Properties

Label 441.6.a.l.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.725083 q^{2} -31.4743 q^{4} -46.7492 q^{5} +46.0241 q^{8} +O(q^{10})\) \(q-0.725083 q^{2} -31.4743 q^{4} -46.7492 q^{5} +46.0241 q^{8} +33.8970 q^{10} -666.090 q^{11} +650.640 q^{13} +973.805 q^{16} +1186.89 q^{17} +1565.05 q^{19} +1471.40 q^{20} +482.970 q^{22} +1100.15 q^{23} -939.515 q^{25} -471.768 q^{26} -2396.72 q^{29} +2048.46 q^{31} -2178.86 q^{32} -860.596 q^{34} +1077.54 q^{37} -1134.79 q^{38} -2151.59 q^{40} +1098.21 q^{41} +16564.3 q^{43} +20964.7 q^{44} -797.702 q^{46} -8298.39 q^{47} +681.226 q^{50} -20478.4 q^{52} -5519.18 q^{53} +31139.1 q^{55} +1737.82 q^{58} -14230.4 q^{59} +14234.7 q^{61} -1485.30 q^{62} -29581.9 q^{64} -30416.9 q^{65} +19730.4 q^{67} -37356.6 q^{68} -64562.7 q^{71} -28567.0 q^{73} -781.309 q^{74} -49258.8 q^{76} -30633.4 q^{79} -45524.6 q^{80} -796.293 q^{82} -675.946 q^{83} -55486.3 q^{85} -12010.5 q^{86} -30656.2 q^{88} +125971. q^{89} -34626.5 q^{92} +6017.02 q^{94} -73164.9 q^{95} +22906.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8} - 204 q^{10} - 396 q^{11} + 350 q^{13} + 113 q^{16} + 1800 q^{17} + 3266 q^{19} + 2520 q^{20} - 1752 q^{22} - 2088 q^{23} - 3238 q^{25} + 2016 q^{26} - 6696 q^{29} + 20 q^{31} + 6129 q^{32} - 5934 q^{34} + 6232 q^{37} - 15210 q^{38} - 3216 q^{40} - 6048 q^{41} - 3020 q^{43} + 30816 q^{44} + 25584 q^{46} + 11700 q^{47} + 19701 q^{50} - 31444 q^{52} - 9468 q^{53} + 38904 q^{55} + 37314 q^{58} - 43938 q^{59} + 64754 q^{61} + 15300 q^{62} - 70783 q^{64} - 39060 q^{65} + 24784 q^{67} - 14994 q^{68} - 97416 q^{71} - 17452 q^{73} - 43434 q^{74} + 12782 q^{76} + 51256 q^{79} - 70272 q^{80} + 58338 q^{82} + 117558 q^{83} - 37860 q^{85} + 150048 q^{86} - 40656 q^{88} + 84276 q^{89} - 150912 q^{92} - 159468 q^{94} - 24264 q^{95} - 20776 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.725083 −0.128178 −0.0640889 0.997944i \(-0.520414\pi\)
−0.0640889 + 0.997944i \(0.520414\pi\)
\(3\) 0 0
\(4\) −31.4743 −0.983570
\(5\) −46.7492 −0.836275 −0.418137 0.908384i \(-0.637317\pi\)
−0.418137 + 0.908384i \(0.637317\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 46.0241 0.254250
\(9\) 0 0
\(10\) 33.8970 0.107192
\(11\) −666.090 −1.65978 −0.829891 0.557926i \(-0.811597\pi\)
−0.829891 + 0.557926i \(0.811597\pi\)
\(12\) 0 0
\(13\) 650.640 1.06778 0.533890 0.845554i \(-0.320729\pi\)
0.533890 + 0.845554i \(0.320729\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 973.805 0.950981
\(17\) 1186.89 0.996069 0.498035 0.867157i \(-0.334055\pi\)
0.498035 + 0.867157i \(0.334055\pi\)
\(18\) 0 0
\(19\) 1565.05 0.994591 0.497296 0.867581i \(-0.334326\pi\)
0.497296 + 0.867581i \(0.334326\pi\)
\(20\) 1471.40 0.822535
\(21\) 0 0
\(22\) 482.970 0.212747
\(23\) 1100.15 0.433644 0.216822 0.976211i \(-0.430431\pi\)
0.216822 + 0.976211i \(0.430431\pi\)
\(24\) 0 0
\(25\) −939.515 −0.300645
\(26\) −471.768 −0.136866
\(27\) 0 0
\(28\) 0 0
\(29\) −2396.72 −0.529203 −0.264602 0.964358i \(-0.585240\pi\)
−0.264602 + 0.964358i \(0.585240\pi\)
\(30\) 0 0
\(31\) 2048.46 0.382844 0.191422 0.981508i \(-0.438690\pi\)
0.191422 + 0.981508i \(0.438690\pi\)
\(32\) −2178.86 −0.376144
\(33\) 0 0
\(34\) −860.596 −0.127674
\(35\) 0 0
\(36\) 0 0
\(37\) 1077.54 0.129399 0.0646995 0.997905i \(-0.479391\pi\)
0.0646995 + 0.997905i \(0.479391\pi\)
\(38\) −1134.79 −0.127484
\(39\) 0 0
\(40\) −2151.59 −0.212622
\(41\) 1098.21 0.102029 0.0510147 0.998698i \(-0.483754\pi\)
0.0510147 + 0.998698i \(0.483754\pi\)
\(42\) 0 0
\(43\) 16564.3 1.36616 0.683081 0.730343i \(-0.260640\pi\)
0.683081 + 0.730343i \(0.260640\pi\)
\(44\) 20964.7 1.63251
\(45\) 0 0
\(46\) −797.702 −0.0555835
\(47\) −8298.39 −0.547960 −0.273980 0.961735i \(-0.588340\pi\)
−0.273980 + 0.961735i \(0.588340\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 681.226 0.0385360
\(51\) 0 0
\(52\) −20478.4 −1.05024
\(53\) −5519.18 −0.269889 −0.134944 0.990853i \(-0.543086\pi\)
−0.134944 + 0.990853i \(0.543086\pi\)
\(54\) 0 0
\(55\) 31139.1 1.38803
\(56\) 0 0
\(57\) 0 0
\(58\) 1737.82 0.0678321
\(59\) −14230.4 −0.532216 −0.266108 0.963943i \(-0.585738\pi\)
−0.266108 + 0.963943i \(0.585738\pi\)
\(60\) 0 0
\(61\) 14234.7 0.489807 0.244904 0.969547i \(-0.421244\pi\)
0.244904 + 0.969547i \(0.421244\pi\)
\(62\) −1485.30 −0.0490721
\(63\) 0 0
\(64\) −29581.9 −0.902768
\(65\) −30416.9 −0.892958
\(66\) 0 0
\(67\) 19730.4 0.536970 0.268485 0.963284i \(-0.413477\pi\)
0.268485 + 0.963284i \(0.413477\pi\)
\(68\) −37356.6 −0.979704
\(69\) 0 0
\(70\) 0 0
\(71\) −64562.7 −1.51997 −0.759986 0.649940i \(-0.774794\pi\)
−0.759986 + 0.649940i \(0.774794\pi\)
\(72\) 0 0
\(73\) −28567.0 −0.627418 −0.313709 0.949519i \(-0.601572\pi\)
−0.313709 + 0.949519i \(0.601572\pi\)
\(74\) −781.309 −0.0165861
\(75\) 0 0
\(76\) −49258.8 −0.978251
\(77\) 0 0
\(78\) 0 0
\(79\) −30633.4 −0.552239 −0.276119 0.961123i \(-0.589049\pi\)
−0.276119 + 0.961123i \(0.589049\pi\)
\(80\) −45524.6 −0.795282
\(81\) 0 0
\(82\) −796.293 −0.0130779
\(83\) −675.946 −0.0107700 −0.00538501 0.999986i \(-0.501714\pi\)
−0.00538501 + 0.999986i \(0.501714\pi\)
\(84\) 0 0
\(85\) −55486.3 −0.832987
\(86\) −12010.5 −0.175111
\(87\) 0 0
\(88\) −30656.2 −0.421999
\(89\) 125971. 1.68576 0.842882 0.538098i \(-0.180857\pi\)
0.842882 + 0.538098i \(0.180857\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −34626.5 −0.426520
\(93\) 0 0
\(94\) 6017.02 0.0702363
\(95\) −73164.9 −0.831751
\(96\) 0 0
\(97\) 22906.8 0.247192 0.123596 0.992333i \(-0.460557\pi\)
0.123596 + 0.992333i \(0.460557\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 29570.5 0.295705
\(101\) −181474. −1.77015 −0.885077 0.465444i \(-0.845895\pi\)
−0.885077 + 0.465444i \(0.845895\pi\)
\(102\) 0 0
\(103\) −64772.0 −0.601581 −0.300791 0.953690i \(-0.597251\pi\)
−0.300791 + 0.953690i \(0.597251\pi\)
\(104\) 29945.1 0.271483
\(105\) 0 0
\(106\) 4001.86 0.0345938
\(107\) 148170. 1.25112 0.625562 0.780175i \(-0.284870\pi\)
0.625562 + 0.780175i \(0.284870\pi\)
\(108\) 0 0
\(109\) −111294. −0.897237 −0.448618 0.893723i \(-0.648084\pi\)
−0.448618 + 0.893723i \(0.648084\pi\)
\(110\) −22578.5 −0.177915
\(111\) 0 0
\(112\) 0 0
\(113\) 43175.5 0.318084 0.159042 0.987272i \(-0.449160\pi\)
0.159042 + 0.987272i \(0.449160\pi\)
\(114\) 0 0
\(115\) −51431.2 −0.362646
\(116\) 75435.0 0.520509
\(117\) 0 0
\(118\) 10318.2 0.0682182
\(119\) 0 0
\(120\) 0 0
\(121\) 282625. 1.75488
\(122\) −10321.4 −0.0627824
\(123\) 0 0
\(124\) −64473.6 −0.376554
\(125\) 190013. 1.08770
\(126\) 0 0
\(127\) −131449. −0.723182 −0.361591 0.932337i \(-0.617766\pi\)
−0.361591 + 0.932337i \(0.617766\pi\)
\(128\) 91172.8 0.491859
\(129\) 0 0
\(130\) 22054.7 0.114457
\(131\) −349458. −1.77916 −0.889582 0.456775i \(-0.849005\pi\)
−0.889582 + 0.456775i \(0.849005\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −14306.2 −0.0688276
\(135\) 0 0
\(136\) 54625.7 0.253250
\(137\) −386434. −1.75903 −0.879516 0.475869i \(-0.842134\pi\)
−0.879516 + 0.475869i \(0.842134\pi\)
\(138\) 0 0
\(139\) −17289.3 −0.0758997 −0.0379498 0.999280i \(-0.512083\pi\)
−0.0379498 + 0.999280i \(0.512083\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 46813.3 0.194827
\(143\) −433384. −1.77228
\(144\) 0 0
\(145\) 112045. 0.442559
\(146\) 20713.4 0.0804210
\(147\) 0 0
\(148\) −33914.9 −0.127273
\(149\) 112171. 0.413917 0.206959 0.978350i \(-0.433643\pi\)
0.206959 + 0.978350i \(0.433643\pi\)
\(150\) 0 0
\(151\) 30495.4 0.108841 0.0544205 0.998518i \(-0.482669\pi\)
0.0544205 + 0.998518i \(0.482669\pi\)
\(152\) 72030.1 0.252874
\(153\) 0 0
\(154\) 0 0
\(155\) −95763.6 −0.320163
\(156\) 0 0
\(157\) −523509. −1.69502 −0.847510 0.530780i \(-0.821899\pi\)
−0.847510 + 0.530780i \(0.821899\pi\)
\(158\) 22211.7 0.0707847
\(159\) 0 0
\(160\) 101860. 0.314560
\(161\) 0 0
\(162\) 0 0
\(163\) −439646. −1.29609 −0.648043 0.761604i \(-0.724412\pi\)
−0.648043 + 0.761604i \(0.724412\pi\)
\(164\) −34565.3 −0.100353
\(165\) 0 0
\(166\) 490.117 0.00138048
\(167\) 279353. 0.775107 0.387554 0.921847i \(-0.373320\pi\)
0.387554 + 0.921847i \(0.373320\pi\)
\(168\) 0 0
\(169\) 52038.8 0.140156
\(170\) 40232.2 0.106770
\(171\) 0 0
\(172\) −521349. −1.34372
\(173\) 99699.4 0.253266 0.126633 0.991950i \(-0.459583\pi\)
0.126633 + 0.991950i \(0.459583\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −648641. −1.57842
\(177\) 0 0
\(178\) −91339.7 −0.216077
\(179\) −329980. −0.769760 −0.384880 0.922967i \(-0.625757\pi\)
−0.384880 + 0.922967i \(0.625757\pi\)
\(180\) 0 0
\(181\) 505810. 1.14760 0.573800 0.818995i \(-0.305469\pi\)
0.573800 + 0.818995i \(0.305469\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 50633.5 0.110254
\(185\) −50374.3 −0.108213
\(186\) 0 0
\(187\) −790578. −1.65326
\(188\) 261186. 0.538958
\(189\) 0 0
\(190\) 53050.6 0.106612
\(191\) 63835.6 0.126613 0.0633067 0.997994i \(-0.479835\pi\)
0.0633067 + 0.997994i \(0.479835\pi\)
\(192\) 0 0
\(193\) 469355. 0.907001 0.453501 0.891256i \(-0.350175\pi\)
0.453501 + 0.891256i \(0.350175\pi\)
\(194\) −16609.3 −0.0316845
\(195\) 0 0
\(196\) 0 0
\(197\) −268021. −0.492043 −0.246021 0.969264i \(-0.579123\pi\)
−0.246021 + 0.969264i \(0.579123\pi\)
\(198\) 0 0
\(199\) 605167. 1.08328 0.541642 0.840609i \(-0.317803\pi\)
0.541642 + 0.840609i \(0.317803\pi\)
\(200\) −43240.3 −0.0764388
\(201\) 0 0
\(202\) 131584. 0.226894
\(203\) 0 0
\(204\) 0 0
\(205\) −51340.4 −0.0853246
\(206\) 46965.1 0.0771094
\(207\) 0 0
\(208\) 633596. 1.01544
\(209\) −1.04246e6 −1.65080
\(210\) 0 0
\(211\) 335389. 0.518612 0.259306 0.965795i \(-0.416506\pi\)
0.259306 + 0.965795i \(0.416506\pi\)
\(212\) 173712. 0.265455
\(213\) 0 0
\(214\) −107435. −0.160366
\(215\) −774367. −1.14249
\(216\) 0 0
\(217\) 0 0
\(218\) 80697.7 0.115006
\(219\) 0 0
\(220\) −980081. −1.36523
\(221\) 772240. 1.06358
\(222\) 0 0
\(223\) −1.02526e6 −1.38061 −0.690305 0.723518i \(-0.742524\pi\)
−0.690305 + 0.723518i \(0.742524\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −31305.8 −0.0407713
\(227\) −504226. −0.649473 −0.324736 0.945805i \(-0.605276\pi\)
−0.324736 + 0.945805i \(0.605276\pi\)
\(228\) 0 0
\(229\) 1.11939e6 1.41057 0.705283 0.708925i \(-0.250820\pi\)
0.705283 + 0.708925i \(0.250820\pi\)
\(230\) 37291.9 0.0464831
\(231\) 0 0
\(232\) −110307. −0.134550
\(233\) −770703. −0.930031 −0.465015 0.885303i \(-0.653951\pi\)
−0.465015 + 0.885303i \(0.653951\pi\)
\(234\) 0 0
\(235\) 387943. 0.458245
\(236\) 447892. 0.523472
\(237\) 0 0
\(238\) 0 0
\(239\) 171646. 0.194374 0.0971871 0.995266i \(-0.469015\pi\)
0.0971871 + 0.995266i \(0.469015\pi\)
\(240\) 0 0
\(241\) 383779. 0.425637 0.212818 0.977092i \(-0.431736\pi\)
0.212818 + 0.977092i \(0.431736\pi\)
\(242\) −204926. −0.224936
\(243\) 0 0
\(244\) −448028. −0.481760
\(245\) 0 0
\(246\) 0 0
\(247\) 1.01828e6 1.06201
\(248\) 94278.3 0.0973380
\(249\) 0 0
\(250\) −137775. −0.139418
\(251\) −1.57046e6 −1.57342 −0.786708 0.617325i \(-0.788216\pi\)
−0.786708 + 0.617325i \(0.788216\pi\)
\(252\) 0 0
\(253\) −732801. −0.719755
\(254\) 95311.4 0.0926959
\(255\) 0 0
\(256\) 880513. 0.839723
\(257\) 790656. 0.746715 0.373357 0.927688i \(-0.378207\pi\)
0.373357 + 0.927688i \(0.378207\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 957348. 0.878287
\(261\) 0 0
\(262\) 253386. 0.228049
\(263\) −464416. −0.414017 −0.207008 0.978339i \(-0.566373\pi\)
−0.207008 + 0.978339i \(0.566373\pi\)
\(264\) 0 0
\(265\) 258017. 0.225701
\(266\) 0 0
\(267\) 0 0
\(268\) −621001. −0.528147
\(269\) 1.99959e6 1.68484 0.842422 0.538818i \(-0.181129\pi\)
0.842422 + 0.538818i \(0.181129\pi\)
\(270\) 0 0
\(271\) −1.61296e6 −1.33414 −0.667070 0.744995i \(-0.732452\pi\)
−0.667070 + 0.744995i \(0.732452\pi\)
\(272\) 1.15580e6 0.947243
\(273\) 0 0
\(274\) 280197. 0.225469
\(275\) 625801. 0.499005
\(276\) 0 0
\(277\) −2.08119e6 −1.62972 −0.814860 0.579658i \(-0.803186\pi\)
−0.814860 + 0.579658i \(0.803186\pi\)
\(278\) 12536.2 0.00972865
\(279\) 0 0
\(280\) 0 0
\(281\) 982035. 0.741927 0.370964 0.928647i \(-0.379027\pi\)
0.370964 + 0.928647i \(0.379027\pi\)
\(282\) 0 0
\(283\) 1.39622e6 1.03630 0.518152 0.855289i \(-0.326620\pi\)
0.518152 + 0.855289i \(0.326620\pi\)
\(284\) 2.03206e6 1.49500
\(285\) 0 0
\(286\) 314240. 0.227167
\(287\) 0 0
\(288\) 0 0
\(289\) −11140.3 −0.00784609
\(290\) −81241.7 −0.0567262
\(291\) 0 0
\(292\) 899124. 0.617110
\(293\) 2.56205e6 1.74348 0.871742 0.489965i \(-0.162990\pi\)
0.871742 + 0.489965i \(0.162990\pi\)
\(294\) 0 0
\(295\) 665260. 0.445078
\(296\) 49593.0 0.0328996
\(297\) 0 0
\(298\) −81333.0 −0.0530550
\(299\) 715803. 0.463037
\(300\) 0 0
\(301\) 0 0
\(302\) −22111.7 −0.0139510
\(303\) 0 0
\(304\) 1.52405e6 0.945838
\(305\) −665463. −0.409613
\(306\) 0 0
\(307\) 884855. 0.535829 0.267915 0.963443i \(-0.413666\pi\)
0.267915 + 0.963443i \(0.413666\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 69436.5 0.0410378
\(311\) 1.80120e6 1.05599 0.527997 0.849246i \(-0.322943\pi\)
0.527997 + 0.849246i \(0.322943\pi\)
\(312\) 0 0
\(313\) −950366. −0.548315 −0.274158 0.961685i \(-0.588399\pi\)
−0.274158 + 0.961685i \(0.588399\pi\)
\(314\) 379587. 0.217264
\(315\) 0 0
\(316\) 964162. 0.543166
\(317\) −3.04277e6 −1.70068 −0.850338 0.526237i \(-0.823602\pi\)
−0.850338 + 0.526237i \(0.823602\pi\)
\(318\) 0 0
\(319\) 1.59643e6 0.878362
\(320\) 1.38293e6 0.754962
\(321\) 0 0
\(322\) 0 0
\(323\) 1.85755e6 0.990682
\(324\) 0 0
\(325\) −611286. −0.321023
\(326\) 318780. 0.166129
\(327\) 0 0
\(328\) 50544.1 0.0259409
\(329\) 0 0
\(330\) 0 0
\(331\) −2.19616e6 −1.10178 −0.550889 0.834579i \(-0.685711\pi\)
−0.550889 + 0.834579i \(0.685711\pi\)
\(332\) 21274.9 0.0105931
\(333\) 0 0
\(334\) −202554. −0.0993515
\(335\) −922382. −0.449054
\(336\) 0 0
\(337\) −2.41491e6 −1.15832 −0.579158 0.815216i \(-0.696618\pi\)
−0.579158 + 0.815216i \(0.696618\pi\)
\(338\) −37732.5 −0.0179648
\(339\) 0 0
\(340\) 1.74639e6 0.819302
\(341\) −1.36446e6 −0.635438
\(342\) 0 0
\(343\) 0 0
\(344\) 762357. 0.347346
\(345\) 0 0
\(346\) −72290.3 −0.0324631
\(347\) 1.08833e6 0.485219 0.242609 0.970124i \(-0.421997\pi\)
0.242609 + 0.970124i \(0.421997\pi\)
\(348\) 0 0
\(349\) −2.79267e6 −1.22731 −0.613657 0.789573i \(-0.710302\pi\)
−0.613657 + 0.789573i \(0.710302\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.45132e6 0.624317
\(353\) 2.53134e6 1.08122 0.540610 0.841273i \(-0.318193\pi\)
0.540610 + 0.841273i \(0.318193\pi\)
\(354\) 0 0
\(355\) 3.01825e6 1.27111
\(356\) −3.96485e6 −1.65807
\(357\) 0 0
\(358\) 239263. 0.0986661
\(359\) −1.09028e6 −0.446480 −0.223240 0.974763i \(-0.571663\pi\)
−0.223240 + 0.974763i \(0.571663\pi\)
\(360\) 0 0
\(361\) −26712.8 −0.0107883
\(362\) −366754. −0.147097
\(363\) 0 0
\(364\) 0 0
\(365\) 1.33548e6 0.524694
\(366\) 0 0
\(367\) −188070. −0.0728879 −0.0364439 0.999336i \(-0.511603\pi\)
−0.0364439 + 0.999336i \(0.511603\pi\)
\(368\) 1.07133e6 0.412387
\(369\) 0 0
\(370\) 36525.6 0.0138705
\(371\) 0 0
\(372\) 0 0
\(373\) −1.79371e6 −0.667545 −0.333772 0.942654i \(-0.608322\pi\)
−0.333772 + 0.942654i \(0.608322\pi\)
\(374\) 573234. 0.211911
\(375\) 0 0
\(376\) −381926. −0.139319
\(377\) −1.55940e6 −0.565073
\(378\) 0 0
\(379\) 3.58806e6 1.28310 0.641551 0.767080i \(-0.278291\pi\)
0.641551 + 0.767080i \(0.278291\pi\)
\(380\) 2.30281e6 0.818086
\(381\) 0 0
\(382\) −46286.1 −0.0162290
\(383\) −3.42457e6 −1.19291 −0.596457 0.802645i \(-0.703425\pi\)
−0.596457 + 0.802645i \(0.703425\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −340321. −0.116257
\(387\) 0 0
\(388\) −720974. −0.243131
\(389\) −8625.27 −0.00289001 −0.00144500 0.999999i \(-0.500460\pi\)
−0.00144500 + 0.999999i \(0.500460\pi\)
\(390\) 0 0
\(391\) 1.30576e6 0.431940
\(392\) 0 0
\(393\) 0 0
\(394\) 194337. 0.0630690
\(395\) 1.43208e6 0.461823
\(396\) 0 0
\(397\) −1.25709e6 −0.400306 −0.200153 0.979765i \(-0.564144\pi\)
−0.200153 + 0.979765i \(0.564144\pi\)
\(398\) −438796. −0.138853
\(399\) 0 0
\(400\) −914904. −0.285908
\(401\) −1.42670e6 −0.443070 −0.221535 0.975152i \(-0.571107\pi\)
−0.221535 + 0.975152i \(0.571107\pi\)
\(402\) 0 0
\(403\) 1.33281e6 0.408794
\(404\) 5.71176e6 1.74107
\(405\) 0 0
\(406\) 0 0
\(407\) −717741. −0.214774
\(408\) 0 0
\(409\) 3.06529e6 0.906073 0.453036 0.891492i \(-0.350341\pi\)
0.453036 + 0.891492i \(0.350341\pi\)
\(410\) 37226.0 0.0109367
\(411\) 0 0
\(412\) 2.03865e6 0.591698
\(413\) 0 0
\(414\) 0 0
\(415\) 31599.9 0.00900670
\(416\) −1.41765e6 −0.401640
\(417\) 0 0
\(418\) 755873. 0.211596
\(419\) −248240. −0.0690776 −0.0345388 0.999403i \(-0.510996\pi\)
−0.0345388 + 0.999403i \(0.510996\pi\)
\(420\) 0 0
\(421\) 5.96280e6 1.63963 0.819814 0.572630i \(-0.194077\pi\)
0.819814 + 0.572630i \(0.194077\pi\)
\(422\) −243185. −0.0664746
\(423\) 0 0
\(424\) −254015. −0.0686192
\(425\) −1.11510e6 −0.299463
\(426\) 0 0
\(427\) 0 0
\(428\) −4.66353e6 −1.23057
\(429\) 0 0
\(430\) 561481. 0.146441
\(431\) 4.93538e6 1.27976 0.639879 0.768476i \(-0.278985\pi\)
0.639879 + 0.768476i \(0.278985\pi\)
\(432\) 0 0
\(433\) −4.15513e6 −1.06504 −0.532519 0.846418i \(-0.678755\pi\)
−0.532519 + 0.846418i \(0.678755\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.50291e6 0.882496
\(437\) 1.72180e6 0.431299
\(438\) 0 0
\(439\) 227955. 0.0564531 0.0282265 0.999602i \(-0.491014\pi\)
0.0282265 + 0.999602i \(0.491014\pi\)
\(440\) 1.43315e6 0.352907
\(441\) 0 0
\(442\) −559938. −0.136328
\(443\) 1.98462e6 0.480472 0.240236 0.970715i \(-0.422775\pi\)
0.240236 + 0.970715i \(0.422775\pi\)
\(444\) 0 0
\(445\) −5.88906e6 −1.40976
\(446\) 743397. 0.176963
\(447\) 0 0
\(448\) 0 0
\(449\) −2.61077e6 −0.611157 −0.305579 0.952167i \(-0.598850\pi\)
−0.305579 + 0.952167i \(0.598850\pi\)
\(450\) 0 0
\(451\) −731506. −0.169347
\(452\) −1.35892e6 −0.312858
\(453\) 0 0
\(454\) 365606. 0.0832480
\(455\) 0 0
\(456\) 0 0
\(457\) −4.09917e6 −0.918132 −0.459066 0.888402i \(-0.651816\pi\)
−0.459066 + 0.888402i \(0.651816\pi\)
\(458\) −811652. −0.180803
\(459\) 0 0
\(460\) 1.61876e6 0.356687
\(461\) −2.62378e6 −0.575009 −0.287505 0.957779i \(-0.592826\pi\)
−0.287505 + 0.957779i \(0.592826\pi\)
\(462\) 0 0
\(463\) −4.28563e6 −0.929100 −0.464550 0.885547i \(-0.653784\pi\)
−0.464550 + 0.885547i \(0.653784\pi\)
\(464\) −2.33394e6 −0.503262
\(465\) 0 0
\(466\) 558824. 0.119209
\(467\) 990118. 0.210085 0.105042 0.994468i \(-0.466502\pi\)
0.105042 + 0.994468i \(0.466502\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −281291. −0.0587369
\(471\) 0 0
\(472\) −654942. −0.135316
\(473\) −1.10333e7 −2.26753
\(474\) 0 0
\(475\) −1.47039e6 −0.299019
\(476\) 0 0
\(477\) 0 0
\(478\) −124457. −0.0249144
\(479\) 2.57976e6 0.513737 0.256868 0.966446i \(-0.417309\pi\)
0.256868 + 0.966446i \(0.417309\pi\)
\(480\) 0 0
\(481\) 701093. 0.138170
\(482\) −278272. −0.0545571
\(483\) 0 0
\(484\) −8.89540e6 −1.72604
\(485\) −1.07087e6 −0.206720
\(486\) 0 0
\(487\) 3.21474e6 0.614219 0.307109 0.951674i \(-0.400638\pi\)
0.307109 + 0.951674i \(0.400638\pi\)
\(488\) 655141. 0.124533
\(489\) 0 0
\(490\) 0 0
\(491\) 7.86108e6 1.47156 0.735781 0.677220i \(-0.236815\pi\)
0.735781 + 0.677220i \(0.236815\pi\)
\(492\) 0 0
\(493\) −2.84465e6 −0.527123
\(494\) −738341. −0.136125
\(495\) 0 0
\(496\) 1.99480e6 0.364078
\(497\) 0 0
\(498\) 0 0
\(499\) −1.35382e6 −0.243395 −0.121697 0.992567i \(-0.538834\pi\)
−0.121697 + 0.992567i \(0.538834\pi\)
\(500\) −5.98051e6 −1.06983
\(501\) 0 0
\(502\) 1.13872e6 0.201677
\(503\) 3.85775e6 0.679851 0.339926 0.940452i \(-0.389598\pi\)
0.339926 + 0.940452i \(0.389598\pi\)
\(504\) 0 0
\(505\) 8.48376e6 1.48034
\(506\) 531341. 0.0922565
\(507\) 0 0
\(508\) 4.13726e6 0.711301
\(509\) −1.06060e7 −1.81451 −0.907253 0.420585i \(-0.861825\pi\)
−0.907253 + 0.420585i \(0.861825\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.55598e6 −0.599493
\(513\) 0 0
\(514\) −573291. −0.0957122
\(515\) 3.02804e6 0.503087
\(516\) 0 0
\(517\) 5.52747e6 0.909495
\(518\) 0 0
\(519\) 0 0
\(520\) −1.39991e6 −0.227034
\(521\) −9.17989e6 −1.48164 −0.740821 0.671703i \(-0.765563\pi\)
−0.740821 + 0.671703i \(0.765563\pi\)
\(522\) 0 0
\(523\) −9.05585e6 −1.44769 −0.723844 0.689964i \(-0.757627\pi\)
−0.723844 + 0.689964i \(0.757627\pi\)
\(524\) 1.09989e7 1.74993
\(525\) 0 0
\(526\) 336740. 0.0530677
\(527\) 2.43130e6 0.381339
\(528\) 0 0
\(529\) −5.22601e6 −0.811953
\(530\) −187084. −0.0289299
\(531\) 0 0
\(532\) 0 0
\(533\) 714539. 0.108945
\(534\) 0 0
\(535\) −6.92681e6 −1.04628
\(536\) 908075. 0.136524
\(537\) 0 0
\(538\) −1.44987e6 −0.215960
\(539\) 0 0
\(540\) 0 0
\(541\) 1.21783e7 1.78894 0.894468 0.447132i \(-0.147555\pi\)
0.894468 + 0.447132i \(0.147555\pi\)
\(542\) 1.16953e6 0.171007
\(543\) 0 0
\(544\) −2.58608e6 −0.374666
\(545\) 5.20292e6 0.750336
\(546\) 0 0
\(547\) −9.00451e6 −1.28674 −0.643372 0.765554i \(-0.722465\pi\)
−0.643372 + 0.765554i \(0.722465\pi\)
\(548\) 1.21627e7 1.73013
\(549\) 0 0
\(550\) −453758. −0.0639613
\(551\) −3.75099e6 −0.526341
\(552\) 0 0
\(553\) 0 0
\(554\) 1.50904e6 0.208894
\(555\) 0 0
\(556\) 544167. 0.0746527
\(557\) −1.54461e6 −0.210950 −0.105475 0.994422i \(-0.533636\pi\)
−0.105475 + 0.994422i \(0.533636\pi\)
\(558\) 0 0
\(559\) 1.07774e7 1.45876
\(560\) 0 0
\(561\) 0 0
\(562\) −712057. −0.0950986
\(563\) −1.18748e7 −1.57890 −0.789449 0.613816i \(-0.789634\pi\)
−0.789449 + 0.613816i \(0.789634\pi\)
\(564\) 0 0
\(565\) −2.01842e6 −0.266005
\(566\) −1.01237e6 −0.132831
\(567\) 0 0
\(568\) −2.97144e6 −0.386452
\(569\) 1.54308e6 0.199806 0.0999031 0.994997i \(-0.468147\pi\)
0.0999031 + 0.994997i \(0.468147\pi\)
\(570\) 0 0
\(571\) −7.01812e6 −0.900804 −0.450402 0.892826i \(-0.648719\pi\)
−0.450402 + 0.892826i \(0.648719\pi\)
\(572\) 1.36404e7 1.74317
\(573\) 0 0
\(574\) 0 0
\(575\) −1.03361e6 −0.130373
\(576\) 0 0
\(577\) 5.37543e6 0.672162 0.336081 0.941833i \(-0.390898\pi\)
0.336081 + 0.941833i \(0.390898\pi\)
\(578\) 8077.66 0.00100569
\(579\) 0 0
\(580\) −3.52652e6 −0.435288
\(581\) 0 0
\(582\) 0 0
\(583\) 3.67627e6 0.447957
\(584\) −1.31477e6 −0.159521
\(585\) 0 0
\(586\) −1.85770e6 −0.223476
\(587\) −2.06682e6 −0.247575 −0.123788 0.992309i \(-0.539504\pi\)
−0.123788 + 0.992309i \(0.539504\pi\)
\(588\) 0 0
\(589\) 3.20594e6 0.380774
\(590\) −482369. −0.0570492
\(591\) 0 0
\(592\) 1.04932e6 0.123056
\(593\) −5.46947e6 −0.638717 −0.319358 0.947634i \(-0.603467\pi\)
−0.319358 + 0.947634i \(0.603467\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.53049e6 −0.407117
\(597\) 0 0
\(598\) −519016. −0.0593510
\(599\) −1.09943e7 −1.25199 −0.625996 0.779826i \(-0.715307\pi\)
−0.625996 + 0.779826i \(0.715307\pi\)
\(600\) 0 0
\(601\) −1.58788e7 −1.79322 −0.896608 0.442826i \(-0.853976\pi\)
−0.896608 + 0.442826i \(0.853976\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −959820. −0.107053
\(605\) −1.32125e7 −1.46756
\(606\) 0 0
\(607\) −5.33262e6 −0.587447 −0.293724 0.955890i \(-0.594895\pi\)
−0.293724 + 0.955890i \(0.594895\pi\)
\(608\) −3.41003e6 −0.374110
\(609\) 0 0
\(610\) 482516. 0.0525033
\(611\) −5.39926e6 −0.585102
\(612\) 0 0
\(613\) −8.91838e6 −0.958594 −0.479297 0.877653i \(-0.659108\pi\)
−0.479297 + 0.877653i \(0.659108\pi\)
\(614\) −641593. −0.0686814
\(615\) 0 0
\(616\) 0 0
\(617\) −9.63586e6 −1.01901 −0.509504 0.860468i \(-0.670171\pi\)
−0.509504 + 0.860468i \(0.670171\pi\)
\(618\) 0 0
\(619\) −1.21747e7 −1.27712 −0.638560 0.769572i \(-0.720470\pi\)
−0.638560 + 0.769572i \(0.720470\pi\)
\(620\) 3.01409e6 0.314903
\(621\) 0 0
\(622\) −1.30602e6 −0.135355
\(623\) 0 0
\(624\) 0 0
\(625\) −5.94695e6 −0.608968
\(626\) 689094. 0.0702818
\(627\) 0 0
\(628\) 1.64770e7 1.66717
\(629\) 1.27893e6 0.128890
\(630\) 0 0
\(631\) −1.30854e7 −1.30832 −0.654161 0.756356i \(-0.726978\pi\)
−0.654161 + 0.756356i \(0.726978\pi\)
\(632\) −1.40987e6 −0.140407
\(633\) 0 0
\(634\) 2.20626e6 0.217989
\(635\) 6.14513e6 0.604779
\(636\) 0 0
\(637\) 0 0
\(638\) −1.15754e6 −0.112586
\(639\) 0 0
\(640\) −4.26226e6 −0.411329
\(641\) 441107. 0.0424032 0.0212016 0.999775i \(-0.493251\pi\)
0.0212016 + 0.999775i \(0.493251\pi\)
\(642\) 0 0
\(643\) 4.18888e6 0.399550 0.199775 0.979842i \(-0.435979\pi\)
0.199775 + 0.979842i \(0.435979\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.34688e6 −0.126983
\(647\) 1.87822e7 1.76395 0.881973 0.471300i \(-0.156215\pi\)
0.881973 + 0.471300i \(0.156215\pi\)
\(648\) 0 0
\(649\) 9.47874e6 0.883362
\(650\) 443233. 0.0411480
\(651\) 0 0
\(652\) 1.38375e7 1.27479
\(653\) 1.51733e6 0.139251 0.0696254 0.997573i \(-0.477820\pi\)
0.0696254 + 0.997573i \(0.477820\pi\)
\(654\) 0 0
\(655\) 1.63368e7 1.48787
\(656\) 1.06944e6 0.0970281
\(657\) 0 0
\(658\) 0 0
\(659\) −1.84809e7 −1.65772 −0.828859 0.559458i \(-0.811009\pi\)
−0.828859 + 0.559458i \(0.811009\pi\)
\(660\) 0 0
\(661\) 1.03952e7 0.925403 0.462702 0.886514i \(-0.346880\pi\)
0.462702 + 0.886514i \(0.346880\pi\)
\(662\) 1.59240e6 0.141223
\(663\) 0 0
\(664\) −31109.8 −0.00273828
\(665\) 0 0
\(666\) 0 0
\(667\) −2.63676e6 −0.229486
\(668\) −8.79242e6 −0.762372
\(669\) 0 0
\(670\) 668803. 0.0575587
\(671\) −9.48162e6 −0.812973
\(672\) 0 0
\(673\) −1.10398e7 −0.939556 −0.469778 0.882785i \(-0.655666\pi\)
−0.469778 + 0.882785i \(0.655666\pi\)
\(674\) 1.75101e6 0.148470
\(675\) 0 0
\(676\) −1.63788e6 −0.137853
\(677\) −8.23485e6 −0.690532 −0.345266 0.938505i \(-0.612211\pi\)
−0.345266 + 0.938505i \(0.612211\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.55371e6 −0.211787
\(681\) 0 0
\(682\) 989343. 0.0814490
\(683\) −1.98437e7 −1.62768 −0.813842 0.581086i \(-0.802628\pi\)
−0.813842 + 0.581086i \(0.802628\pi\)
\(684\) 0 0
\(685\) 1.80655e7 1.47103
\(686\) 0 0
\(687\) 0 0
\(688\) 1.61304e7 1.29919
\(689\) −3.59100e6 −0.288182
\(690\) 0 0
\(691\) 2.37019e6 0.188837 0.0944185 0.995533i \(-0.469901\pi\)
0.0944185 + 0.995533i \(0.469901\pi\)
\(692\) −3.13796e6 −0.249105
\(693\) 0 0
\(694\) −789131. −0.0621943
\(695\) 808259. 0.0634730
\(696\) 0 0
\(697\) 1.30346e6 0.101628
\(698\) 2.02491e6 0.157314
\(699\) 0 0
\(700\) 0 0
\(701\) 1.56833e7 1.20543 0.602714 0.797957i \(-0.294086\pi\)
0.602714 + 0.797957i \(0.294086\pi\)
\(702\) 0 0
\(703\) 1.68641e6 0.128699
\(704\) 1.97042e7 1.49840
\(705\) 0 0
\(706\) −1.83543e6 −0.138588
\(707\) 0 0
\(708\) 0 0
\(709\) 3.68544e6 0.275343 0.137671 0.990478i \(-0.456038\pi\)
0.137671 + 0.990478i \(0.456038\pi\)
\(710\) −2.18848e6 −0.162928
\(711\) 0 0
\(712\) 5.79772e6 0.428605
\(713\) 2.25361e6 0.166018
\(714\) 0 0
\(715\) 2.02604e7 1.48212
\(716\) 1.03859e7 0.757113
\(717\) 0 0
\(718\) 790544. 0.0572289
\(719\) −1.56427e7 −1.12847 −0.564233 0.825615i \(-0.690828\pi\)
−0.564233 + 0.825615i \(0.690828\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19369.0 0.00138282
\(723\) 0 0
\(724\) −1.59200e7 −1.12875
\(725\) 2.25175e6 0.159102
\(726\) 0 0
\(727\) −1.85908e7 −1.30456 −0.652279 0.757979i \(-0.726187\pi\)
−0.652279 + 0.757979i \(0.726187\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −968335. −0.0672541
\(731\) 1.96601e7 1.36079
\(732\) 0 0
\(733\) −2.49466e7 −1.71495 −0.857476 0.514524i \(-0.827969\pi\)
−0.857476 + 0.514524i \(0.827969\pi\)
\(734\) 136367. 0.00934260
\(735\) 0 0
\(736\) −2.39708e6 −0.163113
\(737\) −1.31422e7 −0.891253
\(738\) 0 0
\(739\) −2.42944e7 −1.63642 −0.818211 0.574918i \(-0.805034\pi\)
−0.818211 + 0.574918i \(0.805034\pi\)
\(740\) 1.58549e6 0.106435
\(741\) 0 0
\(742\) 0 0
\(743\) −4.16541e6 −0.276812 −0.138406 0.990376i \(-0.544198\pi\)
−0.138406 + 0.990376i \(0.544198\pi\)
\(744\) 0 0
\(745\) −5.24388e6 −0.346148
\(746\) 1.30059e6 0.0855644
\(747\) 0 0
\(748\) 2.48828e7 1.62610
\(749\) 0 0
\(750\) 0 0
\(751\) −2.36434e7 −1.52972 −0.764858 0.644199i \(-0.777191\pi\)
−0.764858 + 0.644199i \(0.777191\pi\)
\(752\) −8.08101e6 −0.521100
\(753\) 0 0
\(754\) 1.13070e6 0.0724298
\(755\) −1.42564e6 −0.0910209
\(756\) 0 0
\(757\) 1.50108e7 0.952062 0.476031 0.879429i \(-0.342075\pi\)
0.476031 + 0.879429i \(0.342075\pi\)
\(758\) −2.60164e6 −0.164465
\(759\) 0 0
\(760\) −3.36735e6 −0.211472
\(761\) 2.92191e6 0.182897 0.0914483 0.995810i \(-0.470850\pi\)
0.0914483 + 0.995810i \(0.470850\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.00918e6 −0.124533
\(765\) 0 0
\(766\) 2.48310e6 0.152905
\(767\) −9.25887e6 −0.568290
\(768\) 0 0
\(769\) −1.42847e7 −0.871073 −0.435536 0.900171i \(-0.643441\pi\)
−0.435536 + 0.900171i \(0.643441\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.47726e7 −0.892100
\(773\) −1.09012e7 −0.656186 −0.328093 0.944645i \(-0.606406\pi\)
−0.328093 + 0.944645i \(0.606406\pi\)
\(774\) 0 0
\(775\) −1.92455e6 −0.115100
\(776\) 1.05426e6 0.0628485
\(777\) 0 0
\(778\) 6254.04 0.000370434 0
\(779\) 1.71875e6 0.101478
\(780\) 0 0
\(781\) 4.30045e7 2.52282
\(782\) −946787. −0.0553650
\(783\) 0 0
\(784\) 0 0
\(785\) 2.44736e7 1.41750
\(786\) 0 0
\(787\) 2.56449e7 1.47593 0.737963 0.674841i \(-0.235788\pi\)
0.737963 + 0.674841i \(0.235788\pi\)
\(788\) 8.43576e6 0.483959
\(789\) 0 0
\(790\) −1.03838e6 −0.0591955
\(791\) 0 0
\(792\) 0 0
\(793\) 9.26169e6 0.523007
\(794\) 911498. 0.0513103
\(795\) 0 0
\(796\) −1.90472e7 −1.06549
\(797\) 7.73086e6 0.431104 0.215552 0.976492i \(-0.430845\pi\)
0.215552 + 0.976492i \(0.430845\pi\)
\(798\) 0 0
\(799\) −9.84931e6 −0.545807
\(800\) 2.04707e6 0.113086
\(801\) 0 0
\(802\) 1.03448e6 0.0567917
\(803\) 1.90282e7 1.04138
\(804\) 0 0
\(805\) 0 0
\(806\) −966395. −0.0523983
\(807\) 0 0
\(808\) −8.35218e6 −0.450061
\(809\) 1.87811e7 1.00890 0.504452 0.863440i \(-0.331695\pi\)
0.504452 + 0.863440i \(0.331695\pi\)
\(810\) 0 0
\(811\) 9.00729e6 0.480886 0.240443 0.970663i \(-0.422707\pi\)
0.240443 + 0.970663i \(0.422707\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 520422. 0.0275293
\(815\) 2.05531e7 1.08388
\(816\) 0 0
\(817\) 2.59240e7 1.35877
\(818\) −2.22259e6 −0.116138
\(819\) 0 0
\(820\) 1.61590e6 0.0839228
\(821\) 9.27965e6 0.480478 0.240239 0.970714i \(-0.422774\pi\)
0.240239 + 0.970714i \(0.422774\pi\)
\(822\) 0 0
\(823\) −1.08308e7 −0.557393 −0.278697 0.960379i \(-0.589902\pi\)
−0.278697 + 0.960379i \(0.589902\pi\)
\(824\) −2.98107e6 −0.152952
\(825\) 0 0
\(826\) 0 0
\(827\) 2.05230e7 1.04346 0.521731 0.853110i \(-0.325286\pi\)
0.521731 + 0.853110i \(0.325286\pi\)
\(828\) 0 0
\(829\) 1.42216e7 0.718724 0.359362 0.933198i \(-0.382994\pi\)
0.359362 + 0.933198i \(0.382994\pi\)
\(830\) −22912.6 −0.00115446
\(831\) 0 0
\(832\) −1.92472e7 −0.963958
\(833\) 0 0
\(834\) 0 0
\(835\) −1.30595e7 −0.648202
\(836\) 3.28108e7 1.62368
\(837\) 0 0
\(838\) 179995. 0.00885421
\(839\) −9.29934e6 −0.456087 −0.228043 0.973651i \(-0.573233\pi\)
−0.228043 + 0.973651i \(0.573233\pi\)
\(840\) 0 0
\(841\) −1.47669e7 −0.719944
\(842\) −4.32353e6 −0.210164
\(843\) 0 0
\(844\) −1.05561e7 −0.510092
\(845\) −2.43277e6 −0.117209
\(846\) 0 0
\(847\) 0 0
\(848\) −5.37461e6 −0.256659
\(849\) 0 0
\(850\) 808543. 0.0383845
\(851\) 1.18546e6 0.0561131
\(852\) 0 0
\(853\) 3.07436e6 0.144671 0.0723357 0.997380i \(-0.476955\pi\)
0.0723357 + 0.997380i \(0.476955\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.81938e6 0.318098
\(857\) 3.45835e7 1.60848 0.804242 0.594302i \(-0.202572\pi\)
0.804242 + 0.594302i \(0.202572\pi\)
\(858\) 0 0
\(859\) −1.63022e7 −0.753814 −0.376907 0.926251i \(-0.623012\pi\)
−0.376907 + 0.926251i \(0.623012\pi\)
\(860\) 2.43726e7 1.12372
\(861\) 0 0
\(862\) −3.57856e6 −0.164036
\(863\) −2.56962e7 −1.17447 −0.587235 0.809416i \(-0.699784\pi\)
−0.587235 + 0.809416i \(0.699784\pi\)
\(864\) 0 0
\(865\) −4.66087e6 −0.211800
\(866\) 3.01281e6 0.136514
\(867\) 0 0
\(868\) 0 0
\(869\) 2.04046e7 0.916596
\(870\) 0 0
\(871\) 1.28374e7 0.573366
\(872\) −5.12222e6 −0.228122
\(873\) 0 0
\(874\) −1.24844e6 −0.0552829
\(875\) 0 0
\(876\) 0 0
\(877\) 3.30060e7 1.44908 0.724542 0.689230i \(-0.242051\pi\)
0.724542 + 0.689230i \(0.242051\pi\)
\(878\) −165286. −0.00723603
\(879\) 0 0
\(880\) 3.03235e7 1.31999
\(881\) 2.26705e6 0.0984060 0.0492030 0.998789i \(-0.484332\pi\)
0.0492030 + 0.998789i \(0.484332\pi\)
\(882\) 0 0
\(883\) 1.97779e7 0.853649 0.426825 0.904334i \(-0.359632\pi\)
0.426825 + 0.904334i \(0.359632\pi\)
\(884\) −2.43057e7 −1.04611
\(885\) 0 0
\(886\) −1.43901e6 −0.0615858
\(887\) −36468.3 −0.00155635 −0.000778173 1.00000i \(-0.500248\pi\)
−0.000778173 1.00000i \(0.500248\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.27005e6 0.180700
\(891\) 0 0
\(892\) 3.22692e7 1.35793
\(893\) −1.29874e7 −0.544997
\(894\) 0 0
\(895\) 1.54263e7 0.643730
\(896\) 0 0
\(897\) 0 0
\(898\) 1.89303e6 0.0783367
\(899\) −4.90958e6 −0.202602
\(900\) 0 0
\(901\) −6.55068e6 −0.268828
\(902\) 530403. 0.0217065
\(903\) 0 0
\(904\) 1.98711e6 0.0808727
\(905\) −2.36462e7 −0.959709
\(906\) 0 0
\(907\) 1.62298e7 0.655079 0.327540 0.944837i \(-0.393781\pi\)
0.327540 + 0.944837i \(0.393781\pi\)
\(908\) 1.58702e7 0.638802
\(909\) 0 0
\(910\) 0 0
\(911\) 2.61699e7 1.04474 0.522368 0.852720i \(-0.325049\pi\)
0.522368 + 0.852720i \(0.325049\pi\)
\(912\) 0 0
\(913\) 450241. 0.0178759
\(914\) 2.97224e6 0.117684
\(915\) 0 0
\(916\) −3.52320e7 −1.38739
\(917\) 0 0
\(918\) 0 0
\(919\) 4.05973e6 0.158565 0.0792826 0.996852i \(-0.474737\pi\)
0.0792826 + 0.996852i \(0.474737\pi\)
\(920\) −2.36708e6 −0.0922025
\(921\) 0 0
\(922\) 1.90246e6 0.0737034
\(923\) −4.20070e7 −1.62300
\(924\) 0 0
\(925\) −1.01237e6 −0.0389031
\(926\) 3.10744e6 0.119090
\(927\) 0 0
\(928\) 5.22212e6 0.199057
\(929\) 3.69518e6 0.140474 0.0702370 0.997530i \(-0.477624\pi\)
0.0702370 + 0.997530i \(0.477624\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.42573e7 0.914751
\(933\) 0 0
\(934\) −717918. −0.0269282
\(935\) 3.69589e7 1.38258
\(936\) 0 0
\(937\) 1.91384e7 0.712126 0.356063 0.934462i \(-0.384119\pi\)
0.356063 + 0.934462i \(0.384119\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.22102e7 −0.450717
\(941\) 1.20954e7 0.445294 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(942\) 0 0
\(943\) 1.20820e6 0.0442445
\(944\) −1.38577e7 −0.506127
\(945\) 0 0
\(946\) 8.00007e6 0.290647
\(947\) 1.95969e7 0.710087 0.355044 0.934850i \(-0.384466\pi\)
0.355044 + 0.934850i \(0.384466\pi\)
\(948\) 0 0
\(949\) −1.85868e7 −0.669945
\(950\) 1.06615e6 0.0383275
\(951\) 0 0
\(952\) 0 0
\(953\) −4.23265e7 −1.50966 −0.754832 0.655918i \(-0.772282\pi\)
−0.754832 + 0.655918i \(0.772282\pi\)
\(954\) 0 0
\(955\) −2.98426e6 −0.105884
\(956\) −5.40243e6 −0.191181
\(957\) 0 0
\(958\) −1.87054e6 −0.0658496
\(959\) 0 0
\(960\) 0 0
\(961\) −2.44330e7 −0.853430
\(962\) −508351. −0.0177103
\(963\) 0 0
\(964\) −1.20792e7 −0.418644
\(965\) −2.19419e7 −0.758502
\(966\) 0 0
\(967\) 1.39211e6 0.0478749 0.0239375 0.999713i \(-0.492380\pi\)
0.0239375 + 0.999713i \(0.492380\pi\)
\(968\) 1.30075e7 0.446176
\(969\) 0 0
\(970\) 776471. 0.0264970
\(971\) 4.41877e7 1.50402 0.752009 0.659153i \(-0.229085\pi\)
0.752009 + 0.659153i \(0.229085\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.33095e6 −0.0787292
\(975\) 0 0
\(976\) 1.38619e7 0.465798
\(977\) 5.60457e7 1.87848 0.939239 0.343264i \(-0.111533\pi\)
0.939239 + 0.343264i \(0.111533\pi\)
\(978\) 0 0
\(979\) −8.39082e7 −2.79800
\(980\) 0 0
\(981\) 0 0
\(982\) −5.69993e6 −0.188621
\(983\) −5.00560e7 −1.65224 −0.826118 0.563497i \(-0.809456\pi\)
−0.826118 + 0.563497i \(0.809456\pi\)
\(984\) 0 0
\(985\) 1.25298e7 0.411483
\(986\) 2.06261e6 0.0675654
\(987\) 0 0
\(988\) −3.20497e7 −1.04456
\(989\) 1.82233e7 0.592428
\(990\) 0 0
\(991\) 1.59116e7 0.514670 0.257335 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(992\) −4.46330e6 −0.144005
\(993\) 0 0
\(994\) 0 0
\(995\) −2.82911e7 −0.905924
\(996\) 0 0
\(997\) −4.25995e7 −1.35727 −0.678635 0.734476i \(-0.737428\pi\)
−0.678635 + 0.734476i \(0.737428\pi\)
\(998\) 981634. 0.0311978
\(999\) 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 441.6.a.l.1.2 2
3.2 odd 2 49.6.a.f.1.1 2
7.6 odd 2 63.6.a.f.1.2 2
12.11 even 2 784.6.a.v.1.2 2
21.2 odd 6 49.6.c.d.18.2 4
21.5 even 6 49.6.c.e.18.2 4
21.11 odd 6 49.6.c.d.30.2 4
21.17 even 6 49.6.c.e.30.2 4
21.20 even 2 7.6.a.b.1.1 2
28.27 even 2 1008.6.a.bq.1.2 2
84.83 odd 2 112.6.a.h.1.1 2
105.62 odd 4 175.6.b.c.99.3 4
105.83 odd 4 175.6.b.c.99.2 4
105.104 even 2 175.6.a.c.1.2 2
168.83 odd 2 448.6.a.u.1.2 2
168.125 even 2 448.6.a.w.1.1 2
231.230 odd 2 847.6.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.b.1.1 2 21.20 even 2
49.6.a.f.1.1 2 3.2 odd 2
49.6.c.d.18.2 4 21.2 odd 6
49.6.c.d.30.2 4 21.11 odd 6
49.6.c.e.18.2 4 21.5 even 6
49.6.c.e.30.2 4 21.17 even 6
63.6.a.f.1.2 2 7.6 odd 2
112.6.a.h.1.1 2 84.83 odd 2
175.6.a.c.1.2 2 105.104 even 2
175.6.b.c.99.2 4 105.83 odd 4
175.6.b.c.99.3 4 105.62 odd 4
441.6.a.l.1.2 2 1.1 even 1 trivial
448.6.a.u.1.2 2 168.83 odd 2
448.6.a.w.1.1 2 168.125 even 2
784.6.a.v.1.2 2 12.11 even 2
847.6.a.c.1.2 2 231.230 odd 2
1008.6.a.bq.1.2 2 28.27 even 2