Properties

Label 441.7.d.d.244.5
Level $441$
Weight $7$
Character 441.244
Analytic conductor $101.454$
Analytic rank $0$
Dimension $8$
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] = [441,7,Mod(244,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.244");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 441.d (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: \(101.453850876\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 244.5
Root \(-2.30325 + 3.98935i\) of defining polynomial
Character \(\chi\) \(=\) 441.244
Dual form 441.7.d.d.244.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.60650 q^{2} -50.9932 q^{4} -83.0589i q^{5} -414.723 q^{8} +O(q^{10})\) \(q+3.60650 q^{2} -50.9932 q^{4} -83.0589i q^{5} -414.723 q^{8} -299.552i q^{10} +442.608 q^{11} +696.494i q^{13} +1767.86 q^{16} -6289.93i q^{17} -2687.64i q^{19} +4235.44i q^{20} +1596.27 q^{22} -15770.6 q^{23} +8726.22 q^{25} +2511.91i q^{26} +23274.5 q^{29} -47547.1i q^{31} +32918.1 q^{32} -22684.6i q^{34} -10159.8 q^{37} -9692.96i q^{38} +34446.4i q^{40} -38165.4i q^{41} -151197. q^{43} -22570.0 q^{44} -56876.7 q^{46} +50279.2i q^{47} +31471.1 q^{50} -35516.4i q^{52} +199556. q^{53} -36762.6i q^{55} +83939.6 q^{58} +387225. i q^{59} -11782.0i q^{61} -171479. i q^{62} +5575.73 q^{64} +57850.0 q^{65} -384276. q^{67} +320743. i q^{68} -156126. q^{71} -375539. i q^{73} -36641.3 q^{74} +137051. i q^{76} +33183.6 q^{79} -146837. i q^{80} -137644. i q^{82} +984986. i q^{83} -522434. q^{85} -545292. q^{86} -183560. q^{88} +262613. i q^{89} +804193. q^{92} +181332. i q^{94} -223232. q^{95} +575147. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{2} + 346 q^{4} + 454 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{2} + 346 q^{4} + 454 q^{8} - 2140 q^{11} - 7822 q^{16} - 78 q^{22} - 30448 q^{23} - 44548 q^{25} - 32524 q^{29} + 140406 q^{32} + 91340 q^{37} - 445660 q^{43} - 377658 q^{44} - 1051608 q^{46} + 1218884 q^{50} - 26068 q^{53} + 319002 q^{58} - 1410446 q^{64} - 778008 q^{65} - 768188 q^{67} - 225688 q^{71} + 2371060 q^{74} + 1119184 q^{79} + 1953576 q^{85} - 4604804 q^{86} - 609774 q^{88} + 113064 q^{92} + 2320224 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.60650 0.450813 0.225406 0.974265i \(-0.427629\pi\)
0.225406 + 0.974265i \(0.427629\pi\)
\(3\) 0 0
\(4\) −50.9932 −0.796768
\(5\) − 83.0589i − 0.664471i −0.943196 0.332236i \(-0.892197\pi\)
0.943196 0.332236i \(-0.107803\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −414.723 −0.810006
\(9\) 0 0
\(10\) − 299.552i − 0.299552i
\(11\) 442.608 0.332538 0.166269 0.986080i \(-0.446828\pi\)
0.166269 + 0.986080i \(0.446828\pi\)
\(12\) 0 0
\(13\) 696.494i 0.317020i 0.987357 + 0.158510i \(0.0506691\pi\)
−0.987357 + 0.158510i \(0.949331\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1767.86 0.431607
\(17\) − 6289.93i − 1.28026i −0.768266 0.640131i \(-0.778880\pi\)
0.768266 0.640131i \(-0.221120\pi\)
\(18\) 0 0
\(19\) − 2687.64i − 0.391841i −0.980620 0.195920i \(-0.937231\pi\)
0.980620 0.195920i \(-0.0627694\pi\)
\(20\) 4235.44i 0.529430i
\(21\) 0 0
\(22\) 1596.27 0.149912
\(23\) −15770.6 −1.29618 −0.648090 0.761564i \(-0.724432\pi\)
−0.648090 + 0.761564i \(0.724432\pi\)
\(24\) 0 0
\(25\) 8726.22 0.558478
\(26\) 2511.91i 0.142917i
\(27\) 0 0
\(28\) 0 0
\(29\) 23274.5 0.954304 0.477152 0.878821i \(-0.341669\pi\)
0.477152 + 0.878821i \(0.341669\pi\)
\(30\) 0 0
\(31\) − 47547.1i − 1.59602i −0.602642 0.798012i \(-0.705885\pi\)
0.602642 0.798012i \(-0.294115\pi\)
\(32\) 32918.1 1.00458
\(33\) 0 0
\(34\) − 22684.6i − 0.577158i
\(35\) 0 0
\(36\) 0 0
\(37\) −10159.8 −0.200576 −0.100288 0.994958i \(-0.531976\pi\)
−0.100288 + 0.994958i \(0.531976\pi\)
\(38\) − 9692.96i − 0.176647i
\(39\) 0 0
\(40\) 34446.4i 0.538226i
\(41\) − 38165.4i − 0.553756i −0.960905 0.276878i \(-0.910700\pi\)
0.960905 0.276878i \(-0.0892998\pi\)
\(42\) 0 0
\(43\) −151197. −1.90168 −0.950841 0.309679i \(-0.899779\pi\)
−0.950841 + 0.309679i \(0.899779\pi\)
\(44\) −22570.0 −0.264956
\(45\) 0 0
\(46\) −56876.7 −0.584334
\(47\) 50279.2i 0.484278i 0.970242 + 0.242139i \(0.0778491\pi\)
−0.970242 + 0.242139i \(0.922151\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 31471.1 0.251769
\(51\) 0 0
\(52\) − 35516.4i − 0.252592i
\(53\) 199556. 1.34041 0.670206 0.742175i \(-0.266206\pi\)
0.670206 + 0.742175i \(0.266206\pi\)
\(54\) 0 0
\(55\) − 36762.6i − 0.220962i
\(56\) 0 0
\(57\) 0 0
\(58\) 83939.6 0.430212
\(59\) 387225.i 1.88542i 0.333619 + 0.942708i \(0.391730\pi\)
−0.333619 + 0.942708i \(0.608270\pi\)
\(60\) 0 0
\(61\) − 11782.0i − 0.0519073i −0.999663 0.0259536i \(-0.991738\pi\)
0.999663 0.0259536i \(-0.00826223\pi\)
\(62\) − 171479.i − 0.719508i
\(63\) 0 0
\(64\) 5575.73 0.0212697
\(65\) 57850.0 0.210651
\(66\) 0 0
\(67\) −384276. −1.27767 −0.638834 0.769344i \(-0.720583\pi\)
−0.638834 + 0.769344i \(0.720583\pi\)
\(68\) 320743.i 1.02007i
\(69\) 0 0
\(70\) 0 0
\(71\) −156126. −0.436213 −0.218107 0.975925i \(-0.569988\pi\)
−0.218107 + 0.975925i \(0.569988\pi\)
\(72\) 0 0
\(73\) − 375539.i − 0.965355i −0.875798 0.482677i \(-0.839664\pi\)
0.875798 0.482677i \(-0.160336\pi\)
\(74\) −36641.3 −0.0904223
\(75\) 0 0
\(76\) 137051.i 0.312206i
\(77\) 0 0
\(78\) 0 0
\(79\) 33183.6 0.0673042 0.0336521 0.999434i \(-0.489286\pi\)
0.0336521 + 0.999434i \(0.489286\pi\)
\(80\) − 146837.i − 0.286791i
\(81\) 0 0
\(82\) − 137644.i − 0.249640i
\(83\) 984986.i 1.72264i 0.508059 + 0.861322i \(0.330363\pi\)
−0.508059 + 0.861322i \(0.669637\pi\)
\(84\) 0 0
\(85\) −522434. −0.850697
\(86\) −545292. −0.857302
\(87\) 0 0
\(88\) −183560. −0.269358
\(89\) 262613.i 0.372517i 0.982501 + 0.186259i \(0.0596362\pi\)
−0.982501 + 0.186259i \(0.940364\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 804193. 1.03275
\(93\) 0 0
\(94\) 181332.i 0.218319i
\(95\) −223232. −0.260367
\(96\) 0 0
\(97\) 575147.i 0.630179i 0.949062 + 0.315089i \(0.102034\pi\)
−0.949062 + 0.315089i \(0.897966\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −444977. −0.444977
\(101\) 732875.i 0.711321i 0.934615 + 0.355661i \(0.115744\pi\)
−0.934615 + 0.355661i \(0.884256\pi\)
\(102\) 0 0
\(103\) 1.04871e6i 0.959715i 0.877346 + 0.479858i \(0.159312\pi\)
−0.877346 + 0.479858i \(0.840688\pi\)
\(104\) − 288852.i − 0.256788i
\(105\) 0 0
\(106\) 719700. 0.604274
\(107\) −1.17482e6 −0.959006 −0.479503 0.877540i \(-0.659183\pi\)
−0.479503 + 0.877540i \(0.659183\pi\)
\(108\) 0 0
\(109\) 734090. 0.566852 0.283426 0.958994i \(-0.408529\pi\)
0.283426 + 0.958994i \(0.408529\pi\)
\(110\) − 132584.i − 0.0996125i
\(111\) 0 0
\(112\) 0 0
\(113\) −2.04925e6 −1.42023 −0.710116 0.704085i \(-0.751357\pi\)
−0.710116 + 0.704085i \(0.751357\pi\)
\(114\) 0 0
\(115\) 1.30989e6i 0.861274i
\(116\) −1.18684e6 −0.760359
\(117\) 0 0
\(118\) 1.39653e6i 0.849969i
\(119\) 0 0
\(120\) 0 0
\(121\) −1.57566e6 −0.889418
\(122\) − 42491.7i − 0.0234005i
\(123\) 0 0
\(124\) 2.42458e6i 1.27166i
\(125\) − 2.02259e6i − 1.03556i
\(126\) 0 0
\(127\) −1.64688e6 −0.803991 −0.401995 0.915642i \(-0.631683\pi\)
−0.401995 + 0.915642i \(0.631683\pi\)
\(128\) −2.08665e6 −0.994991
\(129\) 0 0
\(130\) 208636. 0.0949641
\(131\) 2.06081e6i 0.916695i 0.888773 + 0.458348i \(0.151559\pi\)
−0.888773 + 0.458348i \(0.848441\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.38589e6 −0.575989
\(135\) 0 0
\(136\) 2.60858e6i 1.03702i
\(137\) −3.22920e6 −1.25584 −0.627919 0.778279i \(-0.716093\pi\)
−0.627919 + 0.778279i \(0.716093\pi\)
\(138\) 0 0
\(139\) − 4.06719e6i − 1.51443i −0.653163 0.757217i \(-0.726558\pi\)
0.653163 0.757217i \(-0.273442\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −563067. −0.196650
\(143\) 308274.i 0.105421i
\(144\) 0 0
\(145\) − 1.93316e6i − 0.634108i
\(146\) − 1.35438e6i − 0.435194i
\(147\) 0 0
\(148\) 518080. 0.159813
\(149\) 2.79474e6 0.844855 0.422427 0.906397i \(-0.361178\pi\)
0.422427 + 0.906397i \(0.361178\pi\)
\(150\) 0 0
\(151\) 603099. 0.175169 0.0875846 0.996157i \(-0.472085\pi\)
0.0875846 + 0.996157i \(0.472085\pi\)
\(152\) 1.11462e6i 0.317393i
\(153\) 0 0
\(154\) 0 0
\(155\) −3.94921e6 −1.06051
\(156\) 0 0
\(157\) 1.42909e6i 0.369285i 0.982806 + 0.184642i \(0.0591127\pi\)
−0.982806 + 0.184642i \(0.940887\pi\)
\(158\) 119677. 0.0303416
\(159\) 0 0
\(160\) − 2.73414e6i − 0.667514i
\(161\) 0 0
\(162\) 0 0
\(163\) −5.57288e6 −1.28682 −0.643408 0.765523i \(-0.722480\pi\)
−0.643408 + 0.765523i \(0.722480\pi\)
\(164\) 1.94617e6i 0.441215i
\(165\) 0 0
\(166\) 3.55235e6i 0.776590i
\(167\) − 2.24388e6i − 0.481781i −0.970552 0.240890i \(-0.922561\pi\)
0.970552 0.240890i \(-0.0774394\pi\)
\(168\) 0 0
\(169\) 4.34171e6 0.899498
\(170\) −1.88416e6 −0.383505
\(171\) 0 0
\(172\) 7.71002e6 1.51520
\(173\) 6.88044e6i 1.32886i 0.747352 + 0.664428i \(0.231325\pi\)
−0.747352 + 0.664428i \(0.768675\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 782471. 0.143526
\(177\) 0 0
\(178\) 947114.i 0.167935i
\(179\) 7.89287e6 1.37618 0.688091 0.725625i \(-0.258449\pi\)
0.688091 + 0.725625i \(0.258449\pi\)
\(180\) 0 0
\(181\) 2.63639e6i 0.444604i 0.974978 + 0.222302i \(0.0713571\pi\)
−0.974978 + 0.222302i \(0.928643\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.54043e6 1.04991
\(185\) 843861.i 0.133277i
\(186\) 0 0
\(187\) − 2.78397e6i − 0.425736i
\(188\) − 2.56390e6i − 0.385858i
\(189\) 0 0
\(190\) −805087. −0.117377
\(191\) −1.23424e7 −1.77133 −0.885665 0.464325i \(-0.846297\pi\)
−0.885665 + 0.464325i \(0.846297\pi\)
\(192\) 0 0
\(193\) −1.33923e7 −1.86287 −0.931435 0.363908i \(-0.881442\pi\)
−0.931435 + 0.363908i \(0.881442\pi\)
\(194\) 2.07427e6i 0.284092i
\(195\) 0 0
\(196\) 0 0
\(197\) 9.42468e6 1.23273 0.616365 0.787460i \(-0.288604\pi\)
0.616365 + 0.787460i \(0.288604\pi\)
\(198\) 0 0
\(199\) − 5.67696e6i − 0.720372i −0.932881 0.360186i \(-0.882713\pi\)
0.932881 0.360186i \(-0.117287\pi\)
\(200\) −3.61896e6 −0.452370
\(201\) 0 0
\(202\) 2.64311e6i 0.320672i
\(203\) 0 0
\(204\) 0 0
\(205\) −3.16998e6 −0.367955
\(206\) 3.78216e6i 0.432652i
\(207\) 0 0
\(208\) 1.23131e6i 0.136828i
\(209\) − 1.18957e6i − 0.130302i
\(210\) 0 0
\(211\) 1.05196e6 0.111983 0.0559914 0.998431i \(-0.482168\pi\)
0.0559914 + 0.998431i \(0.482168\pi\)
\(212\) −1.01760e7 −1.06800
\(213\) 0 0
\(214\) −4.23700e6 −0.432332
\(215\) 1.25583e7i 1.26361i
\(216\) 0 0
\(217\) 0 0
\(218\) 2.64750e6 0.255544
\(219\) 0 0
\(220\) 1.87464e6i 0.176055i
\(221\) 4.38089e6 0.405869
\(222\) 0 0
\(223\) − 9.96615e6i − 0.898696i −0.893357 0.449348i \(-0.851656\pi\)
0.893357 0.449348i \(-0.148344\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.39061e6 −0.640258
\(227\) 1.84709e7i 1.57910i 0.613684 + 0.789552i \(0.289687\pi\)
−0.613684 + 0.789552i \(0.710313\pi\)
\(228\) 0 0
\(229\) 1.36157e7i 1.13379i 0.823790 + 0.566896i \(0.191856\pi\)
−0.823790 + 0.566896i \(0.808144\pi\)
\(230\) 4.72412e6i 0.388273i
\(231\) 0 0
\(232\) −9.65248e6 −0.772992
\(233\) −10206.0 −0.000806843 0 −0.000403422 1.00000i \(-0.500128\pi\)
−0.000403422 1.00000i \(0.500128\pi\)
\(234\) 0 0
\(235\) 4.17614e6 0.321789
\(236\) − 1.97458e7i − 1.50224i
\(237\) 0 0
\(238\) 0 0
\(239\) −2.49567e6 −0.182807 −0.0914037 0.995814i \(-0.529135\pi\)
−0.0914037 + 0.995814i \(0.529135\pi\)
\(240\) 0 0
\(241\) 5.47041e6i 0.390813i 0.980722 + 0.195407i \(0.0626026\pi\)
−0.980722 + 0.195407i \(0.937397\pi\)
\(242\) −5.68261e6 −0.400961
\(243\) 0 0
\(244\) 600800.i 0.0413581i
\(245\) 0 0
\(246\) 0 0
\(247\) 1.87192e6 0.124222
\(248\) 1.97189e7i 1.29279i
\(249\) 0 0
\(250\) − 7.29446e6i − 0.466845i
\(251\) − 2.18424e6i − 0.138127i −0.997612 0.0690637i \(-0.977999\pi\)
0.997612 0.0690637i \(-0.0220012\pi\)
\(252\) 0 0
\(253\) −6.98020e6 −0.431029
\(254\) −5.93948e6 −0.362449
\(255\) 0 0
\(256\) −7.88234e6 −0.469824
\(257\) 3.14815e7i 1.85463i 0.374287 + 0.927313i \(0.377887\pi\)
−0.374287 + 0.927313i \(0.622113\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.94996e6 −0.167840
\(261\) 0 0
\(262\) 7.43233e6i 0.413258i
\(263\) −6.57763e6 −0.361578 −0.180789 0.983522i \(-0.557865\pi\)
−0.180789 + 0.983522i \(0.557865\pi\)
\(264\) 0 0
\(265\) − 1.65749e7i − 0.890665i
\(266\) 0 0
\(267\) 0 0
\(268\) 1.95954e7 1.01801
\(269\) 219878.i 0.0112960i 0.999984 + 0.00564801i \(0.00179783\pi\)
−0.999984 + 0.00564801i \(0.998202\pi\)
\(270\) 0 0
\(271\) − 2.78378e7i − 1.39871i −0.714776 0.699354i \(-0.753471\pi\)
0.714776 0.699354i \(-0.246529\pi\)
\(272\) − 1.11197e7i − 0.552570i
\(273\) 0 0
\(274\) −1.16461e7 −0.566147
\(275\) 3.86229e6 0.185715
\(276\) 0 0
\(277\) −5.75450e6 −0.270750 −0.135375 0.990794i \(-0.543224\pi\)
−0.135375 + 0.990794i \(0.543224\pi\)
\(278\) − 1.46683e7i − 0.682726i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.75833e7 −0.792467 −0.396233 0.918150i \(-0.629683\pi\)
−0.396233 + 0.918150i \(0.629683\pi\)
\(282\) 0 0
\(283\) − 1.47184e7i − 0.649386i −0.945820 0.324693i \(-0.894739\pi\)
0.945820 0.324693i \(-0.105261\pi\)
\(284\) 7.96134e6 0.347561
\(285\) 0 0
\(286\) 1.11179e6i 0.0475253i
\(287\) 0 0
\(288\) 0 0
\(289\) −1.54256e7 −0.639070
\(290\) − 6.97193e6i − 0.285864i
\(291\) 0 0
\(292\) 1.91499e7i 0.769164i
\(293\) 3.49743e6i 0.139042i 0.997580 + 0.0695210i \(0.0221471\pi\)
−0.997580 + 0.0695210i \(0.977853\pi\)
\(294\) 0 0
\(295\) 3.21625e7 1.25280
\(296\) 4.21350e6 0.162468
\(297\) 0 0
\(298\) 1.00792e7 0.380871
\(299\) − 1.09841e7i − 0.410915i
\(300\) 0 0
\(301\) 0 0
\(302\) 2.17508e6 0.0789685
\(303\) 0 0
\(304\) − 4.75138e6i − 0.169121i
\(305\) −978597. −0.0344909
\(306\) 0 0
\(307\) 3.99694e7i 1.38138i 0.723152 + 0.690689i \(0.242692\pi\)
−0.723152 + 0.690689i \(0.757308\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.42428e7 −0.478092
\(311\) − 3.33495e7i − 1.10868i −0.832289 0.554342i \(-0.812970\pi\)
0.832289 0.554342i \(-0.187030\pi\)
\(312\) 0 0
\(313\) 1.87899e7i 0.612761i 0.951909 + 0.306381i \(0.0991180\pi\)
−0.951909 + 0.306381i \(0.900882\pi\)
\(314\) 5.15402e6i 0.166478i
\(315\) 0 0
\(316\) −1.69214e6 −0.0536259
\(317\) 2.03972e7 0.640313 0.320156 0.947365i \(-0.396265\pi\)
0.320156 + 0.947365i \(0.396265\pi\)
\(318\) 0 0
\(319\) 1.03015e7 0.317343
\(320\) − 463114.i − 0.0141331i
\(321\) 0 0
\(322\) 0 0
\(323\) −1.69050e7 −0.501659
\(324\) 0 0
\(325\) 6.07776e6i 0.177049i
\(326\) −2.00986e7 −0.580113
\(327\) 0 0
\(328\) 1.58281e7i 0.448545i
\(329\) 0 0
\(330\) 0 0
\(331\) −2.13549e7 −0.588862 −0.294431 0.955673i \(-0.595130\pi\)
−0.294431 + 0.955673i \(0.595130\pi\)
\(332\) − 5.02275e7i − 1.37255i
\(333\) 0 0
\(334\) − 8.09254e6i − 0.217193i
\(335\) 3.19175e7i 0.848974i
\(336\) 0 0
\(337\) −1.98824e7 −0.519493 −0.259746 0.965677i \(-0.583639\pi\)
−0.259746 + 0.965677i \(0.583639\pi\)
\(338\) 1.56584e7 0.405505
\(339\) 0 0
\(340\) 2.66406e7 0.677808
\(341\) − 2.10448e7i − 0.530739i
\(342\) 0 0
\(343\) 0 0
\(344\) 6.27049e7 1.54037
\(345\) 0 0
\(346\) 2.48143e7i 0.599065i
\(347\) 3.32395e7 0.795548 0.397774 0.917483i \(-0.369783\pi\)
0.397774 + 0.917483i \(0.369783\pi\)
\(348\) 0 0
\(349\) − 2.57850e7i − 0.606584i −0.952898 0.303292i \(-0.901914\pi\)
0.952898 0.303292i \(-0.0980858\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.45698e7 0.334061
\(353\) − 5.56607e7i − 1.26539i −0.774401 0.632695i \(-0.781949\pi\)
0.774401 0.632695i \(-0.218051\pi\)
\(354\) 0 0
\(355\) 1.29676e7i 0.289851i
\(356\) − 1.33915e7i − 0.296810i
\(357\) 0 0
\(358\) 2.84656e7 0.620400
\(359\) 1.61743e7 0.349576 0.174788 0.984606i \(-0.444076\pi\)
0.174788 + 0.984606i \(0.444076\pi\)
\(360\) 0 0
\(361\) 3.98225e7 0.846461
\(362\) 9.50813e6i 0.200433i
\(363\) 0 0
\(364\) 0 0
\(365\) −3.11919e7 −0.641451
\(366\) 0 0
\(367\) − 1.12274e7i − 0.227133i −0.993530 0.113566i \(-0.963773\pi\)
0.993530 0.113566i \(-0.0362274\pi\)
\(368\) −2.78803e7 −0.559441
\(369\) 0 0
\(370\) 3.04339e6i 0.0600830i
\(371\) 0 0
\(372\) 0 0
\(373\) 3.99976e7 0.770739 0.385369 0.922762i \(-0.374074\pi\)
0.385369 + 0.922762i \(0.374074\pi\)
\(374\) − 1.00404e7i − 0.191927i
\(375\) 0 0
\(376\) − 2.08519e7i − 0.392268i
\(377\) 1.62106e7i 0.302534i
\(378\) 0 0
\(379\) 2.92098e7 0.536551 0.268276 0.963342i \(-0.413546\pi\)
0.268276 + 0.963342i \(0.413546\pi\)
\(380\) 1.13833e7 0.207452
\(381\) 0 0
\(382\) −4.45129e7 −0.798538
\(383\) − 4.58625e7i − 0.816321i −0.912910 0.408161i \(-0.866170\pi\)
0.912910 0.408161i \(-0.133830\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.82993e7 −0.839805
\(387\) 0 0
\(388\) − 2.93286e7i − 0.502106i
\(389\) −1.54486e7 −0.262446 −0.131223 0.991353i \(-0.541890\pi\)
−0.131223 + 0.991353i \(0.541890\pi\)
\(390\) 0 0
\(391\) 9.91960e7i 1.65945i
\(392\) 0 0
\(393\) 0 0
\(394\) 3.39901e7 0.555730
\(395\) − 2.75619e6i − 0.0447217i
\(396\) 0 0
\(397\) 2.94145e7i 0.470100i 0.971983 + 0.235050i \(0.0755253\pi\)
−0.971983 + 0.235050i \(0.924475\pi\)
\(398\) − 2.04740e7i − 0.324753i
\(399\) 0 0
\(400\) 1.54268e7 0.241043
\(401\) 1.14011e8 1.76813 0.884064 0.467366i \(-0.154797\pi\)
0.884064 + 0.467366i \(0.154797\pi\)
\(402\) 0 0
\(403\) 3.31163e7 0.505972
\(404\) − 3.73716e7i − 0.566758i
\(405\) 0 0
\(406\) 0 0
\(407\) −4.49681e6 −0.0666992
\(408\) 0 0
\(409\) − 3.45589e7i − 0.505114i −0.967582 0.252557i \(-0.918728\pi\)
0.967582 0.252557i \(-0.0812715\pi\)
\(410\) −1.14325e7 −0.165879
\(411\) 0 0
\(412\) − 5.34769e7i − 0.764671i
\(413\) 0 0
\(414\) 0 0
\(415\) 8.18118e7 1.14465
\(416\) 2.29272e7i 0.318472i
\(417\) 0 0
\(418\) − 4.29018e6i − 0.0587418i
\(419\) 3.73902e7i 0.508294i 0.967166 + 0.254147i \(0.0817948\pi\)
−0.967166 + 0.254147i \(0.918205\pi\)
\(420\) 0 0
\(421\) −5.93910e7 −0.795929 −0.397965 0.917401i \(-0.630283\pi\)
−0.397965 + 0.917401i \(0.630283\pi\)
\(422\) 3.79389e6 0.0504832
\(423\) 0 0
\(424\) −8.27606e7 −1.08574
\(425\) − 5.48872e7i − 0.714998i
\(426\) 0 0
\(427\) 0 0
\(428\) 5.99080e7 0.764105
\(429\) 0 0
\(430\) 4.52914e7i 0.569653i
\(431\) 5.95395e7 0.743658 0.371829 0.928301i \(-0.378731\pi\)
0.371829 + 0.928301i \(0.378731\pi\)
\(432\) 0 0
\(433\) − 6.40730e7i − 0.789244i −0.918843 0.394622i \(-0.870876\pi\)
0.918843 0.394622i \(-0.129124\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.74336e7 −0.451650
\(437\) 4.23857e7i 0.507896i
\(438\) 0 0
\(439\) − 1.52164e8i − 1.79853i −0.437405 0.899265i \(-0.644102\pi\)
0.437405 0.899265i \(-0.355898\pi\)
\(440\) 1.52463e7i 0.178980i
\(441\) 0 0
\(442\) 1.57997e7 0.182971
\(443\) −3.45839e7 −0.397799 −0.198899 0.980020i \(-0.563737\pi\)
−0.198899 + 0.980020i \(0.563737\pi\)
\(444\) 0 0
\(445\) 2.18124e7 0.247527
\(446\) − 3.59429e7i − 0.405143i
\(447\) 0 0
\(448\) 0 0
\(449\) 1.33795e8 1.47809 0.739047 0.673654i \(-0.235276\pi\)
0.739047 + 0.673654i \(0.235276\pi\)
\(450\) 0 0
\(451\) − 1.68923e7i − 0.184145i
\(452\) 1.04498e8 1.13160
\(453\) 0 0
\(454\) 6.66153e7i 0.711880i
\(455\) 0 0
\(456\) 0 0
\(457\) 1.52528e7 0.159809 0.0799043 0.996803i \(-0.474539\pi\)
0.0799043 + 0.996803i \(0.474539\pi\)
\(458\) 4.91050e7i 0.511127i
\(459\) 0 0
\(460\) − 6.67954e7i − 0.686236i
\(461\) 3.05013e7i 0.311326i 0.987810 + 0.155663i \(0.0497514\pi\)
−0.987810 + 0.155663i \(0.950249\pi\)
\(462\) 0 0
\(463\) −1.61661e8 −1.62878 −0.814388 0.580320i \(-0.802927\pi\)
−0.814388 + 0.580320i \(0.802927\pi\)
\(464\) 4.11462e7 0.411885
\(465\) 0 0
\(466\) −36808.1 −0.000363735 0
\(467\) − 6.59141e7i − 0.647184i −0.946197 0.323592i \(-0.895110\pi\)
0.946197 0.323592i \(-0.104890\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.50612e7 0.145067
\(471\) 0 0
\(472\) − 1.60591e8i − 1.52720i
\(473\) −6.69211e7 −0.632382
\(474\) 0 0
\(475\) − 2.34529e7i − 0.218834i
\(476\) 0 0
\(477\) 0 0
\(478\) −9.00064e6 −0.0824119
\(479\) − 9.19759e7i − 0.836889i −0.908242 0.418444i \(-0.862576\pi\)
0.908242 0.418444i \(-0.137424\pi\)
\(480\) 0 0
\(481\) − 7.07623e6i − 0.0635868i
\(482\) 1.97291e7i 0.176183i
\(483\) 0 0
\(484\) 8.03478e7 0.708660
\(485\) 4.77711e7 0.418736
\(486\) 0 0
\(487\) −1.90753e8 −1.65152 −0.825759 0.564023i \(-0.809253\pi\)
−0.825759 + 0.564023i \(0.809253\pi\)
\(488\) 4.88625e6i 0.0420452i
\(489\) 0 0
\(490\) 0 0
\(491\) −8.85504e7 −0.748077 −0.374038 0.927413i \(-0.622027\pi\)
−0.374038 + 0.927413i \(0.622027\pi\)
\(492\) 0 0
\(493\) − 1.46395e8i − 1.22176i
\(494\) 6.75109e6 0.0560006
\(495\) 0 0
\(496\) − 8.40569e7i − 0.688856i
\(497\) 0 0
\(498\) 0 0
\(499\) −5.23508e7 −0.421330 −0.210665 0.977558i \(-0.567563\pi\)
−0.210665 + 0.977558i \(0.567563\pi\)
\(500\) 1.03138e8i 0.825104i
\(501\) 0 0
\(502\) − 7.87748e6i − 0.0622696i
\(503\) 1.34114e8i 1.05383i 0.849919 + 0.526914i \(0.176651\pi\)
−0.849919 + 0.526914i \(0.823349\pi\)
\(504\) 0 0
\(505\) 6.08718e7 0.472653
\(506\) −2.51741e7 −0.194313
\(507\) 0 0
\(508\) 8.39797e7 0.640594
\(509\) 1.87318e8i 1.42045i 0.703976 + 0.710224i \(0.251406\pi\)
−0.703976 + 0.710224i \(0.748594\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.05118e8 0.783188
\(513\) 0 0
\(514\) 1.13538e8i 0.836088i
\(515\) 8.71045e7 0.637703
\(516\) 0 0
\(517\) 2.22540e7i 0.161041i
\(518\) 0 0
\(519\) 0 0
\(520\) −2.39917e7 −0.170628
\(521\) − 4.49210e7i − 0.317641i −0.987307 0.158820i \(-0.949231\pi\)
0.987307 0.158820i \(-0.0507691\pi\)
\(522\) 0 0
\(523\) 1.65990e8i 1.16032i 0.814503 + 0.580160i \(0.197010\pi\)
−0.814503 + 0.580160i \(0.802990\pi\)
\(524\) − 1.05087e8i − 0.730394i
\(525\) 0 0
\(526\) −2.37222e7 −0.163004
\(527\) −2.99068e8 −2.04333
\(528\) 0 0
\(529\) 1.00676e8 0.680081
\(530\) − 5.97775e7i − 0.401523i
\(531\) 0 0
\(532\) 0 0
\(533\) 2.65820e7 0.175552
\(534\) 0 0
\(535\) 9.75796e7i 0.637232i
\(536\) 1.59368e8 1.03492
\(537\) 0 0
\(538\) 792991.i 0.00509238i
\(539\) 0 0
\(540\) 0 0
\(541\) 1.26303e8 0.797666 0.398833 0.917024i \(-0.369415\pi\)
0.398833 + 0.917024i \(0.369415\pi\)
\(542\) − 1.00397e8i − 0.630555i
\(543\) 0 0
\(544\) − 2.07052e8i − 1.28612i
\(545\) − 6.09727e7i − 0.376657i
\(546\) 0 0
\(547\) 2.31931e8 1.41709 0.708545 0.705666i \(-0.249352\pi\)
0.708545 + 0.705666i \(0.249352\pi\)
\(548\) 1.64667e8 1.00061
\(549\) 0 0
\(550\) 1.39294e7 0.0837227
\(551\) − 6.25535e7i − 0.373935i
\(552\) 0 0
\(553\) 0 0
\(554\) −2.07536e7 −0.122057
\(555\) 0 0
\(556\) 2.07399e8i 1.20665i
\(557\) −2.41077e8 −1.39505 −0.697525 0.716560i \(-0.745715\pi\)
−0.697525 + 0.716560i \(0.745715\pi\)
\(558\) 0 0
\(559\) − 1.05308e8i − 0.602872i
\(560\) 0 0
\(561\) 0 0
\(562\) −6.34141e7 −0.357254
\(563\) − 1.17176e8i − 0.656617i −0.944571 0.328308i \(-0.893521\pi\)
0.944571 0.328308i \(-0.106479\pi\)
\(564\) 0 0
\(565\) 1.70208e8i 0.943703i
\(566\) − 5.30821e7i − 0.292751i
\(567\) 0 0
\(568\) 6.47488e7 0.353335
\(569\) 1.20860e8 0.656066 0.328033 0.944666i \(-0.393614\pi\)
0.328033 + 0.944666i \(0.393614\pi\)
\(570\) 0 0
\(571\) 1.07812e8 0.579109 0.289555 0.957161i \(-0.406493\pi\)
0.289555 + 0.957161i \(0.406493\pi\)
\(572\) − 1.57199e7i − 0.0839964i
\(573\) 0 0
\(574\) 0 0
\(575\) −1.37618e8 −0.723887
\(576\) 0 0
\(577\) 7.84554e7i 0.408409i 0.978928 + 0.204205i \(0.0654608\pi\)
−0.978928 + 0.204205i \(0.934539\pi\)
\(578\) −5.56324e7 −0.288101
\(579\) 0 0
\(580\) 9.85778e7i 0.505237i
\(581\) 0 0
\(582\) 0 0
\(583\) 8.83253e7 0.445738
\(584\) 1.55745e8i 0.781943i
\(585\) 0 0
\(586\) 1.26135e7i 0.0626819i
\(587\) − 1.98030e8i − 0.979075i −0.871982 0.489538i \(-0.837166\pi\)
0.871982 0.489538i \(-0.162834\pi\)
\(588\) 0 0
\(589\) −1.27789e8 −0.625387
\(590\) 1.15994e8 0.564780
\(591\) 0 0
\(592\) −1.79611e7 −0.0865702
\(593\) 3.77084e6i 0.0180831i 0.999959 + 0.00904157i \(0.00287806\pi\)
−0.999959 + 0.00904157i \(0.997122\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.42512e8 −0.673153
\(597\) 0 0
\(598\) − 3.96143e7i − 0.185246i
\(599\) −2.67305e8 −1.24373 −0.621867 0.783123i \(-0.713625\pi\)
−0.621867 + 0.783123i \(0.713625\pi\)
\(600\) 0 0
\(601\) 9.26866e7i 0.426966i 0.976947 + 0.213483i \(0.0684809\pi\)
−0.976947 + 0.213483i \(0.931519\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3.07539e7 −0.139569
\(605\) 1.30873e8i 0.590993i
\(606\) 0 0
\(607\) 1.60412e8i 0.717251i 0.933482 + 0.358625i \(0.116754\pi\)
−0.933482 + 0.358625i \(0.883246\pi\)
\(608\) − 8.84718e7i − 0.393635i
\(609\) 0 0
\(610\) −3.52931e6 −0.0155489
\(611\) −3.50192e7 −0.153526
\(612\) 0 0
\(613\) 7.21961e7 0.313424 0.156712 0.987644i \(-0.449911\pi\)
0.156712 + 0.987644i \(0.449911\pi\)
\(614\) 1.44150e8i 0.622742i
\(615\) 0 0
\(616\) 0 0
\(617\) −3.16720e6 −0.0134841 −0.00674203 0.999977i \(-0.502146\pi\)
−0.00674203 + 0.999977i \(0.502146\pi\)
\(618\) 0 0
\(619\) − 1.38699e8i − 0.584791i −0.956297 0.292396i \(-0.905548\pi\)
0.956297 0.292396i \(-0.0944524\pi\)
\(620\) 2.01383e8 0.844982
\(621\) 0 0
\(622\) − 1.20275e8i − 0.499809i
\(623\) 0 0
\(624\) 0 0
\(625\) −3.16467e7 −0.129625
\(626\) 6.77658e7i 0.276240i
\(627\) 0 0
\(628\) − 7.28739e7i − 0.294234i
\(629\) 6.39043e7i 0.256790i
\(630\) 0 0
\(631\) −1.98716e8 −0.790944 −0.395472 0.918478i \(-0.629419\pi\)
−0.395472 + 0.918478i \(0.629419\pi\)
\(632\) −1.37620e7 −0.0545168
\(633\) 0 0
\(634\) 7.35624e7 0.288661
\(635\) 1.36788e8i 0.534229i
\(636\) 0 0
\(637\) 0 0
\(638\) 3.71523e7 0.143062
\(639\) 0 0
\(640\) 1.73315e8i 0.661143i
\(641\) −1.44347e8 −0.548068 −0.274034 0.961720i \(-0.588358\pi\)
−0.274034 + 0.961720i \(0.588358\pi\)
\(642\) 0 0
\(643\) 2.68271e8i 1.00911i 0.863378 + 0.504557i \(0.168344\pi\)
−0.863378 + 0.504557i \(0.831656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.09680e7 −0.226154
\(647\) − 3.91410e8i − 1.44517i −0.691282 0.722585i \(-0.742954\pi\)
0.691282 0.722585i \(-0.257046\pi\)
\(648\) 0 0
\(649\) 1.71389e8i 0.626973i
\(650\) 2.19194e7i 0.0798158i
\(651\) 0 0
\(652\) 2.84179e8 1.02529
\(653\) −4.61183e8 −1.65628 −0.828139 0.560522i \(-0.810600\pi\)
−0.828139 + 0.560522i \(0.810600\pi\)
\(654\) 0 0
\(655\) 1.71169e8 0.609118
\(656\) − 6.74712e7i − 0.239005i
\(657\) 0 0
\(658\) 0 0
\(659\) −2.90612e8 −1.01545 −0.507724 0.861520i \(-0.669513\pi\)
−0.507724 + 0.861520i \(0.669513\pi\)
\(660\) 0 0
\(661\) 2.80781e8i 0.972218i 0.873898 + 0.486109i \(0.161584\pi\)
−0.873898 + 0.486109i \(0.838416\pi\)
\(662\) −7.70164e7 −0.265466
\(663\) 0 0
\(664\) − 4.08496e8i − 1.39535i
\(665\) 0 0
\(666\) 0 0
\(667\) −3.67054e8 −1.23695
\(668\) 1.14422e8i 0.383867i
\(669\) 0 0
\(670\) 1.15111e8i 0.382728i
\(671\) − 5.21480e6i − 0.0172611i
\(672\) 0 0
\(673\) −2.35293e8 −0.771904 −0.385952 0.922519i \(-0.626127\pi\)
−0.385952 + 0.922519i \(0.626127\pi\)
\(674\) −7.17060e7 −0.234194
\(675\) 0 0
\(676\) −2.21397e8 −0.716691
\(677\) 6.18148e8i 1.99217i 0.0884002 + 0.996085i \(0.471825\pi\)
−0.0884002 + 0.996085i \(0.528175\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.16665e8 0.689069
\(681\) 0 0
\(682\) − 7.58979e7i − 0.239264i
\(683\) −1.30744e7 −0.0410356 −0.0205178 0.999789i \(-0.506531\pi\)
−0.0205178 + 0.999789i \(0.506531\pi\)
\(684\) 0 0
\(685\) 2.68214e8i 0.834468i
\(686\) 0 0
\(687\) 0 0
\(688\) −2.67296e8 −0.820780
\(689\) 1.38990e8i 0.424938i
\(690\) 0 0
\(691\) 2.74064e8i 0.830648i 0.909674 + 0.415324i \(0.136332\pi\)
−0.909674 + 0.415324i \(0.863668\pi\)
\(692\) − 3.50855e8i − 1.05879i
\(693\) 0 0
\(694\) 1.19878e8 0.358643
\(695\) −3.37817e8 −1.00630
\(696\) 0 0
\(697\) −2.40058e8 −0.708952
\(698\) − 9.29937e7i − 0.273456i
\(699\) 0 0
\(700\) 0 0
\(701\) −4.35624e8 −1.26461 −0.632307 0.774718i \(-0.717892\pi\)
−0.632307 + 0.774718i \(0.717892\pi\)
\(702\) 0 0
\(703\) 2.73058e7i 0.0785940i
\(704\) 2.46786e6 0.00707299
\(705\) 0 0
\(706\) − 2.00740e8i − 0.570454i
\(707\) 0 0
\(708\) 0 0
\(709\) −3.61813e8 −1.01518 −0.507592 0.861597i \(-0.669464\pi\)
−0.507592 + 0.861597i \(0.669464\pi\)
\(710\) 4.67677e7i 0.130669i
\(711\) 0 0
\(712\) − 1.08912e8i − 0.301741i
\(713\) 7.49848e8i 2.06873i
\(714\) 0 0
\(715\) 2.56049e7 0.0700495
\(716\) −4.02482e8 −1.09650
\(717\) 0 0
\(718\) 5.83325e7 0.157593
\(719\) 2.45377e8i 0.660157i 0.943953 + 0.330079i \(0.107075\pi\)
−0.943953 + 0.330079i \(0.892925\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.43620e8 0.381595
\(723\) 0 0
\(724\) − 1.34438e8i − 0.354246i
\(725\) 2.03099e8 0.532958
\(726\) 0 0
\(727\) 2.59949e8i 0.676526i 0.941052 + 0.338263i \(0.109839\pi\)
−0.941052 + 0.338263i \(0.890161\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.12494e8 −0.289174
\(731\) 9.51018e8i 2.43465i
\(732\) 0 0
\(733\) − 1.25030e8i − 0.317471i −0.987321 0.158735i \(-0.949258\pi\)
0.987321 0.158735i \(-0.0507417\pi\)
\(734\) − 4.04915e7i − 0.102394i
\(735\) 0 0
\(736\) −5.19138e8 −1.30212
\(737\) −1.70084e8 −0.424874
\(738\) 0 0
\(739\) −5.64721e8 −1.39927 −0.699634 0.714502i \(-0.746654\pi\)
−0.699634 + 0.714502i \(0.746654\pi\)
\(740\) − 4.30311e7i − 0.106191i
\(741\) 0 0
\(742\) 0 0
\(743\) −2.68677e8 −0.655035 −0.327518 0.944845i \(-0.606212\pi\)
−0.327518 + 0.944845i \(0.606212\pi\)
\(744\) 0 0
\(745\) − 2.32128e8i − 0.561382i
\(746\) 1.44251e8 0.347459
\(747\) 0 0
\(748\) 1.41964e8i 0.339213i
\(749\) 0 0
\(750\) 0 0
\(751\) 3.90960e8 0.923023 0.461512 0.887134i \(-0.347307\pi\)
0.461512 + 0.887134i \(0.347307\pi\)
\(752\) 8.88868e7i 0.209018i
\(753\) 0 0
\(754\) 5.84634e7i 0.136386i
\(755\) − 5.00927e7i − 0.116395i
\(756\) 0 0
\(757\) −5.67872e8 −1.30907 −0.654535 0.756031i \(-0.727136\pi\)
−0.654535 + 0.756031i \(0.727136\pi\)
\(758\) 1.05345e8 0.241884
\(759\) 0 0
\(760\) 9.25795e7 0.210899
\(761\) 1.92045e8i 0.435762i 0.975975 + 0.217881i \(0.0699144\pi\)
−0.975975 + 0.217881i \(0.930086\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 6.29378e8 1.41134
\(765\) 0 0
\(766\) − 1.65403e8i − 0.368008i
\(767\) −2.69700e8 −0.597715
\(768\) 0 0
\(769\) − 7.75766e8i − 1.70589i −0.521999 0.852946i \(-0.674813\pi\)
0.521999 0.852946i \(-0.325187\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.82914e8 1.48428
\(773\) 2.70408e7i 0.0585438i 0.999571 + 0.0292719i \(0.00931887\pi\)
−0.999571 + 0.0292719i \(0.990681\pi\)
\(774\) 0 0
\(775\) − 4.14907e8i − 0.891344i
\(776\) − 2.38527e8i − 0.510448i
\(777\) 0 0
\(778\) −5.57154e7 −0.118314
\(779\) −1.02575e8 −0.216984
\(780\) 0 0
\(781\) −6.91025e7 −0.145058
\(782\) 3.57750e8i 0.748100i
\(783\) 0 0
\(784\) 0 0
\(785\) 1.18699e8 0.245379
\(786\) 0 0
\(787\) 4.25592e8i 0.873111i 0.899677 + 0.436555i \(0.143802\pi\)
−0.899677 + 0.436555i \(0.856198\pi\)
\(788\) −4.80594e8 −0.982200
\(789\) 0 0
\(790\) − 9.94022e6i − 0.0201611i
\(791\) 0 0
\(792\) 0 0
\(793\) 8.20607e6 0.0164557
\(794\) 1.06083e8i 0.211927i
\(795\) 0 0
\(796\) 2.89486e8i 0.573969i
\(797\) − 7.51386e7i − 0.148419i −0.997243 0.0742093i \(-0.976357\pi\)
0.997243 0.0742093i \(-0.0236433\pi\)
\(798\) 0 0
\(799\) 3.16253e8 0.620003
\(800\) 2.87250e8 0.561035
\(801\) 0 0
\(802\) 4.11181e8 0.797094
\(803\) − 1.66217e8i − 0.321017i
\(804\) 0 0
\(805\) 0 0
\(806\) 1.19434e8 0.228099
\(807\) 0 0
\(808\) − 3.03940e8i − 0.576174i
\(809\) 7.82317e8 1.47753 0.738766 0.673962i \(-0.235409\pi\)
0.738766 + 0.673962i \(0.235409\pi\)
\(810\) 0 0
\(811\) 8.00635e8i 1.50097i 0.660888 + 0.750485i \(0.270180\pi\)
−0.660888 + 0.750485i \(0.729820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.62177e7 −0.0300689
\(815\) 4.62877e8i 0.855053i
\(816\) 0 0
\(817\) 4.06363e8i 0.745157i
\(818\) − 1.24637e8i − 0.227712i
\(819\) 0 0
\(820\) 1.61647e8 0.293175
\(821\) 2.75662e8 0.498135 0.249067 0.968486i \(-0.419876\pi\)
0.249067 + 0.968486i \(0.419876\pi\)
\(822\) 0 0
\(823\) −7.57904e8 −1.35961 −0.679806 0.733392i \(-0.737936\pi\)
−0.679806 + 0.733392i \(0.737936\pi\)
\(824\) − 4.34923e8i − 0.777375i
\(825\) 0 0
\(826\) 0 0
\(827\) −3.06595e8 −0.542061 −0.271030 0.962571i \(-0.587364\pi\)
−0.271030 + 0.962571i \(0.587364\pi\)
\(828\) 0 0
\(829\) − 1.23659e8i − 0.217052i −0.994094 0.108526i \(-0.965387\pi\)
0.994094 0.108526i \(-0.0346130\pi\)
\(830\) 2.95054e8 0.516022
\(831\) 0 0
\(832\) 3.88346e6i 0.00674293i
\(833\) 0 0
\(834\) 0 0
\(835\) −1.86374e8 −0.320129
\(836\) 6.06599e7i 0.103820i
\(837\) 0 0
\(838\) 1.34848e8i 0.229145i
\(839\) − 2.04720e8i − 0.346637i −0.984866 0.173318i \(-0.944551\pi\)
0.984866 0.173318i \(-0.0554489\pi\)
\(840\) 0 0
\(841\) −5.31198e7 −0.0893035
\(842\) −2.14194e8 −0.358815
\(843\) 0 0
\(844\) −5.36427e7 −0.0892243
\(845\) − 3.60617e8i − 0.597691i
\(846\) 0 0
\(847\) 0 0
\(848\) 3.52789e8 0.578532
\(849\) 0 0
\(850\) − 1.97951e8i − 0.322330i
\(851\) 1.60226e8 0.259983
\(852\) 0 0
\(853\) − 1.81851e8i − 0.293001i −0.989211 0.146500i \(-0.953199\pi\)
0.989211 0.146500i \(-0.0468010\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4.87226e8 0.776800
\(857\) − 6.14504e8i − 0.976298i −0.872760 0.488149i \(-0.837672\pi\)
0.872760 0.488149i \(-0.162328\pi\)
\(858\) 0 0
\(859\) − 2.18853e8i − 0.345281i −0.984985 0.172640i \(-0.944770\pi\)
0.984985 0.172640i \(-0.0552299\pi\)
\(860\) − 6.40386e8i − 1.00681i
\(861\) 0 0
\(862\) 2.14729e8 0.335250
\(863\) 6.59431e8 1.02598 0.512988 0.858396i \(-0.328539\pi\)
0.512988 + 0.858396i \(0.328539\pi\)
\(864\) 0 0
\(865\) 5.71482e8 0.882987
\(866\) − 2.31079e8i − 0.355801i
\(867\) 0 0
\(868\) 0 0
\(869\) 1.46873e7 0.0223812
\(870\) 0 0
\(871\) − 2.67646e8i − 0.405047i
\(872\) −3.04444e8 −0.459153
\(873\) 0 0
\(874\) 1.52864e8i 0.228966i
\(875\) 0 0
\(876\) 0 0
\(877\) 1.62366e8 0.240711 0.120355 0.992731i \(-0.461597\pi\)
0.120355 + 0.992731i \(0.461597\pi\)
\(878\) − 5.48778e8i − 0.810800i
\(879\) 0 0
\(880\) − 6.49912e7i − 0.0953688i
\(881\) 8.48365e8i 1.24067i 0.784338 + 0.620333i \(0.213003\pi\)
−0.784338 + 0.620333i \(0.786997\pi\)
\(882\) 0 0
\(883\) 3.14592e8 0.456947 0.228473 0.973550i \(-0.426627\pi\)
0.228473 + 0.973550i \(0.426627\pi\)
\(884\) −2.23396e8 −0.323384
\(885\) 0 0
\(886\) −1.24727e8 −0.179333
\(887\) 8.37224e8i 1.19970i 0.800114 + 0.599848i \(0.204772\pi\)
−0.800114 + 0.599848i \(0.795228\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.86663e7 0.111588
\(891\) 0 0
\(892\) 5.08205e8i 0.716052i
\(893\) 1.35132e8 0.189760
\(894\) 0 0
\(895\) − 6.55573e8i − 0.914433i
\(896\) 0 0
\(897\) 0 0
\(898\) 4.82533e8 0.666344
\(899\) − 1.10664e9i − 1.52309i
\(900\) 0 0
\(901\) − 1.25520e9i − 1.71608i
\(902\) − 6.09222e7i − 0.0830148i
\(903\) 0 0
\(904\) 8.49870e8 1.15040
\(905\) 2.18975e8 0.295427
\(906\) 0 0
\(907\) 6.88047e8 0.922139 0.461070 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(908\) − 9.41890e8i − 1.25818i
\(909\) 0 0
\(910\) 0 0
\(911\) −2.31613e8 −0.306343 −0.153171 0.988200i \(-0.548949\pi\)
−0.153171 + 0.988200i \(0.548949\pi\)
\(912\) 0 0
\(913\) 4.35963e8i 0.572845i
\(914\) 5.50091e7 0.0720437
\(915\) 0 0
\(916\) − 6.94307e8i − 0.903369i
\(917\) 0 0
\(918\) 0 0
\(919\) −1.12516e9 −1.44967 −0.724835 0.688922i \(-0.758084\pi\)
−0.724835 + 0.688922i \(0.758084\pi\)
\(920\) − 5.43241e8i − 0.697637i
\(921\) 0 0
\(922\) 1.10003e8i 0.140350i
\(923\) − 1.08740e8i − 0.138289i
\(924\) 0 0
\(925\) −8.86565e7 −0.112017
\(926\) −5.83029e8 −0.734273
\(927\) 0 0
\(928\) 7.66152e8 0.958675
\(929\) 6.85223e8i 0.854644i 0.904100 + 0.427322i \(0.140543\pi\)
−0.904100 + 0.427322i \(0.859457\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 520438. 0.000642867 0
\(933\) 0 0
\(934\) − 2.37719e8i − 0.291759i
\(935\) −2.31234e8 −0.282889
\(936\) 0 0
\(937\) 7.87984e8i 0.957852i 0.877855 + 0.478926i \(0.158974\pi\)
−0.877855 + 0.478926i \(0.841026\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.12954e8 −0.256391
\(941\) 2.07328e8i 0.248822i 0.992231 + 0.124411i \(0.0397041\pi\)
−0.992231 + 0.124411i \(0.960296\pi\)
\(942\) 0 0
\(943\) 6.01892e8i 0.717767i
\(944\) 6.84561e8i 0.813759i
\(945\) 0 0
\(946\) −2.41351e8 −0.285086
\(947\) 3.54059e8 0.416894 0.208447 0.978034i \(-0.433159\pi\)
0.208447 + 0.978034i \(0.433159\pi\)
\(948\) 0 0
\(949\) 2.61561e8 0.306037
\(950\) − 8.45829e7i − 0.0986533i
\(951\) 0 0
\(952\) 0 0
\(953\) 4.28652e8 0.495252 0.247626 0.968856i \(-0.420350\pi\)
0.247626 + 0.968856i \(0.420350\pi\)
\(954\) 0 0
\(955\) 1.02515e9i 1.17700i
\(956\) 1.27262e8 0.145655
\(957\) 0 0
\(958\) − 3.31711e8i − 0.377280i
\(959\) 0 0
\(960\) 0 0
\(961\) −1.37323e9 −1.54729
\(962\) − 2.55204e7i − 0.0286657i
\(963\) 0 0
\(964\) − 2.78954e8i − 0.311387i
\(965\) 1.11235e9i 1.23782i
\(966\) 0 0
\(967\) 1.16972e9 1.29361 0.646805 0.762655i \(-0.276105\pi\)
0.646805 + 0.762655i \(0.276105\pi\)
\(968\) 6.53462e8 0.720434
\(969\) 0 0
\(970\) 1.72286e8 0.188771
\(971\) − 7.03804e8i − 0.768766i −0.923174 0.384383i \(-0.874414\pi\)
0.923174 0.384383i \(-0.125586\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.87949e8 −0.744525
\(975\) 0 0
\(976\) − 2.08289e7i − 0.0224036i
\(977\) −4.42412e8 −0.474399 −0.237199 0.971461i \(-0.576229\pi\)
−0.237199 + 0.971461i \(0.576229\pi\)
\(978\) 0 0
\(979\) 1.16235e8i 0.123876i
\(980\) 0 0
\(981\) 0 0
\(982\) −3.19357e8 −0.337242
\(983\) − 1.16835e9i − 1.23002i −0.788519 0.615011i \(-0.789152\pi\)
0.788519 0.615011i \(-0.210848\pi\)
\(984\) 0 0
\(985\) − 7.82804e8i − 0.819114i
\(986\) − 5.27974e8i − 0.550784i
\(987\) 0 0
\(988\) −9.54552e7 −0.0989758
\(989\) 2.38447e9 2.46492
\(990\) 0 0
\(991\) −1.00364e9 −1.03123 −0.515615 0.856821i \(-0.672436\pi\)
−0.515615 + 0.856821i \(0.672436\pi\)
\(992\) − 1.56516e9i − 1.60333i
\(993\) 0 0
\(994\) 0 0
\(995\) −4.71522e8 −0.478666
\(996\) 0 0
\(997\) − 1.84885e9i − 1.86559i −0.360409 0.932794i \(-0.617363\pi\)
0.360409 0.932794i \(-0.382637\pi\)
\(998\) −1.88803e8 −0.189941
\(999\) 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 441.7.d.d.244.5 8
3.2 odd 2 147.7.d.a.97.3 8
7.2 even 3 63.7.m.c.10.2 8
7.3 odd 6 63.7.m.c.19.2 8
7.6 odd 2 inner 441.7.d.d.244.6 8
21.2 odd 6 21.7.f.b.10.3 8
21.5 even 6 147.7.f.a.31.3 8
21.11 odd 6 147.7.f.a.19.3 8
21.17 even 6 21.7.f.b.19.3 yes 8
21.20 even 2 147.7.d.a.97.4 8
84.23 even 6 336.7.bh.b.241.3 8
84.59 odd 6 336.7.bh.b.145.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.7.f.b.10.3 8 21.2 odd 6
21.7.f.b.19.3 yes 8 21.17 even 6
63.7.m.c.10.2 8 7.2 even 3
63.7.m.c.19.2 8 7.3 odd 6
147.7.d.a.97.3 8 3.2 odd 2
147.7.d.a.97.4 8 21.20 even 2
147.7.f.a.19.3 8 21.11 odd 6
147.7.f.a.31.3 8 21.5 even 6
336.7.bh.b.145.3 8 84.59 odd 6
336.7.bh.b.241.3 8 84.23 even 6
441.7.d.d.244.5 8 1.1 even 1 trivial
441.7.d.d.244.6 8 7.6 odd 2 inner