Properties

Label 1089.6.a.r.1.1
Level $1089$
Weight $6$
Character 1089.1
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $3$
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] = [1089,6,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.749680\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.3963 q^{2} +76.0833 q^{4} -8.64919 q^{5} -164.454 q^{7} -458.304 q^{8} +O(q^{10})\) \(q-10.3963 q^{2} +76.0833 q^{4} -8.64919 q^{5} -164.454 q^{7} -458.304 q^{8} +89.9197 q^{10} +585.236 q^{13} +1709.71 q^{14} +2330.01 q^{16} -945.333 q^{17} -1148.76 q^{19} -658.060 q^{20} +1346.27 q^{23} -3050.19 q^{25} -6084.30 q^{26} -12512.2 q^{28} +899.585 q^{29} -390.700 q^{31} -9557.75 q^{32} +9827.97 q^{34} +1422.39 q^{35} -4473.41 q^{37} +11942.9 q^{38} +3963.96 q^{40} +16018.7 q^{41} +19905.5 q^{43} -13996.2 q^{46} -1871.38 q^{47} +10238.0 q^{49} +31710.7 q^{50} +44526.7 q^{52} -23565.1 q^{53} +75369.7 q^{56} -9352.37 q^{58} +34709.8 q^{59} -25776.2 q^{61} +4061.84 q^{62} +24805.1 q^{64} -5061.82 q^{65} +55384.6 q^{67} -71924.1 q^{68} -14787.6 q^{70} -56898.4 q^{71} +46871.8 q^{73} +46506.9 q^{74} -87401.5 q^{76} +325.479 q^{79} -20152.7 q^{80} -166536. q^{82} -92908.3 q^{83} +8176.37 q^{85} -206943. q^{86} -23058.0 q^{89} -96244.2 q^{91} +102428. q^{92} +19455.5 q^{94} +9935.84 q^{95} -5013.44 q^{97} -106437. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8} + 414 q^{10} - 486 q^{13} + 1020 q^{14} + 1992 q^{16} + 1086 q^{17} - 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} - 12408 q^{32} + 25320 q^{34} - 24228 q^{35} + 17724 q^{37} + 9240 q^{38} + 15276 q^{40} + 5994 q^{41} + 26208 q^{43} + 16806 q^{46} + 17232 q^{47} + 48531 q^{49} + 41070 q^{50} + 35304 q^{52} - 50586 q^{53} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} - 18486 q^{61} - 19974 q^{62} - 20352 q^{64} - 7668 q^{65} - 47754 q^{67} - 12600 q^{68} - 123372 q^{70} - 39282 q^{71} - 15426 q^{73} + 153294 q^{74} - 103920 q^{76} - 125148 q^{79} - 118680 q^{80} - 255372 q^{82} - 143928 q^{83} + 104040 q^{85} - 243060 q^{86} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 74928 q^{94} - 22200 q^{95} + 9684 q^{97} + 3480 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.3963 −1.83783 −0.918913 0.394460i \(-0.870932\pi\)
−0.918913 + 0.394460i \(0.870932\pi\)
\(3\) 0 0
\(4\) 76.0833 2.37760
\(5\) −8.64919 −0.154721 −0.0773607 0.997003i \(-0.524649\pi\)
−0.0773607 + 0.997003i \(0.524649\pi\)
\(6\) 0 0
\(7\) −164.454 −1.26852 −0.634261 0.773119i \(-0.718696\pi\)
−0.634261 + 0.773119i \(0.718696\pi\)
\(8\) −458.304 −2.53180
\(9\) 0 0
\(10\) 89.9197 0.284351
\(11\) 0 0
\(12\) 0 0
\(13\) 585.236 0.960446 0.480223 0.877147i \(-0.340556\pi\)
0.480223 + 0.877147i \(0.340556\pi\)
\(14\) 1709.71 2.33132
\(15\) 0 0
\(16\) 2330.01 2.27540
\(17\) −945.333 −0.793345 −0.396673 0.917960i \(-0.629835\pi\)
−0.396673 + 0.917960i \(0.629835\pi\)
\(18\) 0 0
\(19\) −1148.76 −0.730037 −0.365019 0.931000i \(-0.618937\pi\)
−0.365019 + 0.931000i \(0.618937\pi\)
\(20\) −658.060 −0.367866
\(21\) 0 0
\(22\) 0 0
\(23\) 1346.27 0.530654 0.265327 0.964158i \(-0.414520\pi\)
0.265327 + 0.964158i \(0.414520\pi\)
\(24\) 0 0
\(25\) −3050.19 −0.976061
\(26\) −6084.30 −1.76513
\(27\) 0 0
\(28\) −12512.2 −3.01604
\(29\) 899.585 0.198631 0.0993155 0.995056i \(-0.468335\pi\)
0.0993155 + 0.995056i \(0.468335\pi\)
\(30\) 0 0
\(31\) −390.700 −0.0730196 −0.0365098 0.999333i \(-0.511624\pi\)
−0.0365098 + 0.999333i \(0.511624\pi\)
\(32\) −9557.75 −1.64999
\(33\) 0 0
\(34\) 9827.97 1.45803
\(35\) 1422.39 0.196268
\(36\) 0 0
\(37\) −4473.41 −0.537198 −0.268599 0.963252i \(-0.586561\pi\)
−0.268599 + 0.963252i \(0.586561\pi\)
\(38\) 11942.9 1.34168
\(39\) 0 0
\(40\) 3963.96 0.391723
\(41\) 16018.7 1.48822 0.744111 0.668056i \(-0.232873\pi\)
0.744111 + 0.668056i \(0.232873\pi\)
\(42\) 0 0
\(43\) 19905.5 1.64173 0.820864 0.571124i \(-0.193492\pi\)
0.820864 + 0.571124i \(0.193492\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −13996.2 −0.975250
\(47\) −1871.38 −0.123571 −0.0617856 0.998089i \(-0.519680\pi\)
−0.0617856 + 0.998089i \(0.519680\pi\)
\(48\) 0 0
\(49\) 10238.0 0.609149
\(50\) 31710.7 1.79383
\(51\) 0 0
\(52\) 44526.7 2.28356
\(53\) −23565.1 −1.15234 −0.576169 0.817330i \(-0.695453\pi\)
−0.576169 + 0.817330i \(0.695453\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 75369.7 3.21164
\(57\) 0 0
\(58\) −9352.37 −0.365049
\(59\) 34709.8 1.29814 0.649071 0.760727i \(-0.275158\pi\)
0.649071 + 0.760727i \(0.275158\pi\)
\(60\) 0 0
\(61\) −25776.2 −0.886940 −0.443470 0.896289i \(-0.646253\pi\)
−0.443470 + 0.896289i \(0.646253\pi\)
\(62\) 4061.84 0.134197
\(63\) 0 0
\(64\) 24805.1 0.756993
\(65\) −5061.82 −0.148602
\(66\) 0 0
\(67\) 55384.6 1.50731 0.753655 0.657271i \(-0.228289\pi\)
0.753655 + 0.657271i \(0.228289\pi\)
\(68\) −71924.1 −1.88626
\(69\) 0 0
\(70\) −14787.6 −0.360706
\(71\) −56898.4 −1.33954 −0.669768 0.742571i \(-0.733606\pi\)
−0.669768 + 0.742571i \(0.733606\pi\)
\(72\) 0 0
\(73\) 46871.8 1.02945 0.514724 0.857356i \(-0.327894\pi\)
0.514724 + 0.857356i \(0.327894\pi\)
\(74\) 46506.9 0.987276
\(75\) 0 0
\(76\) −87401.5 −1.73574
\(77\) 0 0
\(78\) 0 0
\(79\) 325.479 0.00586753 0.00293377 0.999996i \(-0.499066\pi\)
0.00293377 + 0.999996i \(0.499066\pi\)
\(80\) −20152.7 −0.352053
\(81\) 0 0
\(82\) −166536. −2.73509
\(83\) −92908.3 −1.48033 −0.740166 0.672424i \(-0.765253\pi\)
−0.740166 + 0.672424i \(0.765253\pi\)
\(84\) 0 0
\(85\) 8176.37 0.122748
\(86\) −206943. −3.01721
\(87\) 0 0
\(88\) 0 0
\(89\) −23058.0 −0.308565 −0.154283 0.988027i \(-0.549307\pi\)
−0.154283 + 0.988027i \(0.549307\pi\)
\(90\) 0 0
\(91\) −96244.2 −1.21835
\(92\) 102428. 1.26169
\(93\) 0 0
\(94\) 19455.5 0.227103
\(95\) 9935.84 0.112952
\(96\) 0 0
\(97\) −5013.44 −0.0541011 −0.0270506 0.999634i \(-0.508612\pi\)
−0.0270506 + 0.999634i \(0.508612\pi\)
\(98\) −106437. −1.11951
\(99\) 0 0
\(100\) −232069. −2.32069
\(101\) 37928.6 0.369968 0.184984 0.982742i \(-0.440777\pi\)
0.184984 + 0.982742i \(0.440777\pi\)
\(102\) 0 0
\(103\) 180296. 1.67453 0.837265 0.546798i \(-0.184153\pi\)
0.837265 + 0.546798i \(0.184153\pi\)
\(104\) −268216. −2.43165
\(105\) 0 0
\(106\) 244990. 2.11780
\(107\) 92860.5 0.784100 0.392050 0.919944i \(-0.371766\pi\)
0.392050 + 0.919944i \(0.371766\pi\)
\(108\) 0 0
\(109\) −180736. −1.45707 −0.728533 0.685011i \(-0.759797\pi\)
−0.728533 + 0.685011i \(0.759797\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −383178. −2.88639
\(113\) 68275.4 0.503000 0.251500 0.967857i \(-0.419076\pi\)
0.251500 + 0.967857i \(0.419076\pi\)
\(114\) 0 0
\(115\) −11644.1 −0.0821036
\(116\) 68443.4 0.472266
\(117\) 0 0
\(118\) −360854. −2.38576
\(119\) 155463. 1.00638
\(120\) 0 0
\(121\) 0 0
\(122\) 267977. 1.63004
\(123\) 0 0
\(124\) −29725.8 −0.173612
\(125\) 53410.4 0.305739
\(126\) 0 0
\(127\) −27233.1 −0.149826 −0.0749130 0.997190i \(-0.523868\pi\)
−0.0749130 + 0.997190i \(0.523868\pi\)
\(128\) 47966.0 0.258767
\(129\) 0 0
\(130\) 52624.3 0.273104
\(131\) −11887.1 −0.0605199 −0.0302600 0.999542i \(-0.509634\pi\)
−0.0302600 + 0.999542i \(0.509634\pi\)
\(132\) 0 0
\(133\) 188918. 0.926069
\(134\) −575796. −2.77017
\(135\) 0 0
\(136\) 433250. 2.00859
\(137\) −35302.2 −0.160694 −0.0803471 0.996767i \(-0.525603\pi\)
−0.0803471 + 0.996767i \(0.525603\pi\)
\(138\) 0 0
\(139\) −26248.0 −0.115228 −0.0576141 0.998339i \(-0.518349\pi\)
−0.0576141 + 0.998339i \(0.518349\pi\)
\(140\) 108220. 0.466647
\(141\) 0 0
\(142\) 591533. 2.46183
\(143\) 0 0
\(144\) 0 0
\(145\) −7780.68 −0.0307325
\(146\) −487294. −1.89195
\(147\) 0 0
\(148\) −340352. −1.27724
\(149\) −226321. −0.835139 −0.417570 0.908645i \(-0.637118\pi\)
−0.417570 + 0.908645i \(0.637118\pi\)
\(150\) 0 0
\(151\) 301067. 1.07453 0.537267 0.843412i \(-0.319457\pi\)
0.537267 + 0.843412i \(0.319457\pi\)
\(152\) 526481. 1.84831
\(153\) 0 0
\(154\) 0 0
\(155\) 3379.24 0.0112977
\(156\) 0 0
\(157\) 341482. 1.10565 0.552827 0.833296i \(-0.313549\pi\)
0.552827 + 0.833296i \(0.313549\pi\)
\(158\) −3383.78 −0.0107835
\(159\) 0 0
\(160\) 82666.8 0.255289
\(161\) −221398. −0.673147
\(162\) 0 0
\(163\) 604612. 1.78241 0.891205 0.453600i \(-0.149861\pi\)
0.891205 + 0.453600i \(0.149861\pi\)
\(164\) 1.21876e6 3.53840
\(165\) 0 0
\(166\) 965904. 2.72059
\(167\) −159824. −0.443455 −0.221728 0.975109i \(-0.571170\pi\)
−0.221728 + 0.975109i \(0.571170\pi\)
\(168\) 0 0
\(169\) −28791.6 −0.0775442
\(170\) −85004.1 −0.225589
\(171\) 0 0
\(172\) 1.51447e6 3.90338
\(173\) −499771. −1.26957 −0.634783 0.772690i \(-0.718911\pi\)
−0.634783 + 0.772690i \(0.718911\pi\)
\(174\) 0 0
\(175\) 501615. 1.23816
\(176\) 0 0
\(177\) 0 0
\(178\) 239719. 0.567090
\(179\) 626569. 1.46163 0.730813 0.682578i \(-0.239141\pi\)
0.730813 + 0.682578i \(0.239141\pi\)
\(180\) 0 0
\(181\) 393700. 0.893243 0.446621 0.894723i \(-0.352627\pi\)
0.446621 + 0.894723i \(0.352627\pi\)
\(182\) 1.00058e6 2.23911
\(183\) 0 0
\(184\) −617000. −1.34351
\(185\) 38691.4 0.0831160
\(186\) 0 0
\(187\) 0 0
\(188\) −142381. −0.293804
\(189\) 0 0
\(190\) −103296. −0.207587
\(191\) −205468. −0.407531 −0.203766 0.979020i \(-0.565318\pi\)
−0.203766 + 0.979020i \(0.565318\pi\)
\(192\) 0 0
\(193\) 349786. 0.675941 0.337971 0.941157i \(-0.390260\pi\)
0.337971 + 0.941157i \(0.390260\pi\)
\(194\) 52121.3 0.0994285
\(195\) 0 0
\(196\) 778938. 1.44831
\(197\) 863902. 1.58598 0.792992 0.609232i \(-0.208522\pi\)
0.792992 + 0.609232i \(0.208522\pi\)
\(198\) 0 0
\(199\) −610140. −1.09219 −0.546093 0.837725i \(-0.683885\pi\)
−0.546093 + 0.837725i \(0.683885\pi\)
\(200\) 1.39792e6 2.47119
\(201\) 0 0
\(202\) −394318. −0.679936
\(203\) −147940. −0.251968
\(204\) 0 0
\(205\) −138549. −0.230260
\(206\) −1.87441e6 −3.07749
\(207\) 0 0
\(208\) 1.36360e6 2.18540
\(209\) 0 0
\(210\) 0 0
\(211\) −166602. −0.257616 −0.128808 0.991670i \(-0.541115\pi\)
−0.128808 + 0.991670i \(0.541115\pi\)
\(212\) −1.79291e6 −2.73981
\(213\) 0 0
\(214\) −965407. −1.44104
\(215\) −172166. −0.254011
\(216\) 0 0
\(217\) 64252.0 0.0926270
\(218\) 1.87899e6 2.67783
\(219\) 0 0
\(220\) 0 0
\(221\) −553243. −0.761965
\(222\) 0 0
\(223\) −1.05575e6 −1.42167 −0.710836 0.703358i \(-0.751683\pi\)
−0.710836 + 0.703358i \(0.751683\pi\)
\(224\) 1.57181e6 2.09305
\(225\) 0 0
\(226\) −709812. −0.924427
\(227\) 526562. 0.678242 0.339121 0.940743i \(-0.389870\pi\)
0.339121 + 0.940743i \(0.389870\pi\)
\(228\) 0 0
\(229\) 1.11694e6 1.40748 0.703740 0.710458i \(-0.251512\pi\)
0.703740 + 0.710458i \(0.251512\pi\)
\(230\) 121056. 0.150892
\(231\) 0 0
\(232\) −412283. −0.502893
\(233\) −29262.0 −0.0353113 −0.0176557 0.999844i \(-0.505620\pi\)
−0.0176557 + 0.999844i \(0.505620\pi\)
\(234\) 0 0
\(235\) 16185.9 0.0191191
\(236\) 2.64084e6 3.08647
\(237\) 0 0
\(238\) −1.61625e6 −1.84954
\(239\) 822476. 0.931384 0.465692 0.884947i \(-0.345806\pi\)
0.465692 + 0.884947i \(0.345806\pi\)
\(240\) 0 0
\(241\) −762439. −0.845595 −0.422797 0.906224i \(-0.638952\pi\)
−0.422797 + 0.906224i \(0.638952\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.96114e6 −2.10879
\(245\) −88550.1 −0.0942484
\(246\) 0 0
\(247\) −672296. −0.701161
\(248\) 179060. 0.184871
\(249\) 0 0
\(250\) −555272. −0.561895
\(251\) −561364. −0.562419 −0.281209 0.959646i \(-0.590736\pi\)
−0.281209 + 0.959646i \(0.590736\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 283123. 0.275354
\(255\) 0 0
\(256\) −1.29243e6 −1.23256
\(257\) 764965. 0.722451 0.361226 0.932478i \(-0.382358\pi\)
0.361226 + 0.932478i \(0.382358\pi\)
\(258\) 0 0
\(259\) 735668. 0.681447
\(260\) −385120. −0.353316
\(261\) 0 0
\(262\) 123582. 0.111225
\(263\) −763627. −0.680756 −0.340378 0.940289i \(-0.610555\pi\)
−0.340378 + 0.940289i \(0.610555\pi\)
\(264\) 0 0
\(265\) 203819. 0.178292
\(266\) −1.96405e6 −1.70195
\(267\) 0 0
\(268\) 4.21385e6 3.58378
\(269\) −800885. −0.674823 −0.337411 0.941357i \(-0.609551\pi\)
−0.337411 + 0.941357i \(0.609551\pi\)
\(270\) 0 0
\(271\) −98139.7 −0.0811749 −0.0405874 0.999176i \(-0.512923\pi\)
−0.0405874 + 0.999176i \(0.512923\pi\)
\(272\) −2.20263e6 −1.80518
\(273\) 0 0
\(274\) 367013. 0.295328
\(275\) 0 0
\(276\) 0 0
\(277\) 620993. 0.486281 0.243140 0.969991i \(-0.421822\pi\)
0.243140 + 0.969991i \(0.421822\pi\)
\(278\) 272882. 0.211769
\(279\) 0 0
\(280\) −651887. −0.496910
\(281\) 1.31191e6 0.991149 0.495575 0.868565i \(-0.334957\pi\)
0.495575 + 0.868565i \(0.334957\pi\)
\(282\) 0 0
\(283\) 26897.8 0.0199641 0.00998205 0.999950i \(-0.496823\pi\)
0.00998205 + 0.999950i \(0.496823\pi\)
\(284\) −4.32902e6 −3.18488
\(285\) 0 0
\(286\) 0 0
\(287\) −2.63433e6 −1.88784
\(288\) 0 0
\(289\) −526203. −0.370603
\(290\) 80890.4 0.0564810
\(291\) 0 0
\(292\) 3.56617e6 2.44762
\(293\) 638546. 0.434534 0.217267 0.976112i \(-0.430286\pi\)
0.217267 + 0.976112i \(0.430286\pi\)
\(294\) 0 0
\(295\) −300212. −0.200851
\(296\) 2.05018e6 1.36008
\(297\) 0 0
\(298\) 2.35290e6 1.53484
\(299\) 787884. 0.509665
\(300\) 0 0
\(301\) −3.27352e6 −2.08257
\(302\) −3.12998e6 −1.97481
\(303\) 0 0
\(304\) −2.67662e6 −1.66113
\(305\) 222943. 0.137229
\(306\) 0 0
\(307\) −550428. −0.333315 −0.166658 0.986015i \(-0.553297\pi\)
−0.166658 + 0.986015i \(0.553297\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −35131.7 −0.0207632
\(311\) −186775. −0.109501 −0.0547504 0.998500i \(-0.517436\pi\)
−0.0547504 + 0.998500i \(0.517436\pi\)
\(312\) 0 0
\(313\) −934239. −0.539010 −0.269505 0.962999i \(-0.586860\pi\)
−0.269505 + 0.962999i \(0.586860\pi\)
\(314\) −3.55016e6 −2.03200
\(315\) 0 0
\(316\) 24763.5 0.0139507
\(317\) 1.88280e6 1.05234 0.526170 0.850379i \(-0.323628\pi\)
0.526170 + 0.850379i \(0.323628\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −214544. −0.117123
\(321\) 0 0
\(322\) 2.30173e6 1.23713
\(323\) 1.08596e6 0.579172
\(324\) 0 0
\(325\) −1.78508e6 −0.937454
\(326\) −6.28574e6 −3.27576
\(327\) 0 0
\(328\) −7.34144e6 −3.76788
\(329\) 307755. 0.156753
\(330\) 0 0
\(331\) 197056. 0.0988596 0.0494298 0.998778i \(-0.484260\pi\)
0.0494298 + 0.998778i \(0.484260\pi\)
\(332\) −7.06877e6 −3.51964
\(333\) 0 0
\(334\) 1.66158e6 0.814993
\(335\) −479033. −0.233213
\(336\) 0 0
\(337\) −387484. −0.185857 −0.0929285 0.995673i \(-0.529623\pi\)
−0.0929285 + 0.995673i \(0.529623\pi\)
\(338\) 299327. 0.142513
\(339\) 0 0
\(340\) 622085. 0.291845
\(341\) 0 0
\(342\) 0 0
\(343\) 1.08030e6 0.495803
\(344\) −9.12276e6 −4.15652
\(345\) 0 0
\(346\) 5.19577e6 2.33324
\(347\) −2.94793e6 −1.31430 −0.657148 0.753761i \(-0.728238\pi\)
−0.657148 + 0.753761i \(0.728238\pi\)
\(348\) 0 0
\(349\) 924908. 0.406476 0.203238 0.979129i \(-0.434854\pi\)
0.203238 + 0.979129i \(0.434854\pi\)
\(350\) −5.21494e6 −2.27551
\(351\) 0 0
\(352\) 0 0
\(353\) 4.46816e6 1.90850 0.954249 0.299012i \(-0.0966570\pi\)
0.954249 + 0.299012i \(0.0966570\pi\)
\(354\) 0 0
\(355\) 492125. 0.207255
\(356\) −1.75433e6 −0.733647
\(357\) 0 0
\(358\) −6.51401e6 −2.68621
\(359\) −995937. −0.407846 −0.203923 0.978987i \(-0.565369\pi\)
−0.203923 + 0.978987i \(0.565369\pi\)
\(360\) 0 0
\(361\) −1.15645e6 −0.467045
\(362\) −4.09303e6 −1.64162
\(363\) 0 0
\(364\) −7.32258e6 −2.89675
\(365\) −405404. −0.159278
\(366\) 0 0
\(367\) −1.21088e6 −0.469284 −0.234642 0.972082i \(-0.575392\pi\)
−0.234642 + 0.972082i \(0.575392\pi\)
\(368\) 3.13681e6 1.20745
\(369\) 0 0
\(370\) −402248. −0.152753
\(371\) 3.87537e6 1.46177
\(372\) 0 0
\(373\) −1.82235e6 −0.678203 −0.339102 0.940750i \(-0.610123\pi\)
−0.339102 + 0.940750i \(0.610123\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 857662. 0.312857
\(377\) 526470. 0.190774
\(378\) 0 0
\(379\) −419357. −0.149964 −0.0749819 0.997185i \(-0.523890\pi\)
−0.0749819 + 0.997185i \(0.523890\pi\)
\(380\) 755952. 0.268556
\(381\) 0 0
\(382\) 2.13611e6 0.748972
\(383\) −2.95656e6 −1.02989 −0.514943 0.857224i \(-0.672187\pi\)
−0.514943 + 0.857224i \(0.672187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.63648e6 −1.24226
\(387\) 0 0
\(388\) −381439. −0.128631
\(389\) −2.35429e6 −0.788834 −0.394417 0.918932i \(-0.629053\pi\)
−0.394417 + 0.918932i \(0.629053\pi\)
\(390\) 0 0
\(391\) −1.27267e6 −0.420992
\(392\) −4.69210e6 −1.54224
\(393\) 0 0
\(394\) −8.98139e6 −2.91476
\(395\) −2815.13 −0.000907833 0
\(396\) 0 0
\(397\) 3.94809e6 1.25722 0.628609 0.777722i \(-0.283625\pi\)
0.628609 + 0.777722i \(0.283625\pi\)
\(398\) 6.34320e6 2.00725
\(399\) 0 0
\(400\) −7.10697e6 −2.22093
\(401\) 5.76535e6 1.79046 0.895230 0.445604i \(-0.147011\pi\)
0.895230 + 0.445604i \(0.147011\pi\)
\(402\) 0 0
\(403\) −228652. −0.0701314
\(404\) 2.88574e6 0.879637
\(405\) 0 0
\(406\) 1.53803e6 0.463073
\(407\) 0 0
\(408\) 0 0
\(409\) −2.39693e6 −0.708512 −0.354256 0.935148i \(-0.615266\pi\)
−0.354256 + 0.935148i \(0.615266\pi\)
\(410\) 1.44040e6 0.423178
\(411\) 0 0
\(412\) 1.37175e7 3.98137
\(413\) −5.70815e6 −1.64672
\(414\) 0 0
\(415\) 803582. 0.229039
\(416\) −5.59354e6 −1.58472
\(417\) 0 0
\(418\) 0 0
\(419\) 1.41668e6 0.394220 0.197110 0.980381i \(-0.436844\pi\)
0.197110 + 0.980381i \(0.436844\pi\)
\(420\) 0 0
\(421\) −4.80538e6 −1.32136 −0.660682 0.750666i \(-0.729733\pi\)
−0.660682 + 0.750666i \(0.729733\pi\)
\(422\) 1.73204e6 0.473454
\(423\) 0 0
\(424\) 1.08000e7 2.91749
\(425\) 2.88345e6 0.774354
\(426\) 0 0
\(427\) 4.23899e6 1.12510
\(428\) 7.06514e6 1.86428
\(429\) 0 0
\(430\) 1.78989e6 0.466827
\(431\) −2.73465e6 −0.709103 −0.354551 0.935037i \(-0.615366\pi\)
−0.354551 + 0.935037i \(0.615366\pi\)
\(432\) 0 0
\(433\) 2.71922e6 0.696986 0.348493 0.937311i \(-0.386693\pi\)
0.348493 + 0.937311i \(0.386693\pi\)
\(434\) −667984. −0.170232
\(435\) 0 0
\(436\) −1.37510e7 −3.46433
\(437\) −1.54654e6 −0.387397
\(438\) 0 0
\(439\) 4.17101e6 1.03295 0.516476 0.856301i \(-0.327243\pi\)
0.516476 + 0.856301i \(0.327243\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.75169e6 1.40036
\(443\) −6.86870e6 −1.66290 −0.831448 0.555603i \(-0.812487\pi\)
−0.831448 + 0.555603i \(0.812487\pi\)
\(444\) 0 0
\(445\) 199433. 0.0477417
\(446\) 1.09759e7 2.61278
\(447\) 0 0
\(448\) −4.07929e6 −0.960262
\(449\) 693812. 0.162415 0.0812075 0.996697i \(-0.474122\pi\)
0.0812075 + 0.996697i \(0.474122\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 5.19462e6 1.19594
\(453\) 0 0
\(454\) −5.47431e6 −1.24649
\(455\) 832434. 0.188504
\(456\) 0 0
\(457\) −8.12461e6 −1.81975 −0.909876 0.414880i \(-0.863824\pi\)
−0.909876 + 0.414880i \(0.863824\pi\)
\(458\) −1.16121e7 −2.58670
\(459\) 0 0
\(460\) −885924. −0.195210
\(461\) −4.48975e6 −0.983944 −0.491972 0.870611i \(-0.663724\pi\)
−0.491972 + 0.870611i \(0.663724\pi\)
\(462\) 0 0
\(463\) −9.04494e6 −1.96089 −0.980445 0.196793i \(-0.936947\pi\)
−0.980445 + 0.196793i \(0.936947\pi\)
\(464\) 2.09604e6 0.451965
\(465\) 0 0
\(466\) 304217. 0.0648961
\(467\) −7.17275e6 −1.52192 −0.760962 0.648796i \(-0.775273\pi\)
−0.760962 + 0.648796i \(0.775273\pi\)
\(468\) 0 0
\(469\) −9.10820e6 −1.91206
\(470\) −168274. −0.0351376
\(471\) 0 0
\(472\) −1.59077e7 −3.28663
\(473\) 0 0
\(474\) 0 0
\(475\) 3.50394e6 0.712561
\(476\) 1.18282e7 2.39276
\(477\) 0 0
\(478\) −8.55072e6 −1.71172
\(479\) −1.51089e6 −0.300881 −0.150440 0.988619i \(-0.548069\pi\)
−0.150440 + 0.988619i \(0.548069\pi\)
\(480\) 0 0
\(481\) −2.61800e6 −0.515949
\(482\) 7.92655e6 1.55406
\(483\) 0 0
\(484\) 0 0
\(485\) 43362.2 0.00837061
\(486\) 0 0
\(487\) 1.63265e6 0.311940 0.155970 0.987762i \(-0.450150\pi\)
0.155970 + 0.987762i \(0.450150\pi\)
\(488\) 1.18133e7 2.24555
\(489\) 0 0
\(490\) 920595. 0.173212
\(491\) 1.24459e6 0.232982 0.116491 0.993192i \(-0.462835\pi\)
0.116491 + 0.993192i \(0.462835\pi\)
\(492\) 0 0
\(493\) −850407. −0.157583
\(494\) 6.98940e6 1.28861
\(495\) 0 0
\(496\) −910334. −0.166149
\(497\) 9.35714e6 1.69923
\(498\) 0 0
\(499\) −7.66211e6 −1.37752 −0.688759 0.724991i \(-0.741844\pi\)
−0.688759 + 0.724991i \(0.741844\pi\)
\(500\) 4.06364e6 0.726927
\(501\) 0 0
\(502\) 5.83611e6 1.03363
\(503\) −1.07030e7 −1.88619 −0.943097 0.332518i \(-0.892102\pi\)
−0.943097 + 0.332518i \(0.892102\pi\)
\(504\) 0 0
\(505\) −328052. −0.0572420
\(506\) 0 0
\(507\) 0 0
\(508\) −2.07198e6 −0.356227
\(509\) −6.27494e6 −1.07353 −0.536766 0.843731i \(-0.680354\pi\)
−0.536766 + 0.843731i \(0.680354\pi\)
\(510\) 0 0
\(511\) −7.70824e6 −1.30588
\(512\) 1.19016e7 2.00647
\(513\) 0 0
\(514\) −7.95281e6 −1.32774
\(515\) −1.55941e6 −0.259086
\(516\) 0 0
\(517\) 0 0
\(518\) −7.64823e6 −1.25238
\(519\) 0 0
\(520\) 2.31985e6 0.376229
\(521\) −3.60326e6 −0.581570 −0.290785 0.956788i \(-0.593916\pi\)
−0.290785 + 0.956788i \(0.593916\pi\)
\(522\) 0 0
\(523\) −3.56925e6 −0.570589 −0.285294 0.958440i \(-0.592091\pi\)
−0.285294 + 0.958440i \(0.592091\pi\)
\(524\) −904412. −0.143892
\(525\) 0 0
\(526\) 7.93890e6 1.25111
\(527\) 369342. 0.0579298
\(528\) 0 0
\(529\) −4.62391e6 −0.718406
\(530\) −2.11897e6 −0.327669
\(531\) 0 0
\(532\) 1.43735e7 2.20183
\(533\) 9.37473e6 1.42936
\(534\) 0 0
\(535\) −803169. −0.121317
\(536\) −2.53830e7 −3.81620
\(537\) 0 0
\(538\) 8.32625e6 1.24021
\(539\) 0 0
\(540\) 0 0
\(541\) −1.16842e7 −1.71635 −0.858176 0.513355i \(-0.828402\pi\)
−0.858176 + 0.513355i \(0.828402\pi\)
\(542\) 1.02029e6 0.149185
\(543\) 0 0
\(544\) 9.03525e6 1.30901
\(545\) 1.56322e6 0.225439
\(546\) 0 0
\(547\) 4.05598e6 0.579600 0.289800 0.957087i \(-0.406411\pi\)
0.289800 + 0.957087i \(0.406411\pi\)
\(548\) −2.68591e6 −0.382067
\(549\) 0 0
\(550\) 0 0
\(551\) −1.03341e6 −0.145008
\(552\) 0 0
\(553\) −53526.2 −0.00744309
\(554\) −6.45603e6 −0.893699
\(555\) 0 0
\(556\) −1.99703e6 −0.273967
\(557\) −284385. −0.0388390 −0.0194195 0.999811i \(-0.506182\pi\)
−0.0194195 + 0.999811i \(0.506182\pi\)
\(558\) 0 0
\(559\) 1.16494e7 1.57679
\(560\) 3.31418e6 0.446587
\(561\) 0 0
\(562\) −1.36391e7 −1.82156
\(563\) 1.20582e6 0.160329 0.0801646 0.996782i \(-0.474455\pi\)
0.0801646 + 0.996782i \(0.474455\pi\)
\(564\) 0 0
\(565\) −590527. −0.0778249
\(566\) −279637. −0.0366906
\(567\) 0 0
\(568\) 2.60768e7 3.39143
\(569\) 3.94580e6 0.510922 0.255461 0.966819i \(-0.417773\pi\)
0.255461 + 0.966819i \(0.417773\pi\)
\(570\) 0 0
\(571\) −5.88346e6 −0.755166 −0.377583 0.925976i \(-0.623245\pi\)
−0.377583 + 0.925976i \(0.623245\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 2.73874e7 3.46953
\(575\) −4.10637e6 −0.517951
\(576\) 0 0
\(577\) 3.61005e6 0.451413 0.225706 0.974195i \(-0.427531\pi\)
0.225706 + 0.974195i \(0.427531\pi\)
\(578\) 5.47057e6 0.681104
\(579\) 0 0
\(580\) −591980. −0.0730697
\(581\) 1.52791e7 1.87783
\(582\) 0 0
\(583\) 0 0
\(584\) −2.14816e7 −2.60636
\(585\) 0 0
\(586\) −6.63853e6 −0.798597
\(587\) −5.01469e6 −0.600688 −0.300344 0.953831i \(-0.597101\pi\)
−0.300344 + 0.953831i \(0.597101\pi\)
\(588\) 0 0
\(589\) 448821. 0.0533070
\(590\) 3.12110e6 0.369128
\(591\) 0 0
\(592\) −1.04231e7 −1.22234
\(593\) 1.64451e7 1.92044 0.960220 0.279244i \(-0.0900838\pi\)
0.960220 + 0.279244i \(0.0900838\pi\)
\(594\) 0 0
\(595\) −1.34463e6 −0.155708
\(596\) −1.72192e7 −1.98563
\(597\) 0 0
\(598\) −8.19109e6 −0.936675
\(599\) −6.45089e6 −0.734603 −0.367302 0.930102i \(-0.619718\pi\)
−0.367302 + 0.930102i \(0.619718\pi\)
\(600\) 0 0
\(601\) 8.32443e6 0.940087 0.470044 0.882643i \(-0.344238\pi\)
0.470044 + 0.882643i \(0.344238\pi\)
\(602\) 3.40326e7 3.82740
\(603\) 0 0
\(604\) 2.29062e7 2.55482
\(605\) 0 0
\(606\) 0 0
\(607\) −1.47290e7 −1.62256 −0.811279 0.584659i \(-0.801228\pi\)
−0.811279 + 0.584659i \(0.801228\pi\)
\(608\) 1.09796e7 1.20455
\(609\) 0 0
\(610\) −2.31779e6 −0.252202
\(611\) −1.09520e6 −0.118683
\(612\) 0 0
\(613\) 1.14564e7 1.23139 0.615697 0.787983i \(-0.288874\pi\)
0.615697 + 0.787983i \(0.288874\pi\)
\(614\) 5.72243e6 0.612575
\(615\) 0 0
\(616\) 0 0
\(617\) 413797. 0.0437597 0.0218799 0.999761i \(-0.493035\pi\)
0.0218799 + 0.999761i \(0.493035\pi\)
\(618\) 0 0
\(619\) −1.25898e7 −1.32067 −0.660333 0.750973i \(-0.729585\pi\)
−0.660333 + 0.750973i \(0.729585\pi\)
\(620\) 257104. 0.0268615
\(621\) 0 0
\(622\) 1.94177e6 0.201243
\(623\) 3.79198e6 0.391422
\(624\) 0 0
\(625\) 9.06989e6 0.928757
\(626\) 9.71264e6 0.990607
\(627\) 0 0
\(628\) 2.59811e7 2.62881
\(629\) 4.22886e6 0.426183
\(630\) 0 0
\(631\) −1.55648e7 −1.55621 −0.778107 0.628132i \(-0.783820\pi\)
−0.778107 + 0.628132i \(0.783820\pi\)
\(632\) −149168. −0.0148554
\(633\) 0 0
\(634\) −1.95742e7 −1.93402
\(635\) 235544. 0.0231813
\(636\) 0 0
\(637\) 5.99163e6 0.585054
\(638\) 0 0
\(639\) 0 0
\(640\) −414867. −0.0400368
\(641\) 1.23075e7 1.18311 0.591556 0.806264i \(-0.298514\pi\)
0.591556 + 0.806264i \(0.298514\pi\)
\(642\) 0 0
\(643\) −1.44400e6 −0.137733 −0.0688667 0.997626i \(-0.521938\pi\)
−0.0688667 + 0.997626i \(0.521938\pi\)
\(644\) −1.68447e7 −1.60048
\(645\) 0 0
\(646\) −1.12900e7 −1.06442
\(647\) 7.35610e6 0.690855 0.345427 0.938445i \(-0.387734\pi\)
0.345427 + 0.938445i \(0.387734\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.85583e7 1.72288
\(651\) 0 0
\(652\) 4.60009e7 4.23787
\(653\) −3.83734e6 −0.352166 −0.176083 0.984375i \(-0.556343\pi\)
−0.176083 + 0.984375i \(0.556343\pi\)
\(654\) 0 0
\(655\) 102814. 0.00936373
\(656\) 3.73237e7 3.38630
\(657\) 0 0
\(658\) −3.19952e6 −0.288085
\(659\) −1.98049e7 −1.77648 −0.888239 0.459382i \(-0.848071\pi\)
−0.888239 + 0.459382i \(0.848071\pi\)
\(660\) 0 0
\(661\) 1.75724e7 1.56433 0.782164 0.623073i \(-0.214116\pi\)
0.782164 + 0.623073i \(0.214116\pi\)
\(662\) −2.04865e6 −0.181687
\(663\) 0 0
\(664\) 4.25803e7 3.74790
\(665\) −1.63398e6 −0.143283
\(666\) 0 0
\(667\) 1.21108e6 0.105404
\(668\) −1.21599e7 −1.05436
\(669\) 0 0
\(670\) 4.98017e6 0.428605
\(671\) 0 0
\(672\) 0 0
\(673\) −4.88569e6 −0.415804 −0.207902 0.978150i \(-0.566663\pi\)
−0.207902 + 0.978150i \(0.566663\pi\)
\(674\) 4.02840e6 0.341573
\(675\) 0 0
\(676\) −2.19056e6 −0.184369
\(677\) −1.56799e7 −1.31483 −0.657416 0.753528i \(-0.728351\pi\)
−0.657416 + 0.753528i \(0.728351\pi\)
\(678\) 0 0
\(679\) 824478. 0.0686285
\(680\) −3.74726e6 −0.310772
\(681\) 0 0
\(682\) 0 0
\(683\) 1.94703e6 0.159705 0.0798527 0.996807i \(-0.474555\pi\)
0.0798527 + 0.996807i \(0.474555\pi\)
\(684\) 0 0
\(685\) 305336. 0.0248629
\(686\) −1.12311e7 −0.911200
\(687\) 0 0
\(688\) 4.63799e7 3.73558
\(689\) −1.37912e7 −1.10676
\(690\) 0 0
\(691\) −5.39805e6 −0.430073 −0.215036 0.976606i \(-0.568987\pi\)
−0.215036 + 0.976606i \(0.568987\pi\)
\(692\) −3.80242e7 −3.01853
\(693\) 0 0
\(694\) 3.06476e7 2.41545
\(695\) 227024. 0.0178283
\(696\) 0 0
\(697\) −1.51430e7 −1.18067
\(698\) −9.61563e6 −0.747032
\(699\) 0 0
\(700\) 3.81645e7 2.94384
\(701\) 5.48228e6 0.421373 0.210686 0.977554i \(-0.432430\pi\)
0.210686 + 0.977554i \(0.432430\pi\)
\(702\) 0 0
\(703\) 5.13887e6 0.392174
\(704\) 0 0
\(705\) 0 0
\(706\) −4.64524e7 −3.50749
\(707\) −6.23750e6 −0.469312
\(708\) 0 0
\(709\) −1.30530e7 −0.975200 −0.487600 0.873067i \(-0.662128\pi\)
−0.487600 + 0.873067i \(0.662128\pi\)
\(710\) −5.11629e6 −0.380898
\(711\) 0 0
\(712\) 1.05676e7 0.781225
\(713\) −525987. −0.0387482
\(714\) 0 0
\(715\) 0 0
\(716\) 4.76714e7 3.47517
\(717\) 0 0
\(718\) 1.03541e7 0.749550
\(719\) 2.17045e7 1.56577 0.782885 0.622167i \(-0.213748\pi\)
0.782885 + 0.622167i \(0.213748\pi\)
\(720\) 0 0
\(721\) −2.96503e7 −2.12418
\(722\) 1.20228e7 0.858348
\(723\) 0 0
\(724\) 2.99540e7 2.12378
\(725\) −2.74391e6 −0.193876
\(726\) 0 0
\(727\) −1.07681e7 −0.755619 −0.377809 0.925883i \(-0.623323\pi\)
−0.377809 + 0.925883i \(0.623323\pi\)
\(728\) 4.41091e7 3.08461
\(729\) 0 0
\(730\) 4.21470e6 0.292725
\(731\) −1.88173e7 −1.30246
\(732\) 0 0
\(733\) −7.35742e6 −0.505785 −0.252892 0.967494i \(-0.581382\pi\)
−0.252892 + 0.967494i \(0.581382\pi\)
\(734\) 1.25887e7 0.862463
\(735\) 0 0
\(736\) −1.28673e7 −0.875573
\(737\) 0 0
\(738\) 0 0
\(739\) 6.81140e6 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(740\) 2.94377e6 0.197617
\(741\) 0 0
\(742\) −4.02895e7 −2.68647
\(743\) −1.01182e7 −0.672405 −0.336203 0.941790i \(-0.609143\pi\)
−0.336203 + 0.941790i \(0.609143\pi\)
\(744\) 0 0
\(745\) 1.95749e6 0.129214
\(746\) 1.89457e7 1.24642
\(747\) 0 0
\(748\) 0 0
\(749\) −1.52712e7 −0.994649
\(750\) 0 0
\(751\) −1.91140e7 −1.23667 −0.618333 0.785916i \(-0.712192\pi\)
−0.618333 + 0.785916i \(0.712192\pi\)
\(752\) −4.36033e6 −0.281174
\(753\) 0 0
\(754\) −5.47334e6 −0.350610
\(755\) −2.60398e6 −0.166254
\(756\) 0 0
\(757\) 1.48895e7 0.944366 0.472183 0.881501i \(-0.343466\pi\)
0.472183 + 0.881501i \(0.343466\pi\)
\(758\) 4.35977e6 0.275607
\(759\) 0 0
\(760\) −4.55364e6 −0.285973
\(761\) −2.14078e7 −1.34002 −0.670009 0.742353i \(-0.733710\pi\)
−0.670009 + 0.742353i \(0.733710\pi\)
\(762\) 0 0
\(763\) 2.97227e7 1.84832
\(764\) −1.56327e7 −0.968948
\(765\) 0 0
\(766\) 3.07373e7 1.89275
\(767\) 2.03134e7 1.24680
\(768\) 0 0
\(769\) −1.45027e7 −0.884369 −0.442184 0.896924i \(-0.645796\pi\)
−0.442184 + 0.896924i \(0.645796\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.66129e7 1.60712
\(773\) −3.18546e7 −1.91745 −0.958723 0.284342i \(-0.908225\pi\)
−0.958723 + 0.284342i \(0.908225\pi\)
\(774\) 0 0
\(775\) 1.19171e6 0.0712716
\(776\) 2.29768e6 0.136973
\(777\) 0 0
\(778\) 2.44759e7 1.44974
\(779\) −1.84016e7 −1.08646
\(780\) 0 0
\(781\) 0 0
\(782\) 1.32311e7 0.773710
\(783\) 0 0
\(784\) 2.38545e7 1.38606
\(785\) −2.95355e6 −0.171068
\(786\) 0 0
\(787\) 1.68239e7 0.968253 0.484126 0.874998i \(-0.339138\pi\)
0.484126 + 0.874998i \(0.339138\pi\)
\(788\) 6.57285e7 3.77084
\(789\) 0 0
\(790\) 29267.0 0.00166844
\(791\) −1.12281e7 −0.638067
\(792\) 0 0
\(793\) −1.50852e7 −0.851858
\(794\) −4.10455e7 −2.31055
\(795\) 0 0
\(796\) −4.64215e7 −2.59679
\(797\) 2.00376e7 1.11738 0.558690 0.829377i \(-0.311304\pi\)
0.558690 + 0.829377i \(0.311304\pi\)
\(798\) 0 0
\(799\) 1.76908e6 0.0980347
\(800\) 2.91530e7 1.61049
\(801\) 0 0
\(802\) −5.99384e7 −3.29055
\(803\) 0 0
\(804\) 0 0
\(805\) 1.91492e6 0.104150
\(806\) 2.37714e6 0.128889
\(807\) 0 0
\(808\) −1.73829e7 −0.936683
\(809\) −1.59711e7 −0.857954 −0.428977 0.903315i \(-0.641126\pi\)
−0.428977 + 0.903315i \(0.641126\pi\)
\(810\) 0 0
\(811\) −1.37309e7 −0.733074 −0.366537 0.930403i \(-0.619457\pi\)
−0.366537 + 0.930403i \(0.619457\pi\)
\(812\) −1.12558e7 −0.599080
\(813\) 0 0
\(814\) 0 0
\(815\) −5.22941e6 −0.275777
\(816\) 0 0
\(817\) −2.28666e7 −1.19852
\(818\) 2.49192e7 1.30212
\(819\) 0 0
\(820\) −1.05413e7 −0.547467
\(821\) 2.26402e7 1.17226 0.586128 0.810218i \(-0.300652\pi\)
0.586128 + 0.810218i \(0.300652\pi\)
\(822\) 0 0
\(823\) 1.16510e7 0.599603 0.299802 0.954002i \(-0.403079\pi\)
0.299802 + 0.954002i \(0.403079\pi\)
\(824\) −8.26304e7 −4.23957
\(825\) 0 0
\(826\) 5.93438e7 3.02639
\(827\) 5.48883e6 0.279072 0.139536 0.990217i \(-0.455439\pi\)
0.139536 + 0.990217i \(0.455439\pi\)
\(828\) 0 0
\(829\) 1.81680e7 0.918164 0.459082 0.888394i \(-0.348178\pi\)
0.459082 + 0.888394i \(0.348178\pi\)
\(830\) −8.35429e6 −0.420934
\(831\) 0 0
\(832\) 1.45169e7 0.727050
\(833\) −9.67828e6 −0.483265
\(834\) 0 0
\(835\) 1.38234e6 0.0686120
\(836\) 0 0
\(837\) 0 0
\(838\) −1.47283e7 −0.724507
\(839\) 1.83237e7 0.898689 0.449344 0.893359i \(-0.351658\pi\)
0.449344 + 0.893359i \(0.351658\pi\)
\(840\) 0 0
\(841\) −1.97019e7 −0.960546
\(842\) 4.99582e7 2.42844
\(843\) 0 0
\(844\) −1.26756e7 −0.612509
\(845\) 249024. 0.0119978
\(846\) 0 0
\(847\) 0 0
\(848\) −5.49069e7 −2.62203
\(849\) 0 0
\(850\) −2.99772e7 −1.42313
\(851\) −6.02240e6 −0.285066
\(852\) 0 0
\(853\) −3.66987e7 −1.72694 −0.863471 0.504398i \(-0.831715\pi\)
−0.863471 + 0.504398i \(0.831715\pi\)
\(854\) −4.40698e7 −2.06774
\(855\) 0 0
\(856\) −4.25584e7 −1.98518
\(857\) 3.80021e7 1.76749 0.883743 0.467972i \(-0.155015\pi\)
0.883743 + 0.467972i \(0.155015\pi\)
\(858\) 0 0
\(859\) −7.40664e6 −0.342482 −0.171241 0.985229i \(-0.554778\pi\)
−0.171241 + 0.985229i \(0.554778\pi\)
\(860\) −1.30990e7 −0.603937
\(861\) 0 0
\(862\) 2.84303e7 1.30321
\(863\) 9.79342e6 0.447618 0.223809 0.974633i \(-0.428151\pi\)
0.223809 + 0.974633i \(0.428151\pi\)
\(864\) 0 0
\(865\) 4.32261e6 0.196429
\(866\) −2.82698e7 −1.28094
\(867\) 0 0
\(868\) 4.88851e6 0.220230
\(869\) 0 0
\(870\) 0 0
\(871\) 3.24131e7 1.44769
\(872\) 8.28322e7 3.68899
\(873\) 0 0
\(874\) 1.60783e7 0.711969
\(875\) −8.78353e6 −0.387837
\(876\) 0 0
\(877\) 3.57898e7 1.57130 0.785652 0.618669i \(-0.212328\pi\)
0.785652 + 0.618669i \(0.212328\pi\)
\(878\) −4.33632e7 −1.89839
\(879\) 0 0
\(880\) 0 0
\(881\) −7.14038e6 −0.309943 −0.154971 0.987919i \(-0.549529\pi\)
−0.154971 + 0.987919i \(0.549529\pi\)
\(882\) 0 0
\(883\) −2.77023e7 −1.19568 −0.597838 0.801617i \(-0.703973\pi\)
−0.597838 + 0.801617i \(0.703973\pi\)
\(884\) −4.20926e7 −1.81165
\(885\) 0 0
\(886\) 7.14091e7 3.05611
\(887\) 387870. 0.0165530 0.00827651 0.999966i \(-0.497365\pi\)
0.00827651 + 0.999966i \(0.497365\pi\)
\(888\) 0 0
\(889\) 4.47857e6 0.190058
\(890\) −2.07337e6 −0.0877410
\(891\) 0 0
\(892\) −8.03250e7 −3.38017
\(893\) 2.14977e6 0.0902117
\(894\) 0 0
\(895\) −5.41932e6 −0.226145
\(896\) −7.88818e6 −0.328251
\(897\) 0 0
\(898\) −7.21309e6 −0.298491
\(899\) −351468. −0.0145040
\(900\) 0 0
\(901\) 2.22769e7 0.914203
\(902\) 0 0
\(903\) 0 0
\(904\) −3.12909e7 −1.27349
\(905\) −3.40519e6 −0.138204
\(906\) 0 0
\(907\) 5.05051e6 0.203853 0.101926 0.994792i \(-0.467499\pi\)
0.101926 + 0.994792i \(0.467499\pi\)
\(908\) 4.00626e7 1.61259
\(909\) 0 0
\(910\) −8.65425e6 −0.346438
\(911\) 1.72283e7 0.687774 0.343887 0.939011i \(-0.388256\pi\)
0.343887 + 0.939011i \(0.388256\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 8.44660e7 3.34439
\(915\) 0 0
\(916\) 8.49807e7 3.34643
\(917\) 1.95488e6 0.0767709
\(918\) 0 0
\(919\) −3.89056e7 −1.51958 −0.759789 0.650170i \(-0.774698\pi\)
−0.759789 + 0.650170i \(0.774698\pi\)
\(920\) 5.33655e6 0.207870
\(921\) 0 0
\(922\) 4.66769e7 1.80832
\(923\) −3.32990e7 −1.28655
\(924\) 0 0
\(925\) 1.36447e7 0.524338
\(926\) 9.40340e7 3.60377
\(927\) 0 0
\(928\) −8.59801e6 −0.327739
\(929\) −3.18602e7 −1.21118 −0.605590 0.795777i \(-0.707063\pi\)
−0.605590 + 0.795777i \(0.707063\pi\)
\(930\) 0 0
\(931\) −1.17610e7 −0.444701
\(932\) −2.22635e6 −0.0839564
\(933\) 0 0
\(934\) 7.45701e7 2.79703
\(935\) 0 0
\(936\) 0 0
\(937\) 4.09748e7 1.52464 0.762322 0.647198i \(-0.224059\pi\)
0.762322 + 0.647198i \(0.224059\pi\)
\(938\) 9.46917e7 3.51402
\(939\) 0 0
\(940\) 1.23148e6 0.0454577
\(941\) −1.76369e7 −0.649304 −0.324652 0.945833i \(-0.605247\pi\)
−0.324652 + 0.945833i \(0.605247\pi\)
\(942\) 0 0
\(943\) 2.15655e7 0.789732
\(944\) 8.08741e7 2.95379
\(945\) 0 0
\(946\) 0 0
\(947\) −4.07232e7 −1.47559 −0.737797 0.675022i \(-0.764134\pi\)
−0.737797 + 0.675022i \(0.764134\pi\)
\(948\) 0 0
\(949\) 2.74311e7 0.988730
\(950\) −3.64280e7 −1.30956
\(951\) 0 0
\(952\) −7.12495e7 −2.54794
\(953\) −1.02210e7 −0.364553 −0.182276 0.983247i \(-0.558347\pi\)
−0.182276 + 0.983247i \(0.558347\pi\)
\(954\) 0 0
\(955\) 1.77713e6 0.0630538
\(956\) 6.25767e7 2.21446
\(957\) 0 0
\(958\) 1.57077e7 0.552967
\(959\) 5.80557e6 0.203844
\(960\) 0 0
\(961\) −2.84765e7 −0.994668
\(962\) 2.72175e7 0.948225
\(963\) 0 0
\(964\) −5.80089e7 −2.01049
\(965\) −3.02536e6 −0.104583
\(966\) 0 0
\(967\) 7.46133e6 0.256596 0.128298 0.991736i \(-0.459049\pi\)
0.128298 + 0.991736i \(0.459049\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −450807. −0.0153837
\(971\) −4.34819e7 −1.47999 −0.739997 0.672610i \(-0.765173\pi\)
−0.739997 + 0.672610i \(0.765173\pi\)
\(972\) 0 0
\(973\) 4.31657e6 0.146170
\(974\) −1.69736e7 −0.573292
\(975\) 0 0
\(976\) −6.00587e7 −2.01814
\(977\) 1.52214e7 0.510173 0.255087 0.966918i \(-0.417896\pi\)
0.255087 + 0.966918i \(0.417896\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −6.73719e6 −0.224085
\(981\) 0 0
\(982\) −1.29392e7 −0.428181
\(983\) −2.90192e7 −0.957858 −0.478929 0.877854i \(-0.658975\pi\)
−0.478929 + 0.877854i \(0.658975\pi\)
\(984\) 0 0
\(985\) −7.47205e6 −0.245386
\(986\) 8.84110e6 0.289610
\(987\) 0 0
\(988\) −5.11505e7 −1.66708
\(989\) 2.67981e7 0.871190
\(990\) 0 0
\(991\) 3.46744e7 1.12156 0.560782 0.827963i \(-0.310500\pi\)
0.560782 + 0.827963i \(0.310500\pi\)
\(992\) 3.73422e6 0.120481
\(993\) 0 0
\(994\) −9.72798e7 −3.12289
\(995\) 5.27722e6 0.168985
\(996\) 0 0
\(997\) −1.05268e7 −0.335397 −0.167699 0.985838i \(-0.553634\pi\)
−0.167699 + 0.985838i \(0.553634\pi\)
\(998\) 7.96577e7 2.53164
\(999\) 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 1089.6.a.r.1.1 3
3.2 odd 2 121.6.a.d.1.3 3
11.10 odd 2 99.6.a.g.1.3 3
33.32 even 2 11.6.a.b.1.1 3
132.131 odd 2 176.6.a.i.1.1 3
165.32 odd 4 275.6.b.b.199.1 6
165.98 odd 4 275.6.b.b.199.6 6
165.164 even 2 275.6.a.b.1.3 3
231.230 odd 2 539.6.a.e.1.1 3
264.131 odd 2 704.6.a.t.1.3 3
264.197 even 2 704.6.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.1 3 33.32 even 2
99.6.a.g.1.3 3 11.10 odd 2
121.6.a.d.1.3 3 3.2 odd 2
176.6.a.i.1.1 3 132.131 odd 2
275.6.a.b.1.3 3 165.164 even 2
275.6.b.b.199.1 6 165.32 odd 4
275.6.b.b.199.6 6 165.98 odd 4
539.6.a.e.1.1 3 231.230 odd 2
704.6.a.q.1.1 3 264.197 even 2
704.6.a.t.1.3 3 264.131 odd 2
1089.6.a.r.1.1 3 1.1 even 1 trivial