\) 0 0
\(493\) 1.06660e6i 0.197644i
\(494\) 0 0
\(495\) −1.53845e7 + 8.54952e6i −2.82209 + 1.56830i
\(496\) 0 0
\(497\) 1.78910e6i 0.324895i
\(498\) 0 0
\(499\) 7.20861e6 1.29599 0.647993 0.761646i \(-0.275609\pi\)
0.647993 + 0.761646i \(0.275609\pi\)
\(500\) 0 0
\(501\) 6.94376e6 1.23595
\(502\) 0 0
\(503\) 7.24297e6i 1.27643i −0.769858 0.638215i \(-0.779673\pi\)
0.769858 0.638215i \(-0.220327\pi\)
\(504\) 0 0
\(505\) −3.78653e6 + 2.10426e6i −0.660714 + 0.367173i
\(506\) 0 0
\(507\) 1.85277e7i 3.20111i
\(508\) 0 0
\(509\) −5.34344e6 −0.914169 −0.457085 0.889423i \(-0.651106\pi\)
−0.457085 + 0.889423i \(0.651106\pi\)
\(510\) 0 0
\(511\) −544337. −0.0922180
\(512\) 0 0
\(513\) 6.92135e6i 1.16118i
\(514\) 0 0
\(515\) 1.86391e6 + 3.35404e6i 0.309676 + 0.557249i
\(516\) 0 0
\(517\) 9.11184e6i 1.49927i
\(518\) 0 0
\(519\) 1.18131e7 1.92507
\(520\) 0 0
\(521\) −3.91899e6 −0.632529 −0.316264 0.948671i \(-0.602429\pi\)
−0.316264 + 0.948671i \(0.602429\pi\)
\(522\) 0 0
\(523\) 3.74591e6i 0.598829i 0.954123 + 0.299414i \(0.0967913\pi\)
−0.954123 + 0.299414i \(0.903209\pi\)
\(524\) 0 0
\(525\) 1.03752e6 1.66840e6i 0.164286 0.264181i
\(526\) 0 0
\(527\) 90355.5i 0.0141719i
\(528\) 0 0
\(529\) −1.43866e6 −0.223522
\(530\) 0 0
\(531\) 1.07519e7 1.65481
\(532\) 0 0
\(533\) 1.70185e7i 2.59480i
\(534\) 0 0
\(535\) −2.28817e6 4.11747e6i −0.345624 0.621937i
\(536\) 0 0
\(537\) 1.72680e7i 2.58408i
\(538\) 0 0
\(539\) 1.14295e7 1.69455
\(540\) 0 0
\(541\) −1.32207e6 −0.194206 −0.0971031 0.995274i \(-0.530958\pi\)
−0.0971031 + 0.995274i \(0.530958\pi\)
\(542\) 0 0
\(543\) 1.68250e7i 2.44881i
\(544\) 0 0
\(545\) 2.52334e6 1.40227e6i 0.363901 0.202228i
\(546\) 0 0
\(547\) 8.82591e6i 1.26122i 0.776100 + 0.630610i \(0.217195\pi\)
−0.776100 + 0.630610i \(0.782805\pi\)
\(548\) 0 0
\(549\) 463862. 0.0656837
\(550\) 0 0
\(551\) −6.32862e6 −0.888035
\(552\) 0 0
\(553\) 1.53668e6i 0.213683i
\(554\) 0 0
\(555\) −8.28097e6 + 4.60192e6i −1.14117 + 0.634171i
\(556\) 0 0
\(557\) 446795.i 0.0610197i −0.999534 0.0305099i \(-0.990287\pi\)
0.999534 0.0305099i \(-0.00971310\pi\)
\(558\) 0 0
\(559\) 1.42895e7 1.93414
\(560\) 0 0
\(561\) 4.02168e6 0.539511
\(562\) 0 0
\(563\) 1.61652e6i 0.214937i −0.994208 0.107468i \(-0.965726\pi\)
0.994208 0.107468i \(-0.0342745\pi\)
\(564\) 0 0
\(565\) 2.40981e6 + 4.33636e6i 0.317586 + 0.571484i
\(566\) 0 0
\(567\) 772318.i 0.100888i
\(568\) 0 0
\(569\) 1.25416e7 1.62395 0.811974 0.583693i \(-0.198393\pi\)
0.811974 + 0.583693i \(0.198393\pi\)
\(570\) 0 0
\(571\) −1.19556e6 −0.153455 −0.0767276 0.997052i \(-0.524447\pi\)
−0.0767276 + 0.997052i \(0.524447\pi\)
\(572\) 0 0
\(573\) 1.81173e7i 2.30519i
\(574\) 0 0
\(575\) 7.44699e6 + 4.63105e6i 0.939315 + 0.584131i
\(576\) 0 0
\(577\) 1.15431e7i 1.44339i −0.692211 0.721695i \(-0.743363\pi\)
0.692211 0.721695i \(-0.256637\pi\)
\(578\) 0 0
\(579\) −1.04652e7 −1.29733
\(580\) 0 0
\(581\) 538990. 0.0662430
\(582\) 0 0
\(583\) 3.68860e6i 0.449459i
\(584\) 0 0
\(585\) −1.25997e7 2.26727e7i −1.52220 2.73914i
\(586\) 0 0
\(587\) 4.96527e6i 0.594768i 0.954758 + 0.297384i \(0.0961142\pi\)
−0.954758 + 0.297384i \(0.903886\pi\)
\(588\) 0 0
\(589\) 536119. 0.0636756
\(590\) 0 0
\(591\) −507837. −0.0598076
\(592\) 0 0
\(593\) 3.44208e6i 0.401962i 0.979595 + 0.200981i \(0.0644129\pi\)
−0.979595 + 0.200981i \(0.935587\pi\)
\(594\) 0 0
\(595\) −254252. + 141293.i −0.0294423 + 0.0163617i
\(596\) 0 0
\(597\) 8.26787e6i 0.949419i
\(598\) 0 0
\(599\) 7.32967e6 0.834675 0.417338 0.908752i \(-0.362963\pi\)
0.417338 + 0.908752i \(0.362963\pi\)
\(600\) 0 0
\(601\) −5.96589e6 −0.673735 −0.336868 0.941552i \(-0.609368\pi\)
−0.336868 + 0.941552i \(0.609368\pi\)
\(602\) 0 0
\(603\) 2.14223e7i 2.39923i
\(604\) 0 0
\(605\) 1.63502e7 9.08617e6i 1.81608 1.00924i
\(606\) 0 0
\(607\) 1.35750e7i 1.49543i −0.664018 0.747716i \(-0.731150\pi\)
0.664018 0.747716i \(-0.268850\pi\)
\(608\) 0 0
\(609\) −3.08404e6 −0.336959
\(610\) 0 0
\(611\) −1.34284e7 −1.45520
\(612\) 0 0
\(613\) 1.11070e7i 1.19384i 0.802302 + 0.596918i \(0.203608\pi\)
−0.802302 + 0.596918i \(0.796392\pi\)
\(614\) 0 0
\(615\) −1.17015e7 2.10564e7i −1.24754 2.24489i
\(616\) 0 0
\(617\) 5.62274e6i 0.594614i −0.954782 0.297307i \(-0.903912\pi\)
0.954782 0.297307i \(-0.0960884\pi\)
\(618\) 0 0
\(619\) −2.37545e6 −0.249184 −0.124592 0.992208i \(-0.539762\pi\)
−0.124592 + 0.992208i \(0.539762\pi\)
\(620\) 0 0
\(621\) 1.50551e7 1.56659
\(622\) 0 0
\(623\) 906646.i 0.0935874i
\(624\) 0 0
\(625\) −4.31886e6 8.75870e6i −0.442251 0.896891i
\(626\) 0 0
\(627\) 2.38624e7i 2.42407i
\(628\) 0 0
\(629\) 1.40260e6 0.141354
\(630\) 0 0
\(631\) 3.70995e6 0.370932 0.185466 0.982651i \(-0.440621\pi\)
0.185466 + 0.982651i \(0.440621\pi\)
\(632\) 0 0
\(633\) 1.25707e7i 1.24695i
\(634\) 0 0
\(635\) 4.85735e6 + 8.74061e6i 0.478041 + 0.860216i
\(636\) 0 0
\(637\) 1.68440e7i 1.64474i
\(638\) 0 0
\(639\) −3.34340e7 −3.23918
\(640\) 0 0
\(641\) 7.35252e6 0.706791 0.353396 0.935474i \(-0.385027\pi\)
0.353396 + 0.935474i \(0.385027\pi\)
\(642\) 0 0
\(643\) 3.03451e6i 0.289442i 0.989473 + 0.144721i \(0.0462284\pi\)
−0.989473 + 0.144721i \(0.953772\pi\)
\(644\) 0 0
\(645\) 1.76799e7 9.82511e6i 1.67333 0.929904i
\(646\) 0 0
\(647\) 4.69808e6i 0.441224i 0.975362 + 0.220612i \(0.0708055\pi\)
−0.975362 + 0.220612i \(0.929194\pi\)
\(648\) 0 0
\(649\) −1.69266e7 −1.57746
\(650\) 0 0
\(651\) 261259. 0.0241613
\(652\) 0 0
\(653\) 1.29779e7i 1.19103i 0.803346 + 0.595513i \(0.203051\pi\)
−0.803346 + 0.595513i \(0.796949\pi\)
\(654\) 0 0
\(655\) 1.14924e7 6.38656e6i 1.04666 0.581653i
\(656\) 0 0
\(657\) 1.01724e7i 0.919407i
\(658\) 0 0
\(659\) −1.46461e6 −0.131374 −0.0656868 0.997840i \(-0.520924\pi\)
−0.0656868 + 0.997840i \(0.520924\pi\)
\(660\) 0 0
\(661\) 1.06947e7 0.952061 0.476031 0.879429i \(-0.342075\pi\)
0.476031 + 0.879429i \(0.342075\pi\)
\(662\) 0 0
\(663\) 5.92689e6i 0.523653i
\(664\) 0 0
\(665\) −838357. 1.50859e6i −0.0735148 0.132287i
\(666\) 0 0
\(667\) 1.37658e7i 1.19808i
\(668\) 0 0
\(669\) −5.39001e6 −0.465612
\(670\) 0 0
\(671\) −730253. −0.0626134
\(672\) 0 0
\(673\) 8.91777e6i 0.758959i −0.925200 0.379480i \(-0.876103\pi\)
0.925200 0.379480i \(-0.123897\pi\)
\(674\) 0 0
\(675\) −1.42368e7 8.85345e6i −1.20269 0.747916i
\(676\) 0 0
\(677\) 1.93032e7i 1.61867i −0.587349 0.809334i \(-0.699828\pi\)
0.587349 0.809334i \(-0.300172\pi\)
\(678\) 0 0
\(679\) −2.01304e6 −0.167563
\(680\) 0 0
\(681\) −1.07476e7 −0.888063
\(682\) 0 0
\(683\) 1.15277e7i 0.945565i 0.881179 + 0.472783i \(0.156750\pi\)
−0.881179 + 0.472783i \(0.843250\pi\)
\(684\) 0 0
\(685\) −5.89079e6 1.06002e7i −0.479675 0.863157i
\(686\) 0 0
\(687\) 1.06563e7i 0.861421i
\(688\) 0 0
\(689\) 5.43601e6 0.436247
\(690\) 0 0
\(691\) 3.95573e6 0.315160 0.157580 0.987506i \(-0.449631\pi\)
0.157580 + 0.987506i \(0.449631\pi\)
\(692\) 0 0
\(693\) 7.53450e6i 0.595966i
\(694\) 0 0
\(695\) 1.70014e7 9.44808e6i 1.33513 0.741962i
\(696\) 0 0
\(697\) 3.56645e6i 0.278070i
\(698\) 0 0
\(699\) −1.68646e6 −0.130552
\(700\) 0 0
\(701\) −2.05492e7 −1.57943 −0.789715 0.613474i \(-0.789771\pi\)
−0.789715 + 0.613474i \(0.789771\pi\)
\(702\) 0 0
\(703\) 8.32225e6i 0.635115i
\(704\) 0 0
\(705\) −1.66145e7 + 9.23304e6i −1.25897 + 0.699636i
\(706\) 0 0
\(707\) 1.85444e6i 0.139529i
\(708\) 0 0
\(709\) 1.07226e7 0.801100 0.400550 0.916275i \(-0.368819\pi\)
0.400550 + 0.916275i \(0.368819\pi\)
\(710\) 0 0
\(711\) 2.87168e7 2.13040
\(712\) 0 0
\(713\) 1.16615e6i 0.0859073i
\(714\) 0 0
\(715\) 1.98356e7 + 3.56934e7i 1.45105 + 2.61110i
\(716\) 0 0
\(717\) 3.57463e7i 2.59677i
\(718\) 0 0
\(719\) 8.12405e6 0.586071 0.293035 0.956102i \(-0.405335\pi\)
0.293035 + 0.956102i \(0.405335\pi\)
\(720\) 0 0
\(721\) 1.64262e6 0.117679
\(722\) 0 0
\(723\) 3.53307e7i 2.51366i
\(724\) 0 0
\(725\) −8.09525e6 + 1.30176e7i −0.571986 + 0.919786i
\(726\) 0 0
\(727\) 1.87419e7i 1.31516i 0.753386 + 0.657578i \(0.228419\pi\)
−0.753386 + 0.657578i \(0.771581\pi\)
\(728\) 0 0
\(729\) 1.98167e7 1.38106
\(730\) 0 0
\(731\) −2.99456e6 −0.207271
\(732\) 0 0
\(733\) 1.43325e6i 0.0985284i 0.998786 + 0.0492642i \(0.0156876\pi\)
−0.998786 + 0.0492642i \(0.984312\pi\)
\(734\) 0 0
\(735\) 1.15815e7 + 2.08405e7i 0.790764 + 1.42295i
\(736\) 0 0
\(737\) 3.37249e7i 2.28708i
\(738\) 0 0
\(739\) 3.87933e6 0.261304 0.130652 0.991428i \(-0.458293\pi\)
0.130652 + 0.991428i \(0.458293\pi\)
\(740\) 0 0
\(741\) −3.51669e7 −2.35282
\(742\) 0 0
\(743\) 2.51128e7i 1.66887i 0.551104 + 0.834437i \(0.314207\pi\)
−0.551104 + 0.834437i \(0.685793\pi\)
\(744\) 0 0
\(745\) −6.66963e6 + 3.70646e6i −0.440262 + 0.244663i
\(746\) 0 0
\(747\) 1.00724e7i 0.660439i
\(748\) 0 0
\(749\) −2.01651e6 −0.131340
\(750\) 0 0
\(751\) −1.68579e7 −1.09070 −0.545349 0.838209i \(-0.683603\pi\)
−0.545349 + 0.838209i \(0.683603\pi\)
\(752\) 0 0
\(753\) 4.51498e6i 0.290180i
\(754\) 0 0
\(755\) 1.55845e7 8.66065e6i 0.995005 0.552946i
\(756\) 0 0
\(757\) 2.64597e6i 0.167821i 0.996473 + 0.0839104i \(0.0267410\pi\)
−0.996473 + 0.0839104i \(0.973259\pi\)
\(758\) 0 0
\(759\) −5.19048e7 −3.27042
\(760\) 0 0
\(761\) 778479. 0.0487288 0.0243644 0.999703i \(-0.492244\pi\)
0.0243644 + 0.999703i \(0.492244\pi\)
\(762\) 0 0
\(763\) 1.23579e6i 0.0768483i
\(764\) 0 0
\(765\) 2.64043e6 + 4.75135e6i 0.163125 + 0.293538i
\(766\) 0 0
\(767\) 2.49454e7i 1.53109i
\(768\) 0 0
\(769\) −1.11827e7 −0.681919 −0.340959 0.940078i \(-0.610752\pi\)
−0.340959 + 0.940078i \(0.610752\pi\)
\(770\) 0 0
\(771\) 4.68915e7 2.84092
\(772\) 0 0
\(773\) 2.00821e7i 1.20882i −0.796675 0.604408i \(-0.793410\pi\)
0.796675 0.604408i \(-0.206590\pi\)
\(774\) 0 0
\(775\) 685777. 1.10277e6i 0.0410137 0.0659523i
\(776\) 0 0
\(777\) 4.05556e6i 0.240990i
\(778\) 0 0
\(779\) −2.11613e7 −1.24939
\(780\) 0 0
\(781\) 5.26347e7 3.08777
\(782\) 0 0
\(783\) 2.63169e7i 1.53402i
\(784\) 0 0
\(785\) 1.47306e6 + 2.65071e6i 0.0853190 + 0.153528i
\(786\) 0 0
\(787\) 2.62920e6i 0.151317i 0.997134 + 0.0756584i \(0.0241059\pi\)
−0.997134 + 0.0756584i \(0.975894\pi\)
\(788\) 0 0
\(789\) −3.23687e7 −1.85111
\(790\) 0 0
\(791\) 2.12371e6 0.120685
\(792\) 0 0
\(793\) 1.07620e6i 0.0607729i
\(794\) 0 0
\(795\) 6.72577e6 3.73766e6i 0.377419 0.209740i
\(796\) 0 0
\(797\) 6.44461e6i 0.359377i −0.983724 0.179689i \(-0.942491\pi\)
0.983724 0.179689i \(-0.0575090\pi\)
\(798\) 0 0
\(799\) 2.81410e6 0.155945
\(800\) 0 0
\(801\) −1.69430e7 −0.933061
\(802\) 0 0
\(803\) 1.60142e7i 0.876430i
\(804\) 0 0
\(805\) 3.28143e6 1.82357e6i 0.178474 0.0991817i
\(806\) 0 0
\(807\) 8.12405e6i 0.439126i
\(808\) 0 0
\(809\) −3.59388e6 −0.193060 −0.0965299 0.995330i \(-0.530774\pi\)
−0.0965299 + 0.995330i \(0.530774\pi\)
\(810\) 0 0
\(811\) −2.79497e7 −1.49220 −0.746098 0.665837i \(-0.768075\pi\)
−0.746098 + 0.665837i \(0.768075\pi\)
\(812\) 0 0
\(813\) 3.77065e6i 0.200074i
\(814\) 0 0
\(815\) −1.37120e7 2.46741e7i −0.723111 1.30121i
\(816\) 0 0
\(817\) 1.77680e7i 0.931289i
\(818\) 0 0
\(819\) −1.11039e7 −0.578448
\(820\) 0 0
\(821\) −8.41956e6 −0.435945 −0.217972 0.975955i \(-0.569944\pi\)
−0.217972 + 0.975955i \(0.569944\pi\)
\(822\) 0 0
\(823\) 1.51378e7i 0.779048i −0.921016 0.389524i \(-0.872640\pi\)
0.921016 0.389524i \(-0.127360\pi\)
\(824\) 0 0
\(825\) 4.90837e7 + 3.05236e7i 2.51074 + 1.56135i
\(826\) 0 0
\(827\) 5.14412e6i 0.261545i −0.991412 0.130773i \(-0.958254\pi\)
0.991412 0.130773i \(-0.0417458\pi\)
\(828\) 0 0
\(829\) 1.00711e7 0.508970 0.254485 0.967077i \(-0.418094\pi\)
0.254485 + 0.967077i \(0.418094\pi\)
\(830\) 0 0
\(831\) 1.67011e7 0.838963
\(832\) 0 0
\(833\) 3.52988e6i 0.176258i
\(834\) 0 0
\(835\) −7.17706e6 1.29148e7i −0.356230 0.641022i
\(836\) 0 0
\(837\) 2.22939e6i 0.109995i
\(838\) 0 0
\(839\) −933470. −0.0457821 −0.0228910 0.999738i \(-0.507287\pi\)
−0.0228910 + 0.999738i \(0.507287\pi\)
\(840\) 0 0
\(841\) 3.55199e6 0.173173
\(842\) 0 0
\(843\) 6.31679e6i 0.306145i
\(844\) 0 0
\(845\) −3.44600e7 + 1.91502e7i −1.66025 + 0.922638i
\(846\) 0 0
\(847\) 8.00743e6i 0.383517i
\(848\) 0 0
\(849\) 1.16265e7 0.553580
\(850\) 0 0
\(851\) −1.81023e7 −0.856859
\(852\) 0 0
\(853\) 1.96289e7i 0.923682i 0.886963 + 0.461841i \(0.152811\pi\)
−0.886963 + 0.461841i \(0.847189\pi\)
\(854\) 0 0
\(855\) −2.81919e7 + 1.56669e7i −1.31889 + 0.732938i
\(856\) 0 0
\(857\) 2.93615e7i 1.36561i 0.730601 + 0.682805i \(0.239240\pi\)
−0.730601 + 0.682805i \(0.760760\pi\)
\(858\) 0 0
\(859\) −2.22206e7 −1.02748 −0.513739 0.857947i \(-0.671740\pi\)
−0.513739 + 0.857947i \(0.671740\pi\)
\(860\) 0 0
\(861\) −1.03123e7 −0.474074
\(862\) 0 0
\(863\) 1.40411e7i 0.641763i −0.947119 0.320881i \(-0.896021\pi\)
0.947119 0.320881i \(-0.103979\pi\)
\(864\) 0 0
\(865\) −1.22100e7 2.19714e7i −0.554851 0.998432i
\(866\) 0 0
\(867\) 3.60601e7i 1.62922i
\(868\) 0 0
\(869\) −4.52086e7 −2.03082
\(870\) 0 0
\(871\) −4.97015e7 −2.21985
\(872\) 0 0
\(873\) 3.76189e7i 1.67059i
\(874\) 0 0
\(875\) −4.17547e6 205258.i −0.184368 0.00906318i
\(876\) 0 0
\(877\) 3.11419e7i 1.36724i −0.729836 0.683622i \(-0.760404\pi\)
0.729836 0.683622i \(-0.239596\pi\)
\(878\) 0 0
\(879\) 5.32882e7 2.32626
\(880\) 0 0
\(881\) −6.80425e6 −0.295352 −0.147676 0.989036i \(-0.547179\pi\)
−0.147676 + 0.989036i \(0.547179\pi\)
\(882\) 0 0
\(883\) 1.02459e7i 0.442230i −0.975248 0.221115i \(-0.929030\pi\)
0.975248 0.221115i \(-0.0709697\pi\)
\(884\) 0 0
\(885\) −1.71518e7 3.08639e7i −0.736123 1.32462i
\(886\) 0 0
\(887\) 3.03507e7i 1.29527i 0.761951 + 0.647635i \(0.224242\pi\)
−0.761951 + 0.647635i \(0.775758\pi\)
\(888\) 0 0
\(889\) 4.28067e6 0.181659
\(890\) 0 0
\(891\) 2.27214e7 0.958827
\(892\) 0 0
\(893\) 1.66973e7i 0.700677i
\(894\) 0 0
\(895\) 3.21171e7 1.78482e7i 1.34023 0.744794i
\(896\) 0 0
\(897\) 7.64939e7i 3.17428i
\(898\) 0 0
\(899\) −2.03847e6 −0.0841212
\(900\) 0 0
\(901\) −1.13919e6 −0.0467501
\(902\) 0 0
\(903\) 8.65865e6i 0.353371i
\(904\) 0 0
\(905\) 3.12932e7 1.73903e7i 1.27007 0.705807i
\(906\) 0 0
\(907\) 5.03000e6i 0.203025i 0.994834 + 0.101513i \(0.0323682\pi\)
−0.994834 + 0.101513i \(0.967632\pi\)
\(908\) 0 0
\(909\) 3.46550e7 1.39109
\(910\) 0 0
\(911\) −6.17959e6 −0.246697 −0.123348 0.992363i \(-0.539363\pi\)
−0.123348 + 0.992363i \(0.539363\pi\)
\(912\) 0 0
\(913\) 1.58569e7i 0.629567i
\(914\) 0 0
\(915\) −739966. 1.33154e6i −0.0292186 0.0525777i
\(916\) 0 0
\(917\) 5.62833e6i 0.221033i
\(918\) 0 0
\(919\) 4.27546e7 1.66992 0.834958 0.550314i \(-0.185492\pi\)
0.834958 + 0.550314i \(0.185492\pi\)
\(920\) 0 0
\(921\) 6.09239e7 2.36667
\(922\) 0 0
\(923\) 7.75697e7i 2.99701i
\(924\) 0 0
\(925\) 1.71184e7 + 1.06454e7i 0.657823 + 0.409080i
\(926\) 0 0
\(927\) 3.06967e7i 1.17325i
\(928\) 0 0
\(929\) 1.75757e7 0.668151 0.334075 0.942546i \(-0.391576\pi\)
0.334075 + 0.942546i \(0.391576\pi\)
\(930\) 0 0
\(931\) 2.09444e7 0.791941
\(932\) 0 0
\(933\) 5.70761e7i 2.14660i
\(934\) 0 0
\(935\) −4.15681e6 7.48001e6i −0.155500 0.279816i
\(936\) 0 0
\(937\) 4.25313e7i 1.58256i −0.611455 0.791279i \(-0.709415\pi\)
0.611455 0.791279i \(-0.290585\pi\)
\(938\) 0 0
\(939\) 5.07091e7 1.87682
\(940\) 0 0
\(941\) 1.23345e7 0.454095 0.227047 0.973884i \(-0.427093\pi\)
0.227047 + 0.973884i \(0.427093\pi\)
\(942\) 0 0
\(943\) 4.60294e7i 1.68561i
\(944\) 0 0
\(945\) −6.27330e6 + 3.48621e6i −0.228516 + 0.126991i
\(946\) 0 0
\(947\) 2.28532e7i 0.828081i −0.910259 0.414040i \(-0.864117\pi\)
0.910259 0.414040i \(-0.135883\pi\)
\(948\) 0 0
\(949\) 2.36007e7 0.850668
\(950\) 0 0
\(951\) 5.68894e7 2.03976
\(952\) 0 0
\(953\) 3.08757e7i 1.10125i 0.834754 + 0.550623i \(0.185610\pi\)
−0.834754 + 0.550623i \(0.814390\pi\)
\(954\) 0 0
\(955\) −3.36968e7 + 1.87261e7i −1.19558 + 0.664413i
\(956\) 0 0
\(957\) 9.07315e7i 3.20242i
\(958\) 0 0
\(959\) −5.19142e6 −0.182280
\(960\) 0 0
\(961\) −2.84565e7 −0.993968
\(962\) 0 0
\(963\) 3.76838e7i 1.30945i
\(964\) 0 0
\(965\) 1.08168e7 + 1.94644e7i 0.373921 + 0.672855i
\(966\) 0 0
\(967\) 1.99276e7i 0.685313i 0.939461 + 0.342657i \(0.111327\pi\)
−0.939461 + 0.342657i \(0.888673\pi\)
\(968\) 0 0
\(969\) 7.36967e6 0.252138
\(970\) 0 0
\(971\) −3.53795e7 −1.20421 −0.602107 0.798415i \(-0.705672\pi\)
−0.602107 + 0.798415i \(0.705672\pi\)
\(972\) 0 0
\(973\) 8.32638e6i 0.281951i
\(974\) 0 0
\(975\) −4.49837e7 + 7.23364e7i −1.51546 + 2.43694i
\(976\) 0 0
\(977\) 3.28357e7i 1.10055i 0.834983 + 0.550275i \(0.185477\pi\)
−0.834983 + 0.550275i \(0.814523\pi\)
\(978\) 0 0
\(979\) 2.66732e7 0.889445
\(980\) 0 0
\(981\) −2.30940e7 −0.766172
\(982\) 0 0
\(983\) 2.83290e7i 0.935077i −0.883973 0.467539i \(-0.845141\pi\)
0.883973 0.467539i \(-0.154859\pi\)
\(984\) 0 0
\(985\) 524900. + 944537.i 0.0172380 + 0.0310191i
\(986\) 0 0
\(987\) 8.13687e6i 0.265867i
\(988\) 0 0
\(989\) 3.86484e7 1.25644
\(990\) 0 0
\(991\) −5.98087e6 −0.193455 −0.0967276 0.995311i \(-0.530838\pi\)
−0.0967276 + 0.995311i \(0.530838\pi\)
\(992\) 0 0
\(993\) 3.34028e7i 1.07500i
\(994\) 0 0
\(995\) 1.53776e7 8.54567e6i 0.492414 0.273646i
\(996\) 0 0
\(997\) 1.87665e6i 0.0597922i 0.999553 + 0.0298961i \(0.00951764\pi\)
−0.999553 + 0.0298961i \(0.990482\pi\)
\(998\) 0 0
\(999\) 3.46072e7 1.09711
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 160.6.c.c.129.11 yes 12
4.3 odd 2 inner 160.6.c.c.129.1 12
5.2 odd 4 800.6.a.ba.1.6 6
5.3 odd 4 800.6.a.z.1.1 6
5.4 even 2 inner 160.6.c.c.129.2 yes 12
8.3 odd 2 320.6.c.k.129.12 12
8.5 even 2 320.6.c.k.129.2 12
20.3 even 4 800.6.a.z.1.6 6
20.7 even 4 800.6.a.ba.1.1 6
20.19 odd 2 inner 160.6.c.c.129.12 yes 12
40.19 odd 2 320.6.c.k.129.1 12
40.29 even 2 320.6.c.k.129.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.c.c.129.1 12 4.3 odd 2 inner
160.6.c.c.129.2 yes 12 5.4 even 2 inner
160.6.c.c.129.11 yes 12 1.1 even 1 trivial
160.6.c.c.129.12 yes 12 20.19 odd 2 inner
320.6.c.k.129.1 12 40.19 odd 2
320.6.c.k.129.2 12 8.5 even 2
320.6.c.k.129.11 12 40.29 even 2
320.6.c.k.129.12 12 8.3 odd 2
800.6.a.z.1.1 6 5.3 odd 4
800.6.a.z.1.6 6 20.3 even 4
800.6.a.ba.1.1 6 20.7 even 4
800.6.a.ba.1.6 6 5.2 odd 4