Properties

Label 162.7.b.c.161.8
Level $162$
Weight $7$
Character 162.161
Analytic conductor $37.269$
Analytic rank $0$
Dimension $12$
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] = [162,7,Mod(161,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.2687615464\)
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: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{42} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.8
Root \(-7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.7.b.c.161.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.65685i q^{2} -32.0000 q^{4} -45.6802i q^{5} +490.195 q^{7} -181.019i q^{8} +O(q^{10})\) \(q+5.65685i q^{2} -32.0000 q^{4} -45.6802i q^{5} +490.195 q^{7} -181.019i q^{8} +258.406 q^{10} +1008.44i q^{11} -933.602 q^{13} +2772.96i q^{14} +1024.00 q^{16} -8090.59i q^{17} -7727.36 q^{19} +1461.77i q^{20} -5704.61 q^{22} -13680.9i q^{23} +13538.3 q^{25} -5281.25i q^{26} -15686.2 q^{28} +2268.65i q^{29} +34125.1 q^{31} +5792.62i q^{32} +45767.3 q^{34} -22392.2i q^{35} +92058.0 q^{37} -43712.6i q^{38} -8269.00 q^{40} -35820.6i q^{41} -69141.8 q^{43} -32270.1i q^{44} +77390.9 q^{46} -15254.9i q^{47} +122642. q^{49} +76584.3i q^{50} +29875.3 q^{52} +236591. i q^{53} +46065.9 q^{55} -88734.7i q^{56} -12833.4 q^{58} -256217. i q^{59} +39839.2 q^{61} +193041. i q^{62} -32768.0 q^{64} +42647.1i q^{65} +320409. q^{67} +258899. i q^{68} +126669. q^{70} -404593. i q^{71} +393719. q^{73} +520759. i q^{74} +247276. q^{76} +494333. i q^{77} +898368. q^{79} -46776.5i q^{80} +202632. q^{82} -178882. i q^{83} -369580. q^{85} -391125. i q^{86} +182548. q^{88} -826458. i q^{89} -457647. q^{91} +437789. i q^{92} +86294.8 q^{94} +352988. i q^{95} +635962. q^{97} +693768. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 384 q^{4} - 480 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49} + 107520 q^{52} + 580392 q^{55} - 318528 q^{58} - 271488 q^{61} - 393216 q^{64} + 579876 q^{67} - 311904 q^{70} - 977700 q^{73} + 90240 q^{76} + 1529592 q^{79} + 1073088 q^{82} - 3239136 q^{85} - 230400 q^{88} + 355584 q^{91} + 1473696 q^{94} + 77748 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.65685i 0.707107i
\(3\) 0 0
\(4\) −32.0000 −0.500000
\(5\) − 45.6802i − 0.365442i −0.983165 0.182721i \(-0.941510\pi\)
0.983165 0.182721i \(-0.0584905\pi\)
\(6\) 0 0
\(7\) 490.195 1.42914 0.714570 0.699564i \(-0.246623\pi\)
0.714570 + 0.699564i \(0.246623\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) 0 0
\(10\) 258.406 0.258406
\(11\) 1008.44i 0.757657i 0.925467 + 0.378829i \(0.123673\pi\)
−0.925467 + 0.378829i \(0.876327\pi\)
\(12\) 0 0
\(13\) −933.602 −0.424944 −0.212472 0.977167i \(-0.568151\pi\)
−0.212472 + 0.977167i \(0.568151\pi\)
\(14\) 2772.96i 1.01055i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) − 8090.59i − 1.64677i −0.567482 0.823386i \(-0.692082\pi\)
0.567482 0.823386i \(-0.307918\pi\)
\(18\) 0 0
\(19\) −7727.36 −1.12660 −0.563301 0.826252i \(-0.690469\pi\)
−0.563301 + 0.826252i \(0.690469\pi\)
\(20\) 1461.77i 0.182721i
\(21\) 0 0
\(22\) −5704.61 −0.535745
\(23\) − 13680.9i − 1.12443i −0.826992 0.562213i \(-0.809950\pi\)
0.826992 0.562213i \(-0.190050\pi\)
\(24\) 0 0
\(25\) 13538.3 0.866452
\(26\) − 5281.25i − 0.300481i
\(27\) 0 0
\(28\) −15686.2 −0.714570
\(29\) 2268.65i 0.0930192i 0.998918 + 0.0465096i \(0.0148098\pi\)
−0.998918 + 0.0465096i \(0.985190\pi\)
\(30\) 0 0
\(31\) 34125.1 1.14548 0.572742 0.819735i \(-0.305880\pi\)
0.572742 + 0.819735i \(0.305880\pi\)
\(32\) 5792.62i 0.176777i
\(33\) 0 0
\(34\) 45767.3 1.16444
\(35\) − 22392.2i − 0.522267i
\(36\) 0 0
\(37\) 92058.0 1.81743 0.908713 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) − 43712.6i − 0.796628i
\(39\) 0 0
\(40\) −8269.00 −0.129203
\(41\) − 35820.6i − 0.519734i −0.965644 0.259867i \(-0.916321\pi\)
0.965644 0.259867i \(-0.0836788\pi\)
\(42\) 0 0
\(43\) −69141.8 −0.869631 −0.434816 0.900520i \(-0.643186\pi\)
−0.434816 + 0.900520i \(0.643186\pi\)
\(44\) − 32270.1i − 0.378829i
\(45\) 0 0
\(46\) 77390.9 0.795090
\(47\) − 15254.9i − 0.146932i −0.997298 0.0734659i \(-0.976594\pi\)
0.997298 0.0734659i \(-0.0234060\pi\)
\(48\) 0 0
\(49\) 122642. 1.04244
\(50\) 76584.3i 0.612674i
\(51\) 0 0
\(52\) 29875.3 0.212472
\(53\) 236591.i 1.58917i 0.607153 + 0.794585i \(0.292312\pi\)
−0.607153 + 0.794585i \(0.707688\pi\)
\(54\) 0 0
\(55\) 46065.9 0.276880
\(56\) − 88734.7i − 0.505277i
\(57\) 0 0
\(58\) −12833.4 −0.0657745
\(59\) − 256217.i − 1.24753i −0.781611 0.623766i \(-0.785602\pi\)
0.781611 0.623766i \(-0.214398\pi\)
\(60\) 0 0
\(61\) 39839.2 0.175518 0.0877589 0.996142i \(-0.472029\pi\)
0.0877589 + 0.996142i \(0.472029\pi\)
\(62\) 193041.i 0.809980i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) 42647.1i 0.155292i
\(66\) 0 0
\(67\) 320409. 1.06532 0.532660 0.846330i \(-0.321193\pi\)
0.532660 + 0.846330i \(0.321193\pi\)
\(68\) 258899.i 0.823386i
\(69\) 0 0
\(70\) 126669. 0.369299
\(71\) − 404593.i − 1.13043i −0.824944 0.565215i \(-0.808793\pi\)
0.824944 0.565215i \(-0.191207\pi\)
\(72\) 0 0
\(73\) 393719. 1.01209 0.506044 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) 520759.i 1.28511i
\(75\) 0 0
\(76\) 247276. 0.563301
\(77\) 494333.i 1.08280i
\(78\) 0 0
\(79\) 898368. 1.82210 0.911052 0.412292i \(-0.135272\pi\)
0.911052 + 0.412292i \(0.135272\pi\)
\(80\) − 46776.5i − 0.0913604i
\(81\) 0 0
\(82\) 202632. 0.367508
\(83\) − 178882.i − 0.312847i −0.987690 0.156424i \(-0.950003\pi\)
0.987690 0.156424i \(-0.0499965\pi\)
\(84\) 0 0
\(85\) −369580. −0.601799
\(86\) − 391125.i − 0.614922i
\(87\) 0 0
\(88\) 182548. 0.267872
\(89\) − 826458.i − 1.17233i −0.810191 0.586166i \(-0.800636\pi\)
0.810191 0.586166i \(-0.199364\pi\)
\(90\) 0 0
\(91\) −457647. −0.607304
\(92\) 437789.i 0.562213i
\(93\) 0 0
\(94\) 86294.8 0.103897
\(95\) 352988.i 0.411707i
\(96\) 0 0
\(97\) 635962. 0.696813 0.348406 0.937344i \(-0.386723\pi\)
0.348406 + 0.937344i \(0.386723\pi\)
\(98\) 693768.i 0.737116i
\(99\) 0 0
\(100\) −433226. −0.433226
\(101\) 262465.i 0.254746i 0.991855 + 0.127373i \(0.0406545\pi\)
−0.991855 + 0.127373i \(0.959346\pi\)
\(102\) 0 0
\(103\) −82476.0 −0.0754772 −0.0377386 0.999288i \(-0.512015\pi\)
−0.0377386 + 0.999288i \(0.512015\pi\)
\(104\) 169000.i 0.150240i
\(105\) 0 0
\(106\) −1.33836e6 −1.12371
\(107\) 1.03624e6i 0.845881i 0.906158 + 0.422940i \(0.139002\pi\)
−0.906158 + 0.422940i \(0.860998\pi\)
\(108\) 0 0
\(109\) 420018. 0.324331 0.162165 0.986764i \(-0.448152\pi\)
0.162165 + 0.986764i \(0.448152\pi\)
\(110\) 260588.i 0.195784i
\(111\) 0 0
\(112\) 501959. 0.357285
\(113\) − 1.52972e6i − 1.06017i −0.847944 0.530086i \(-0.822160\pi\)
0.847944 0.530086i \(-0.177840\pi\)
\(114\) 0 0
\(115\) −624947. −0.410913
\(116\) − 72596.7i − 0.0465096i
\(117\) 0 0
\(118\) 1.44938e6 0.882139
\(119\) − 3.96596e6i − 2.35347i
\(120\) 0 0
\(121\) 754606. 0.425955
\(122\) 225365.i 0.124110i
\(123\) 0 0
\(124\) −1.09200e6 −0.572742
\(125\) − 1.33219e6i − 0.682080i
\(126\) 0 0
\(127\) −949267. −0.463422 −0.231711 0.972785i \(-0.574432\pi\)
−0.231711 + 0.972785i \(0.574432\pi\)
\(128\) − 185364.i − 0.0883883i
\(129\) 0 0
\(130\) −241249. −0.109808
\(131\) − 917060.i − 0.407928i −0.978978 0.203964i \(-0.934617\pi\)
0.978978 0.203964i \(-0.0653826\pi\)
\(132\) 0 0
\(133\) −3.78791e6 −1.61007
\(134\) 1.81250e6i 0.753294i
\(135\) 0 0
\(136\) −1.46455e6 −0.582222
\(137\) 477205.i 0.185585i 0.995685 + 0.0927926i \(0.0295794\pi\)
−0.995685 + 0.0927926i \(0.970421\pi\)
\(138\) 0 0
\(139\) −4.08452e6 −1.52089 −0.760443 0.649405i \(-0.775018\pi\)
−0.760443 + 0.649405i \(0.775018\pi\)
\(140\) 716551.i 0.261134i
\(141\) 0 0
\(142\) 2.28872e6 0.799334
\(143\) − 941484.i − 0.321962i
\(144\) 0 0
\(145\) 103632. 0.0339931
\(146\) 2.22721e6i 0.715654i
\(147\) 0 0
\(148\) −2.94586e6 −0.908713
\(149\) − 3.68008e6i − 1.11250i −0.831017 0.556248i \(-0.812241\pi\)
0.831017 0.556248i \(-0.187759\pi\)
\(150\) 0 0
\(151\) −667485. −0.193870 −0.0969350 0.995291i \(-0.530904\pi\)
−0.0969350 + 0.995291i \(0.530904\pi\)
\(152\) 1.39880e6i 0.398314i
\(153\) 0 0
\(154\) −2.79637e6 −0.765654
\(155\) − 1.55884e6i − 0.418608i
\(156\) 0 0
\(157\) −1.50716e6 −0.389458 −0.194729 0.980857i \(-0.562383\pi\)
−0.194729 + 0.980857i \(0.562383\pi\)
\(158\) 5.08194e6i 1.28842i
\(159\) 0 0
\(160\) 264608. 0.0646016
\(161\) − 6.70631e6i − 1.60696i
\(162\) 0 0
\(163\) −5.90805e6 −1.36421 −0.682105 0.731254i \(-0.738935\pi\)
−0.682105 + 0.731254i \(0.738935\pi\)
\(164\) 1.14626e6i 0.259867i
\(165\) 0 0
\(166\) 1.01191e6 0.221216
\(167\) 2.69176e6i 0.577945i 0.957337 + 0.288972i \(0.0933136\pi\)
−0.957337 + 0.288972i \(0.906686\pi\)
\(168\) 0 0
\(169\) −3.95520e6 −0.819423
\(170\) − 2.09066e6i − 0.425536i
\(171\) 0 0
\(172\) 2.21254e6 0.434816
\(173\) 6.34579e6i 1.22560i 0.790239 + 0.612798i \(0.209956\pi\)
−0.790239 + 0.612798i \(0.790044\pi\)
\(174\) 0 0
\(175\) 6.63641e6 1.23828
\(176\) 1.03264e6i 0.189414i
\(177\) 0 0
\(178\) 4.67515e6 0.828964
\(179\) 9.31786e6i 1.62464i 0.583212 + 0.812320i \(0.301796\pi\)
−0.583212 + 0.812320i \(0.698204\pi\)
\(180\) 0 0
\(181\) 1.28997e6 0.217542 0.108771 0.994067i \(-0.465308\pi\)
0.108771 + 0.994067i \(0.465308\pi\)
\(182\) − 2.58884e6i − 0.429429i
\(183\) 0 0
\(184\) −2.47651e6 −0.397545
\(185\) − 4.20523e6i − 0.664163i
\(186\) 0 0
\(187\) 8.15889e6 1.24769
\(188\) 488157.i 0.0734659i
\(189\) 0 0
\(190\) −1.99680e6 −0.291121
\(191\) 7.23490e6i 1.03832i 0.854676 + 0.519161i \(0.173756\pi\)
−0.854676 + 0.519161i \(0.826244\pi\)
\(192\) 0 0
\(193\) −4.67637e6 −0.650485 −0.325243 0.945631i \(-0.605446\pi\)
−0.325243 + 0.945631i \(0.605446\pi\)
\(194\) 3.59755e6i 0.492721i
\(195\) 0 0
\(196\) −3.92454e6 −0.521220
\(197\) − 2.50022e6i − 0.327024i −0.986541 0.163512i \(-0.947718\pi\)
0.986541 0.163512i \(-0.0522822\pi\)
\(198\) 0 0
\(199\) −3.28474e6 −0.416814 −0.208407 0.978042i \(-0.566828\pi\)
−0.208407 + 0.978042i \(0.566828\pi\)
\(200\) − 2.45070e6i − 0.306337i
\(201\) 0 0
\(202\) −1.48472e6 −0.180132
\(203\) 1.11208e6i 0.132937i
\(204\) 0 0
\(205\) −1.63629e6 −0.189933
\(206\) − 466555.i − 0.0533704i
\(207\) 0 0
\(208\) −956008. −0.106236
\(209\) − 7.79260e6i − 0.853578i
\(210\) 0 0
\(211\) 1.66888e6 0.177655 0.0888276 0.996047i \(-0.471688\pi\)
0.0888276 + 0.996047i \(0.471688\pi\)
\(212\) − 7.57091e6i − 0.794585i
\(213\) 0 0
\(214\) −5.86186e6 −0.598128
\(215\) 3.15841e6i 0.317800i
\(216\) 0 0
\(217\) 1.67280e7 1.63706
\(218\) 2.37598e6i 0.229337i
\(219\) 0 0
\(220\) −1.47411e6 −0.138440
\(221\) 7.55339e6i 0.699786i
\(222\) 0 0
\(223\) −1.13073e7 −1.01963 −0.509815 0.860284i \(-0.670286\pi\)
−0.509815 + 0.860284i \(0.670286\pi\)
\(224\) 2.83951e6i 0.252639i
\(225\) 0 0
\(226\) 8.65340e6 0.749655
\(227\) − 1.53264e7i − 1.31028i −0.755508 0.655139i \(-0.772610\pi\)
0.755508 0.655139i \(-0.227390\pi\)
\(228\) 0 0
\(229\) 1.27105e6 0.105842 0.0529208 0.998599i \(-0.483147\pi\)
0.0529208 + 0.998599i \(0.483147\pi\)
\(230\) − 3.53523e6i − 0.290559i
\(231\) 0 0
\(232\) 410669. 0.0328873
\(233\) 1.45867e6i 0.115316i 0.998336 + 0.0576578i \(0.0183632\pi\)
−0.998336 + 0.0576578i \(0.981637\pi\)
\(234\) 0 0
\(235\) −696848. −0.0536951
\(236\) 8.19894e6i 0.623766i
\(237\) 0 0
\(238\) 2.24349e7 1.66415
\(239\) 5.02512e6i 0.368089i 0.982918 + 0.184044i \(0.0589190\pi\)
−0.982918 + 0.184044i \(0.941081\pi\)
\(240\) 0 0
\(241\) 5.13489e6 0.366843 0.183421 0.983034i \(-0.441283\pi\)
0.183421 + 0.983034i \(0.441283\pi\)
\(242\) 4.26869e6i 0.301196i
\(243\) 0 0
\(244\) −1.27485e6 −0.0877589
\(245\) − 5.60231e6i − 0.380951i
\(246\) 0 0
\(247\) 7.21428e6 0.478743
\(248\) − 6.17731e6i − 0.404990i
\(249\) 0 0
\(250\) 7.53599e6 0.482303
\(251\) − 1.72705e6i − 0.109216i −0.998508 0.0546078i \(-0.982609\pi\)
0.998508 0.0546078i \(-0.0173909\pi\)
\(252\) 0 0
\(253\) 1.37964e7 0.851930
\(254\) − 5.36986e6i − 0.327689i
\(255\) 0 0
\(256\) 1.04858e6 0.0625000
\(257\) − 1.32178e7i − 0.778682i −0.921094 0.389341i \(-0.872703\pi\)
0.921094 0.389341i \(-0.127297\pi\)
\(258\) 0 0
\(259\) 4.51264e7 2.59735
\(260\) − 1.36471e6i − 0.0776462i
\(261\) 0 0
\(262\) 5.18767e6 0.288449
\(263\) 1.19615e7i 0.657537i 0.944411 + 0.328768i \(0.106634\pi\)
−0.944411 + 0.328768i \(0.893366\pi\)
\(264\) 0 0
\(265\) 1.08075e7 0.580749
\(266\) − 2.14277e7i − 1.13849i
\(267\) 0 0
\(268\) −1.02531e7 −0.532660
\(269\) − 210238.i − 0.0108007i −0.999985 0.00540037i \(-0.998281\pi\)
0.999985 0.00540037i \(-0.00171900\pi\)
\(270\) 0 0
\(271\) −1.71422e6 −0.0861309 −0.0430654 0.999072i \(-0.513712\pi\)
−0.0430654 + 0.999072i \(0.513712\pi\)
\(272\) − 8.28476e6i − 0.411693i
\(273\) 0 0
\(274\) −2.69948e6 −0.131229
\(275\) 1.36526e7i 0.656474i
\(276\) 0 0
\(277\) 1.46890e7 0.691118 0.345559 0.938397i \(-0.387689\pi\)
0.345559 + 0.938397i \(0.387689\pi\)
\(278\) − 2.31055e7i − 1.07543i
\(279\) 0 0
\(280\) −4.05342e6 −0.184649
\(281\) 1.30860e7i 0.589776i 0.955532 + 0.294888i \(0.0952824\pi\)
−0.955532 + 0.294888i \(0.904718\pi\)
\(282\) 0 0
\(283\) 3.05680e6 0.134868 0.0674338 0.997724i \(-0.478519\pi\)
0.0674338 + 0.997724i \(0.478519\pi\)
\(284\) 1.29470e7i 0.565215i
\(285\) 0 0
\(286\) 5.32584e6 0.227662
\(287\) − 1.75591e7i − 0.742773i
\(288\) 0 0
\(289\) −4.13200e7 −1.71186
\(290\) 586232.i 0.0240368i
\(291\) 0 0
\(292\) −1.25990e7 −0.506044
\(293\) 2.94163e7i 1.16946i 0.811228 + 0.584729i \(0.198799\pi\)
−0.811228 + 0.584729i \(0.801201\pi\)
\(294\) 0 0
\(295\) −1.17040e7 −0.455901
\(296\) − 1.66643e7i − 0.642557i
\(297\) 0 0
\(298\) 2.08177e7 0.786653
\(299\) 1.27725e7i 0.477818i
\(300\) 0 0
\(301\) −3.38929e7 −1.24282
\(302\) − 3.77587e6i − 0.137087i
\(303\) 0 0
\(304\) −7.91282e6 −0.281650
\(305\) − 1.81986e6i − 0.0641415i
\(306\) 0 0
\(307\) 3.82307e7 1.32129 0.660643 0.750700i \(-0.270284\pi\)
0.660643 + 0.750700i \(0.270284\pi\)
\(308\) − 1.58187e7i − 0.541399i
\(309\) 0 0
\(310\) 8.81815e6 0.296001
\(311\) − 736808.i − 0.0244948i −0.999925 0.0122474i \(-0.996101\pi\)
0.999925 0.0122474i \(-0.00389856\pi\)
\(312\) 0 0
\(313\) −1.42553e7 −0.464883 −0.232442 0.972610i \(-0.574671\pi\)
−0.232442 + 0.972610i \(0.574671\pi\)
\(314\) − 8.52579e6i − 0.275388i
\(315\) 0 0
\(316\) −2.87478e7 −0.911052
\(317\) 5.31100e7i 1.66724i 0.552337 + 0.833621i \(0.313736\pi\)
−0.552337 + 0.833621i \(0.686264\pi\)
\(318\) 0 0
\(319\) −2.28780e6 −0.0704767
\(320\) 1.49685e6i 0.0456802i
\(321\) 0 0
\(322\) 3.79366e7 1.13629
\(323\) 6.25189e7i 1.85526i
\(324\) 0 0
\(325\) −1.26394e7 −0.368194
\(326\) − 3.34210e7i − 0.964643i
\(327\) 0 0
\(328\) −6.48422e6 −0.183754
\(329\) − 7.47788e6i − 0.209986i
\(330\) 0 0
\(331\) −5.09562e7 −1.40512 −0.702559 0.711625i \(-0.747959\pi\)
−0.702559 + 0.711625i \(0.747959\pi\)
\(332\) 5.72422e6i 0.156424i
\(333\) 0 0
\(334\) −1.52269e7 −0.408669
\(335\) − 1.46363e7i − 0.389312i
\(336\) 0 0
\(337\) 5.42292e7 1.41691 0.708457 0.705754i \(-0.249392\pi\)
0.708457 + 0.705754i \(0.249392\pi\)
\(338\) − 2.23740e7i − 0.579419i
\(339\) 0 0
\(340\) 1.18266e7 0.300900
\(341\) 3.44132e7i 0.867885i
\(342\) 0 0
\(343\) 2.44751e6 0.0606517
\(344\) 1.25160e7i 0.307461i
\(345\) 0 0
\(346\) −3.58972e7 −0.866628
\(347\) − 2.06891e7i − 0.495169i −0.968866 0.247584i \(-0.920363\pi\)
0.968866 0.247584i \(-0.0796368\pi\)
\(348\) 0 0
\(349\) −3.45539e7 −0.812871 −0.406435 0.913680i \(-0.633228\pi\)
−0.406435 + 0.913680i \(0.633228\pi\)
\(350\) 3.75412e7i 0.875597i
\(351\) 0 0
\(352\) −5.84152e6 −0.133936
\(353\) − 8.06874e7i − 1.83435i −0.398487 0.917174i \(-0.630464\pi\)
0.398487 0.917174i \(-0.369536\pi\)
\(354\) 0 0
\(355\) −1.84819e7 −0.413106
\(356\) 2.64467e7i 0.586166i
\(357\) 0 0
\(358\) −5.27098e7 −1.14879
\(359\) 1.64156e7i 0.354792i 0.984140 + 0.177396i \(0.0567674\pi\)
−0.984140 + 0.177396i \(0.943233\pi\)
\(360\) 0 0
\(361\) 1.26662e7 0.269231
\(362\) 7.29717e6i 0.153826i
\(363\) 0 0
\(364\) 1.46447e7 0.303652
\(365\) − 1.79852e7i − 0.369859i
\(366\) 0 0
\(367\) −5.05783e7 −1.02321 −0.511606 0.859220i \(-0.670949\pi\)
−0.511606 + 0.859220i \(0.670949\pi\)
\(368\) − 1.40092e7i − 0.281107i
\(369\) 0 0
\(370\) 2.37884e7 0.469634
\(371\) 1.15976e8i 2.27115i
\(372\) 0 0
\(373\) 1.40213e7 0.270185 0.135092 0.990833i \(-0.456867\pi\)
0.135092 + 0.990833i \(0.456867\pi\)
\(374\) 4.61536e7i 0.882249i
\(375\) 0 0
\(376\) −2.76143e6 −0.0519483
\(377\) − 2.11801e6i − 0.0395280i
\(378\) 0 0
\(379\) −1.76705e7 −0.324587 −0.162294 0.986743i \(-0.551889\pi\)
−0.162294 + 0.986743i \(0.551889\pi\)
\(380\) − 1.12956e7i − 0.205854i
\(381\) 0 0
\(382\) −4.09268e7 −0.734205
\(383\) 9.08579e7i 1.61721i 0.588352 + 0.808605i \(0.299777\pi\)
−0.588352 + 0.808605i \(0.700223\pi\)
\(384\) 0 0
\(385\) 2.25812e7 0.395700
\(386\) − 2.64536e7i − 0.459962i
\(387\) 0 0
\(388\) −2.03508e7 −0.348406
\(389\) − 8.10099e7i − 1.37622i −0.725604 0.688112i \(-0.758440\pi\)
0.725604 0.688112i \(-0.241560\pi\)
\(390\) 0 0
\(391\) −1.10687e8 −1.85167
\(392\) − 2.22006e7i − 0.368558i
\(393\) 0 0
\(394\) 1.41434e7 0.231241
\(395\) − 4.10377e7i − 0.665873i
\(396\) 0 0
\(397\) −5.78058e7 −0.923846 −0.461923 0.886920i \(-0.652840\pi\)
−0.461923 + 0.886920i \(0.652840\pi\)
\(398\) − 1.85813e7i − 0.294732i
\(399\) 0 0
\(400\) 1.38632e7 0.216613
\(401\) − 1.22792e8i − 1.90431i −0.305622 0.952153i \(-0.598864\pi\)
0.305622 0.952153i \(-0.401136\pi\)
\(402\) 0 0
\(403\) −3.18593e7 −0.486767
\(404\) − 8.39887e6i − 0.127373i
\(405\) 0 0
\(406\) −6.29086e6 −0.0940009
\(407\) 9.28352e7i 1.37699i
\(408\) 0 0
\(409\) −1.80741e7 −0.264172 −0.132086 0.991238i \(-0.542168\pi\)
−0.132086 + 0.991238i \(0.542168\pi\)
\(410\) − 9.25627e6i − 0.134303i
\(411\) 0 0
\(412\) 2.63923e6 0.0377386
\(413\) − 1.25596e8i − 1.78290i
\(414\) 0 0
\(415\) −8.17137e6 −0.114327
\(416\) − 5.40800e6i − 0.0751202i
\(417\) 0 0
\(418\) 4.40816e7 0.603571
\(419\) − 1.00741e7i − 0.136950i −0.997653 0.0684751i \(-0.978187\pi\)
0.997653 0.0684751i \(-0.0218134\pi\)
\(420\) 0 0
\(421\) 1.40856e8 1.88768 0.943839 0.330406i \(-0.107186\pi\)
0.943839 + 0.330406i \(0.107186\pi\)
\(422\) 9.44061e6i 0.125621i
\(423\) 0 0
\(424\) 4.28275e7 0.561857
\(425\) − 1.09533e8i − 1.42685i
\(426\) 0 0
\(427\) 1.95290e7 0.250839
\(428\) − 3.31597e7i − 0.422940i
\(429\) 0 0
\(430\) −1.78667e7 −0.224718
\(431\) 6.81157e6i 0.0850777i 0.999095 + 0.0425388i \(0.0135446\pi\)
−0.999095 + 0.0425388i \(0.986455\pi\)
\(432\) 0 0
\(433\) −1.17807e8 −1.45113 −0.725566 0.688153i \(-0.758422\pi\)
−0.725566 + 0.688153i \(0.758422\pi\)
\(434\) 9.46277e7i 1.15757i
\(435\) 0 0
\(436\) −1.34406e7 −0.162165
\(437\) 1.05717e8i 1.26678i
\(438\) 0 0
\(439\) 7.10305e7 0.839559 0.419780 0.907626i \(-0.362107\pi\)
0.419780 + 0.907626i \(0.362107\pi\)
\(440\) − 8.33881e6i − 0.0978918i
\(441\) 0 0
\(442\) −4.27284e7 −0.494823
\(443\) − 2.11414e7i − 0.243177i −0.992581 0.121588i \(-0.961201\pi\)
0.992581 0.121588i \(-0.0387988\pi\)
\(444\) 0 0
\(445\) −3.77528e7 −0.428419
\(446\) − 6.39635e7i − 0.720987i
\(447\) 0 0
\(448\) −1.60627e7 −0.178642
\(449\) 5.48751e7i 0.606228i 0.952954 + 0.303114i \(0.0980263\pi\)
−0.952954 + 0.303114i \(0.901974\pi\)
\(450\) 0 0
\(451\) 3.61230e7 0.393780
\(452\) 4.89510e7i 0.530086i
\(453\) 0 0
\(454\) 8.66994e7 0.926507
\(455\) 2.09054e7i 0.221934i
\(456\) 0 0
\(457\) −1.96565e7 −0.205948 −0.102974 0.994684i \(-0.532836\pi\)
−0.102974 + 0.994684i \(0.532836\pi\)
\(458\) 7.19015e6i 0.0748413i
\(459\) 0 0
\(460\) 1.99983e7 0.205456
\(461\) − 5.73967e7i − 0.585847i −0.956136 0.292923i \(-0.905372\pi\)
0.956136 0.292923i \(-0.0946281\pi\)
\(462\) 0 0
\(463\) 1.01215e8 1.01977 0.509885 0.860242i \(-0.329688\pi\)
0.509885 + 0.860242i \(0.329688\pi\)
\(464\) 2.32309e6i 0.0232548i
\(465\) 0 0
\(466\) −8.25146e6 −0.0815404
\(467\) − 1.26426e8i − 1.24133i −0.784077 0.620663i \(-0.786863\pi\)
0.784077 0.620663i \(-0.213137\pi\)
\(468\) 0 0
\(469\) 1.57063e8 1.52249
\(470\) − 3.94197e6i − 0.0379681i
\(471\) 0 0
\(472\) −4.63802e7 −0.441069
\(473\) − 6.97255e7i − 0.658882i
\(474\) 0 0
\(475\) −1.04615e8 −0.976147
\(476\) 1.26911e8i 1.17673i
\(477\) 0 0
\(478\) −2.84264e7 −0.260278
\(479\) 6.67030e7i 0.606931i 0.952843 + 0.303465i \(0.0981436\pi\)
−0.952843 + 0.303465i \(0.901856\pi\)
\(480\) 0 0
\(481\) −8.59456e7 −0.772304
\(482\) 2.90473e7i 0.259397i
\(483\) 0 0
\(484\) −2.41474e7 −0.212978
\(485\) − 2.90509e7i − 0.254645i
\(486\) 0 0
\(487\) 6.97650e7 0.604019 0.302009 0.953305i \(-0.402343\pi\)
0.302009 + 0.953305i \(0.402343\pi\)
\(488\) − 7.21167e6i − 0.0620549i
\(489\) 0 0
\(490\) 3.16915e7 0.269373
\(491\) 1.06576e8i 0.900356i 0.892939 + 0.450178i \(0.148640\pi\)
−0.892939 + 0.450178i \(0.851360\pi\)
\(492\) 0 0
\(493\) 1.83547e7 0.153181
\(494\) 4.08101e7i 0.338522i
\(495\) 0 0
\(496\) 3.49441e7 0.286371
\(497\) − 1.98329e8i − 1.61554i
\(498\) 0 0
\(499\) 1.35114e8 1.08743 0.543713 0.839271i \(-0.317018\pi\)
0.543713 + 0.839271i \(0.317018\pi\)
\(500\) 4.26300e7i 0.341040i
\(501\) 0 0
\(502\) 9.76969e6 0.0772271
\(503\) 2.31473e8i 1.81885i 0.415869 + 0.909425i \(0.363477\pi\)
−0.415869 + 0.909425i \(0.636523\pi\)
\(504\) 0 0
\(505\) 1.19894e7 0.0930947
\(506\) 7.80442e7i 0.602406i
\(507\) 0 0
\(508\) 3.03765e7 0.231711
\(509\) − 1.31287e8i − 0.995559i −0.867304 0.497780i \(-0.834149\pi\)
0.867304 0.497780i \(-0.165851\pi\)
\(510\) 0 0
\(511\) 1.92999e8 1.44641
\(512\) 5.93164e6i 0.0441942i
\(513\) 0 0
\(514\) 7.47712e7 0.550611
\(515\) 3.76752e6i 0.0275825i
\(516\) 0 0
\(517\) 1.53837e7 0.111324
\(518\) 2.55273e8i 1.83661i
\(519\) 0 0
\(520\) 7.71996e6 0.0549041
\(521\) − 1.14057e8i − 0.806509i −0.915088 0.403255i \(-0.867879\pi\)
0.915088 0.403255i \(-0.132121\pi\)
\(522\) 0 0
\(523\) 6.52785e7 0.456315 0.228158 0.973624i \(-0.426730\pi\)
0.228158 + 0.973624i \(0.426730\pi\)
\(524\) 2.93459e7i 0.203964i
\(525\) 0 0
\(526\) −6.76647e7 −0.464949
\(527\) − 2.76092e8i − 1.88635i
\(528\) 0 0
\(529\) −3.91312e7 −0.264336
\(530\) 6.11366e7i 0.410652i
\(531\) 0 0
\(532\) 1.21213e8 0.805035
\(533\) 3.34422e7i 0.220858i
\(534\) 0 0
\(535\) 4.73357e7 0.309120
\(536\) − 5.80001e7i − 0.376647i
\(537\) 0 0
\(538\) 1.18928e6 0.00763728
\(539\) 1.23677e8i 0.789812i
\(540\) 0 0
\(541\) 1.71142e8 1.08085 0.540423 0.841394i \(-0.318264\pi\)
0.540423 + 0.841394i \(0.318264\pi\)
\(542\) − 9.69710e6i − 0.0609037i
\(543\) 0 0
\(544\) 4.68657e7 0.291111
\(545\) − 1.91865e7i − 0.118524i
\(546\) 0 0
\(547\) 5.80956e7 0.354961 0.177481 0.984124i \(-0.443205\pi\)
0.177481 + 0.984124i \(0.443205\pi\)
\(548\) − 1.52706e7i − 0.0927926i
\(549\) 0 0
\(550\) −7.72308e7 −0.464197
\(551\) − 1.75306e7i − 0.104796i
\(552\) 0 0
\(553\) 4.40375e8 2.60404
\(554\) 8.30934e7i 0.488694i
\(555\) 0 0
\(556\) 1.30705e8 0.760443
\(557\) − 1.36639e8i − 0.790696i −0.918532 0.395348i \(-0.870624\pi\)
0.918532 0.395348i \(-0.129376\pi\)
\(558\) 0 0
\(559\) 6.45509e7 0.369544
\(560\) − 2.29296e7i − 0.130567i
\(561\) 0 0
\(562\) −7.40255e7 −0.417035
\(563\) 2.77071e8i 1.55262i 0.630350 + 0.776311i \(0.282912\pi\)
−0.630350 + 0.776311i \(0.717088\pi\)
\(564\) 0 0
\(565\) −6.98779e7 −0.387431
\(566\) 1.72919e7i 0.0953657i
\(567\) 0 0
\(568\) −7.32392e7 −0.399667
\(569\) 2.43372e8i 1.32109i 0.750785 + 0.660546i \(0.229675\pi\)
−0.750785 + 0.660546i \(0.770325\pi\)
\(570\) 0 0
\(571\) −1.39062e8 −0.746965 −0.373483 0.927637i \(-0.621836\pi\)
−0.373483 + 0.927637i \(0.621836\pi\)
\(572\) 3.01275e7i 0.160981i
\(573\) 0 0
\(574\) 9.93291e7 0.525220
\(575\) − 1.85216e8i − 0.974262i
\(576\) 0 0
\(577\) −3.45574e8 −1.79893 −0.899464 0.436994i \(-0.856043\pi\)
−0.899464 + 0.436994i \(0.856043\pi\)
\(578\) − 2.33741e8i − 1.21046i
\(579\) 0 0
\(580\) −3.31623e6 −0.0169966
\(581\) − 8.76870e7i − 0.447102i
\(582\) 0 0
\(583\) −2.38588e8 −1.20405
\(584\) − 7.12708e7i − 0.357827i
\(585\) 0 0
\(586\) −1.66404e8 −0.826932
\(587\) 1.65925e8i 0.820349i 0.912007 + 0.410174i \(0.134532\pi\)
−0.912007 + 0.410174i \(0.865468\pi\)
\(588\) 0 0
\(589\) −2.63697e8 −1.29051
\(590\) − 6.62081e7i − 0.322370i
\(591\) 0 0
\(592\) 9.42674e7 0.454356
\(593\) 2.87880e8i 1.38054i 0.723553 + 0.690268i \(0.242508\pi\)
−0.723553 + 0.690268i \(0.757492\pi\)
\(594\) 0 0
\(595\) −1.81166e8 −0.860055
\(596\) 1.17762e8i 0.556248i
\(597\) 0 0
\(598\) −7.22523e7 −0.337869
\(599\) 4.24365e7i 0.197451i 0.995115 + 0.0987254i \(0.0314765\pi\)
−0.995115 + 0.0987254i \(0.968523\pi\)
\(600\) 0 0
\(601\) −1.78000e7 −0.0819968 −0.0409984 0.999159i \(-0.513054\pi\)
−0.0409984 + 0.999159i \(0.513054\pi\)
\(602\) − 1.91727e8i − 0.878809i
\(603\) 0 0
\(604\) 2.13595e7 0.0969350
\(605\) − 3.44706e7i − 0.155662i
\(606\) 0 0
\(607\) −3.44970e8 −1.54246 −0.771232 0.636554i \(-0.780359\pi\)
−0.771232 + 0.636554i \(0.780359\pi\)
\(608\) − 4.47617e7i − 0.199157i
\(609\) 0 0
\(610\) 1.02947e7 0.0453549
\(611\) 1.42420e7i 0.0624378i
\(612\) 0 0
\(613\) 1.15533e8 0.501562 0.250781 0.968044i \(-0.419313\pi\)
0.250781 + 0.968044i \(0.419313\pi\)
\(614\) 2.16265e8i 0.934291i
\(615\) 0 0
\(616\) 8.94838e7 0.382827
\(617\) − 2.30584e8i − 0.981689i −0.871247 0.490845i \(-0.836688\pi\)
0.871247 0.490845i \(-0.163312\pi\)
\(618\) 0 0
\(619\) 2.86629e8 1.20851 0.604253 0.796792i \(-0.293472\pi\)
0.604253 + 0.796792i \(0.293472\pi\)
\(620\) 4.98830e7i 0.209304i
\(621\) 0 0
\(622\) 4.16802e6 0.0173204
\(623\) − 4.05126e8i − 1.67543i
\(624\) 0 0
\(625\) 1.50682e8 0.617192
\(626\) − 8.06402e7i − 0.328722i
\(627\) 0 0
\(628\) 4.82291e7 0.194729
\(629\) − 7.44804e8i − 2.99288i
\(630\) 0 0
\(631\) −2.03780e8 −0.811098 −0.405549 0.914073i \(-0.632920\pi\)
−0.405549 + 0.914073i \(0.632920\pi\)
\(632\) − 1.62622e8i − 0.644211i
\(633\) 0 0
\(634\) −3.00435e8 −1.17892
\(635\) 4.33627e7i 0.169354i
\(636\) 0 0
\(637\) −1.14499e8 −0.442978
\(638\) − 1.29417e7i − 0.0498345i
\(639\) 0 0
\(640\) −8.46746e6 −0.0323008
\(641\) 4.76661e8i 1.80982i 0.425602 + 0.904910i \(0.360062\pi\)
−0.425602 + 0.904910i \(0.639938\pi\)
\(642\) 0 0
\(643\) 1.61187e8 0.606315 0.303157 0.952941i \(-0.401959\pi\)
0.303157 + 0.952941i \(0.401959\pi\)
\(644\) 2.14602e8i 0.803481i
\(645\) 0 0
\(646\) −3.53660e8 −1.31186
\(647\) 3.18171e8i 1.17476i 0.809313 + 0.587378i \(0.199840\pi\)
−0.809313 + 0.587378i \(0.800160\pi\)
\(648\) 0 0
\(649\) 2.58380e8 0.945202
\(650\) − 7.14992e7i − 0.260352i
\(651\) 0 0
\(652\) 1.89058e8 0.682105
\(653\) 1.79024e8i 0.642943i 0.946919 + 0.321471i \(0.104177\pi\)
−0.946919 + 0.321471i \(0.895823\pi\)
\(654\) 0 0
\(655\) −4.18915e7 −0.149074
\(656\) − 3.66803e7i − 0.129934i
\(657\) 0 0
\(658\) 4.23013e7 0.148483
\(659\) − 3.16743e8i − 1.10675i −0.832932 0.553376i \(-0.813339\pi\)
0.832932 0.553376i \(-0.186661\pi\)
\(660\) 0 0
\(661\) −3.76093e8 −1.30224 −0.651119 0.758976i \(-0.725700\pi\)
−0.651119 + 0.758976i \(0.725700\pi\)
\(662\) − 2.88252e8i − 0.993569i
\(663\) 0 0
\(664\) −3.23811e7 −0.110608
\(665\) 1.73033e8i 0.588387i
\(666\) 0 0
\(667\) 3.10371e7 0.104593
\(668\) − 8.61362e7i − 0.288972i
\(669\) 0 0
\(670\) 8.27956e7 0.275285
\(671\) 4.01755e7i 0.132982i
\(672\) 0 0
\(673\) −2.37679e8 −0.779734 −0.389867 0.920871i \(-0.627479\pi\)
−0.389867 + 0.920871i \(0.627479\pi\)
\(674\) 3.06767e8i 1.00191i
\(675\) 0 0
\(676\) 1.26566e8 0.409711
\(677\) − 3.89681e8i − 1.25587i −0.778267 0.627933i \(-0.783901\pi\)
0.778267 0.627933i \(-0.216099\pi\)
\(678\) 0 0
\(679\) 3.11745e8 0.995843
\(680\) 6.69011e7i 0.212768i
\(681\) 0 0
\(682\) −1.94671e8 −0.613687
\(683\) 3.91877e8i 1.22995i 0.788546 + 0.614975i \(0.210834\pi\)
−0.788546 + 0.614975i \(0.789166\pi\)
\(684\) 0 0
\(685\) 2.17988e7 0.0678206
\(686\) 1.38452e7i 0.0428872i
\(687\) 0 0
\(688\) −7.08012e7 −0.217408
\(689\) − 2.20882e8i − 0.675308i
\(690\) 0 0
\(691\) −2.52910e8 −0.766534 −0.383267 0.923638i \(-0.625201\pi\)
−0.383267 + 0.923638i \(0.625201\pi\)
\(692\) − 2.03065e8i − 0.612798i
\(693\) 0 0
\(694\) 1.17035e8 0.350137
\(695\) 1.86582e8i 0.555795i
\(696\) 0 0
\(697\) −2.89810e8 −0.855883
\(698\) − 1.95467e8i − 0.574786i
\(699\) 0 0
\(700\) −2.12365e8 −0.619141
\(701\) − 2.15054e8i − 0.624299i −0.950033 0.312150i \(-0.898951\pi\)
0.950033 0.312150i \(-0.101049\pi\)
\(702\) 0 0
\(703\) −7.11366e8 −2.04751
\(704\) − 3.30446e7i − 0.0947072i
\(705\) 0 0
\(706\) 4.56437e8 1.29708
\(707\) 1.28659e8i 0.364067i
\(708\) 0 0
\(709\) −1.40902e8 −0.395346 −0.197673 0.980268i \(-0.563338\pi\)
−0.197673 + 0.980268i \(0.563338\pi\)
\(710\) − 1.04549e8i − 0.292110i
\(711\) 0 0
\(712\) −1.49605e8 −0.414482
\(713\) − 4.66863e8i − 1.28801i
\(714\) 0 0
\(715\) −4.30072e7 −0.117658
\(716\) − 2.98172e8i − 0.812320i
\(717\) 0 0
\(718\) −9.28608e7 −0.250876
\(719\) 4.59075e8i 1.23508i 0.786538 + 0.617542i \(0.211872\pi\)
−0.786538 + 0.617542i \(0.788128\pi\)
\(720\) 0 0
\(721\) −4.04293e7 −0.107867
\(722\) 7.16510e7i 0.190375i
\(723\) 0 0
\(724\) −4.12790e7 −0.108771
\(725\) 3.07136e7i 0.0805967i
\(726\) 0 0
\(727\) 4.52533e8 1.17773 0.588867 0.808230i \(-0.299574\pi\)
0.588867 + 0.808230i \(0.299574\pi\)
\(728\) 8.28429e7i 0.214714i
\(729\) 0 0
\(730\) 1.01740e8 0.261530
\(731\) 5.59397e8i 1.43208i
\(732\) 0 0
\(733\) −4.75576e8 −1.20756 −0.603779 0.797152i \(-0.706339\pi\)
−0.603779 + 0.797152i \(0.706339\pi\)
\(734\) − 2.86114e8i − 0.723520i
\(735\) 0 0
\(736\) 7.92482e7 0.198772
\(737\) 3.23113e8i 0.807147i
\(738\) 0 0
\(739\) 8.91913e7 0.220999 0.110499 0.993876i \(-0.464755\pi\)
0.110499 + 0.993876i \(0.464755\pi\)
\(740\) 1.34567e8i 0.332082i
\(741\) 0 0
\(742\) −6.56057e8 −1.60594
\(743\) 7.28521e8i 1.77613i 0.459715 + 0.888066i \(0.347951\pi\)
−0.459715 + 0.888066i \(0.652049\pi\)
\(744\) 0 0
\(745\) −1.68107e8 −0.406552
\(746\) 7.93163e7i 0.191049i
\(747\) 0 0
\(748\) −2.61084e8 −0.623844
\(749\) 5.07960e8i 1.20888i
\(750\) 0 0
\(751\) −3.58845e8 −0.847201 −0.423601 0.905849i \(-0.639234\pi\)
−0.423601 + 0.905849i \(0.639234\pi\)
\(752\) − 1.56210e7i − 0.0367330i
\(753\) 0 0
\(754\) 1.19813e7 0.0279505
\(755\) 3.04909e7i 0.0708482i
\(756\) 0 0
\(757\) 3.38232e8 0.779700 0.389850 0.920878i \(-0.372527\pi\)
0.389850 + 0.920878i \(0.372527\pi\)
\(758\) − 9.99595e7i − 0.229518i
\(759\) 0 0
\(760\) 6.38976e7 0.145561
\(761\) 6.74492e8i 1.53046i 0.643755 + 0.765232i \(0.277376\pi\)
−0.643755 + 0.765232i \(0.722624\pi\)
\(762\) 0 0
\(763\) 2.05891e8 0.463514
\(764\) − 2.31517e8i − 0.519161i
\(765\) 0 0
\(766\) −5.13970e8 −1.14354
\(767\) 2.39205e8i 0.530132i
\(768\) 0 0
\(769\) 7.62772e8 1.67732 0.838660 0.544656i \(-0.183340\pi\)
0.838660 + 0.544656i \(0.183340\pi\)
\(770\) 1.27739e8i 0.279802i
\(771\) 0 0
\(772\) 1.49644e8 0.325243
\(773\) − 6.42100e8i − 1.39016i −0.718933 0.695080i \(-0.755369\pi\)
0.718933 0.695080i \(-0.244631\pi\)
\(774\) 0 0
\(775\) 4.61997e8 0.992508
\(776\) − 1.15121e8i − 0.246361i
\(777\) 0 0
\(778\) 4.58261e8 0.973137
\(779\) 2.76799e8i 0.585533i
\(780\) 0 0
\(781\) 4.08009e8 0.856478
\(782\) − 6.26138e8i − 1.30933i
\(783\) 0 0
\(784\) 1.25585e8 0.260610
\(785\) 6.88474e7i 0.142324i
\(786\) 0 0
\(787\) 3.41525e8 0.700646 0.350323 0.936629i \(-0.386072\pi\)
0.350323 + 0.936629i \(0.386072\pi\)
\(788\) 8.00070e7i 0.163512i
\(789\) 0 0
\(790\) 2.32144e8 0.470843
\(791\) − 7.49861e8i − 1.51513i
\(792\) 0 0
\(793\) −3.71940e7 −0.0745852
\(794\) − 3.26999e8i − 0.653258i
\(795\) 0 0
\(796\) 1.05112e8 0.208407
\(797\) − 5.29044e8i − 1.04500i −0.852639 0.522501i \(-0.824999\pi\)
0.852639 0.522501i \(-0.175001\pi\)
\(798\) 0 0
\(799\) −1.23421e8 −0.241963
\(800\) 7.84223e7i 0.153169i
\(801\) 0 0
\(802\) 6.94616e8 1.34655
\(803\) 3.97043e8i 0.766816i
\(804\) 0 0
\(805\) −3.06346e8 −0.587251
\(806\) − 1.80223e8i − 0.344196i
\(807\) 0 0
\(808\) 4.75112e7 0.0900662
\(809\) 7.38055e8i 1.39394i 0.717102 + 0.696968i \(0.245468\pi\)
−0.717102 + 0.696968i \(0.754532\pi\)
\(810\) 0 0
\(811\) −4.57151e8 −0.857031 −0.428516 0.903534i \(-0.640963\pi\)
−0.428516 + 0.903534i \(0.640963\pi\)
\(812\) − 3.55865e7i − 0.0664687i
\(813\) 0 0
\(814\) −5.25155e8 −0.973676
\(815\) 2.69881e8i 0.498540i
\(816\) 0 0
\(817\) 5.34283e8 0.979728
\(818\) − 1.02243e8i − 0.186798i
\(819\) 0 0
\(820\) 5.23614e7 0.0949663
\(821\) 1.40803e8i 0.254438i 0.991875 + 0.127219i \(0.0406051\pi\)
−0.991875 + 0.127219i \(0.959395\pi\)
\(822\) 0 0
\(823\) 2.97363e8 0.533443 0.266721 0.963774i \(-0.414060\pi\)
0.266721 + 0.963774i \(0.414060\pi\)
\(824\) 1.49297e7i 0.0266852i
\(825\) 0 0
\(826\) 7.10480e8 1.26070
\(827\) 6.71385e8i 1.18701i 0.804830 + 0.593506i \(0.202257\pi\)
−0.804830 + 0.593506i \(0.797743\pi\)
\(828\) 0 0
\(829\) −4.09596e8 −0.718939 −0.359470 0.933157i \(-0.617042\pi\)
−0.359470 + 0.933157i \(0.617042\pi\)
\(830\) − 4.62242e7i − 0.0808417i
\(831\) 0 0
\(832\) 3.05923e7 0.0531180
\(833\) − 9.92245e8i − 1.71666i
\(834\) 0 0
\(835\) 1.22960e8 0.211205
\(836\) 2.49363e8i 0.426789i
\(837\) 0 0
\(838\) 5.69876e7 0.0968385
\(839\) 3.01060e8i 0.509761i 0.966973 + 0.254881i \(0.0820362\pi\)
−0.966973 + 0.254881i \(0.917964\pi\)
\(840\) 0 0
\(841\) 5.89677e8 0.991347
\(842\) 7.96800e8i 1.33479i
\(843\) 0 0
\(844\) −5.34042e7 −0.0888276
\(845\) 1.80674e8i 0.299451i
\(846\) 0 0
\(847\) 3.69904e8 0.608749
\(848\) 2.42269e8i 0.397293i
\(849\) 0 0
\(850\) 6.19612e8 1.00893
\(851\) − 1.25944e9i − 2.04356i
\(852\) 0 0
\(853\) −3.30806e8 −0.532998 −0.266499 0.963835i \(-0.585867\pi\)
−0.266499 + 0.963835i \(0.585867\pi\)
\(854\) 1.10473e8i 0.177370i
\(855\) 0 0
\(856\) 1.87580e8 0.299064
\(857\) − 7.28944e8i − 1.15811i −0.815287 0.579057i \(-0.803421\pi\)
0.815287 0.579057i \(-0.196579\pi\)
\(858\) 0 0
\(859\) −3.36404e7 −0.0530739 −0.0265370 0.999648i \(-0.508448\pi\)
−0.0265370 + 0.999648i \(0.508448\pi\)
\(860\) − 1.01069e8i − 0.158900i
\(861\) 0 0
\(862\) −3.85321e7 −0.0601590
\(863\) − 3.65765e8i − 0.569075i −0.958665 0.284537i \(-0.908160\pi\)
0.958665 0.284537i \(-0.0918400\pi\)
\(864\) 0 0
\(865\) 2.89877e8 0.447884
\(866\) − 6.66416e8i − 1.02610i
\(867\) 0 0
\(868\) −5.35295e8 −0.818529
\(869\) 9.05952e8i 1.38053i
\(870\) 0 0
\(871\) −2.99134e8 −0.452701
\(872\) − 7.60314e7i − 0.114668i
\(873\) 0 0
\(874\) −5.98027e8 −0.895750
\(875\) − 6.53031e8i − 0.974787i
\(876\) 0 0
\(877\) 7.33254e8 1.08707 0.543533 0.839388i \(-0.317086\pi\)
0.543533 + 0.839388i \(0.317086\pi\)
\(878\) 4.01809e8i 0.593658i
\(879\) 0 0
\(880\) 4.71714e7 0.0692199
\(881\) − 1.64762e8i − 0.240951i −0.992716 0.120475i \(-0.961558\pi\)
0.992716 0.120475i \(-0.0384419\pi\)
\(882\) 0 0
\(883\) 4.55533e8 0.661665 0.330832 0.943690i \(-0.392671\pi\)
0.330832 + 0.943690i \(0.392671\pi\)
\(884\) − 2.41708e8i − 0.349893i
\(885\) 0 0
\(886\) 1.19594e8 0.171952
\(887\) − 1.03986e9i − 1.49005i −0.667034 0.745027i \(-0.732436\pi\)
0.667034 0.745027i \(-0.267564\pi\)
\(888\) 0 0
\(889\) −4.65326e8 −0.662295
\(890\) − 2.13562e8i − 0.302938i
\(891\) 0 0
\(892\) 3.61832e8 0.509815
\(893\) 1.17880e8i 0.165534i
\(894\) 0 0
\(895\) 4.25642e8 0.593712
\(896\) − 9.08644e7i − 0.126319i
\(897\) 0 0
\(898\) −3.10420e8 −0.428668
\(899\) 7.74178e7i 0.106552i
\(900\) 0 0
\(901\) 1.91416e9 2.61700
\(902\) 2.04343e8i 0.278445i
\(903\) 0 0
\(904\) −2.76909e8 −0.374828
\(905\) − 5.89261e7i − 0.0794991i
\(906\) 0 0
\(907\) −7.16242e8 −0.959927 −0.479963 0.877289i \(-0.659350\pi\)
−0.479963 + 0.877289i \(0.659350\pi\)
\(908\) 4.90446e8i 0.655139i
\(909\) 0 0
\(910\) −1.18259e8 −0.156931
\(911\) − 8.69074e8i − 1.14948i −0.818336 0.574740i \(-0.805103\pi\)
0.818336 0.574740i \(-0.194897\pi\)
\(912\) 0 0
\(913\) 1.80392e8 0.237031
\(914\) − 1.11194e8i − 0.145627i
\(915\) 0 0
\(916\) −4.06736e7 −0.0529208
\(917\) − 4.49538e8i − 0.582986i
\(918\) 0 0
\(919\) 6.50303e8 0.837855 0.418928 0.908020i \(-0.362406\pi\)
0.418928 + 0.908020i \(0.362406\pi\)
\(920\) 1.13127e8i 0.145280i
\(921\) 0 0
\(922\) 3.24685e8 0.414256
\(923\) 3.77729e8i 0.480369i
\(924\) 0 0
\(925\) 1.24631e9 1.57471
\(926\) 5.72559e8i 0.721087i
\(927\) 0 0
\(928\) −1.31414e7 −0.0164436
\(929\) 3.39542e8i 0.423493i 0.977325 + 0.211746i \(0.0679150\pi\)
−0.977325 + 0.211746i \(0.932085\pi\)
\(930\) 0 0
\(931\) −9.47699e8 −1.17441
\(932\) − 4.66773e7i − 0.0576578i
\(933\) 0 0
\(934\) 7.15174e8 0.877750
\(935\) − 3.72700e8i − 0.455957i
\(936\) 0 0
\(937\) −1.59228e9 −1.93554 −0.967769 0.251838i \(-0.918965\pi\)
−0.967769 + 0.251838i \(0.918965\pi\)
\(938\) 8.88480e8i 1.07656i
\(939\) 0 0
\(940\) 2.22991e7 0.0268475
\(941\) 1.35993e9i 1.63210i 0.577981 + 0.816051i \(0.303841\pi\)
−0.577981 + 0.816051i \(0.696159\pi\)
\(942\) 0 0
\(943\) −4.90058e8 −0.584403
\(944\) − 2.62366e8i − 0.311883i
\(945\) 0 0
\(946\) 3.94427e8 0.465900
\(947\) − 3.20462e8i − 0.377334i −0.982041 0.188667i \(-0.939583\pi\)
0.982041 0.188667i \(-0.0604167\pi\)
\(948\) 0 0
\(949\) −3.67577e8 −0.430081
\(950\) − 5.91794e8i − 0.690240i
\(951\) 0 0
\(952\) −7.17916e8 −0.832076
\(953\) 6.65905e8i 0.769367i 0.923049 + 0.384683i \(0.125689\pi\)
−0.923049 + 0.384683i \(0.874311\pi\)
\(954\) 0 0
\(955\) 3.30492e8 0.379446
\(956\) − 1.60804e8i − 0.184044i
\(957\) 0 0
\(958\) −3.77329e8 −0.429165
\(959\) 2.33924e8i 0.265227i
\(960\) 0 0
\(961\) 2.77021e8 0.312135
\(962\) − 4.86182e8i − 0.546101i
\(963\) 0 0
\(964\) −1.64316e8 −0.183421
\(965\) 2.13618e8i 0.237714i
\(966\) 0 0
\(967\) 3.58370e8 0.396325 0.198163 0.980169i \(-0.436503\pi\)
0.198163 + 0.980169i \(0.436503\pi\)
\(968\) − 1.36598e8i − 0.150598i
\(969\) 0 0
\(970\) 1.64337e8 0.180061
\(971\) − 3.10565e8i − 0.339230i −0.985510 0.169615i \(-0.945748\pi\)
0.985510 0.169615i \(-0.0542525\pi\)
\(972\) 0 0
\(973\) −2.00221e9 −2.17356
\(974\) 3.94650e8i 0.427106i
\(975\) 0 0
\(976\) 4.07953e7 0.0438794
\(977\) − 1.37591e9i − 1.47539i −0.675135 0.737694i \(-0.735915\pi\)
0.675135 0.737694i \(-0.264085\pi\)
\(978\) 0 0
\(979\) 8.33435e8 0.888227
\(980\) 1.79274e8i 0.190475i
\(981\) 0 0
\(982\) −6.02884e8 −0.636648
\(983\) 5.64536e8i 0.594335i 0.954825 + 0.297168i \(0.0960420\pi\)
−0.954825 + 0.297168i \(0.903958\pi\)
\(984\) 0 0
\(985\) −1.14211e8 −0.119508
\(986\) 1.03830e8i 0.108316i
\(987\) 0 0
\(988\) −2.30857e8 −0.239371
\(989\) 9.45922e8i 0.977836i
\(990\) 0 0
\(991\) 1.94100e8 0.199436 0.0997182 0.995016i \(-0.468206\pi\)
0.0997182 + 0.995016i \(0.468206\pi\)
\(992\) 1.97674e8i 0.202495i
\(993\) 0 0
\(994\) 1.12192e9 1.14236
\(995\) 1.50048e8i 0.152321i
\(996\) 0 0
\(997\) 4.16571e7 0.0420342 0.0210171 0.999779i \(-0.493310\pi\)
0.0210171 + 0.999779i \(0.493310\pi\)
\(998\) 7.64322e8i 0.768926i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 162.7.b.c.161.8 12
3.2 odd 2 inner 162.7.b.c.161.5 12
9.2 odd 6 18.7.d.a.5.2 12
9.4 even 3 18.7.d.a.11.2 yes 12
9.5 odd 6 54.7.d.a.35.6 12
9.7 even 3 54.7.d.a.17.6 12
36.7 odd 6 432.7.q.b.17.5 12
36.11 even 6 144.7.q.c.113.3 12
36.23 even 6 432.7.q.b.305.5 12
36.31 odd 6 144.7.q.c.65.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 12 9.2 odd 6
18.7.d.a.11.2 yes 12 9.4 even 3
54.7.d.a.17.6 12 9.7 even 3
54.7.d.a.35.6 12 9.5 odd 6
144.7.q.c.65.3 12 36.31 odd 6
144.7.q.c.113.3 12 36.11 even 6
162.7.b.c.161.5 12 3.2 odd 2 inner
162.7.b.c.161.8 12 1.1 even 1 trivial
432.7.q.b.17.5 12 36.7 odd 6
432.7.q.b.305.5 12 36.23 even 6