Properties

Label 121.6.a.d.1.3
Level $121$
Weight $6$
Character 121.1
Self dual yes
Analytic conductor $19.406$
Analytic rank $0$
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] = [121,6,Mod(1,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 121.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: \(19.4064421974\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.749680\) of defining polynomial
Character \(\chi\) \(=\) 121.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.3963 q^{2} +20.6466 q^{3} +76.0833 q^{4} +8.64919 q^{5} +214.649 q^{6} -164.454 q^{7} +458.304 q^{8} +183.283 q^{9} +O(q^{10})\) \(q+10.3963 q^{2} +20.6466 q^{3} +76.0833 q^{4} +8.64919 q^{5} +214.649 q^{6} -164.454 q^{7} +458.304 q^{8} +183.283 q^{9} +89.9197 q^{10} +1570.86 q^{12} +585.236 q^{13} -1709.71 q^{14} +178.577 q^{15} +2330.01 q^{16} +945.333 q^{17} +1905.47 q^{18} -1148.76 q^{19} +658.060 q^{20} -3395.41 q^{21} -1346.27 q^{23} +9462.44 q^{24} -3050.19 q^{25} +6084.30 q^{26} -1232.95 q^{27} -12512.2 q^{28} -899.585 q^{29} +1856.54 q^{30} -390.700 q^{31} +9557.75 q^{32} +9827.97 q^{34} -1422.39 q^{35} +13944.8 q^{36} -4473.41 q^{37} -11942.9 q^{38} +12083.2 q^{39} +3963.96 q^{40} -16018.7 q^{41} -35299.8 q^{42} +19905.5 q^{43} +1585.25 q^{45} -13996.2 q^{46} +1871.38 q^{47} +48106.8 q^{48} +10238.0 q^{49} -31710.7 q^{50} +19517.9 q^{51} +44526.7 q^{52} +23565.1 q^{53} -12818.1 q^{54} -75369.7 q^{56} -23718.0 q^{57} -9352.37 q^{58} -34709.8 q^{59} +13586.7 q^{60} -25776.2 q^{61} -4061.84 q^{62} -30141.6 q^{63} +24805.1 q^{64} +5061.82 q^{65} +55384.6 q^{67} +71924.1 q^{68} -27795.9 q^{69} -14787.6 q^{70} +56898.4 q^{71} +83999.6 q^{72} +46871.8 q^{73} -46506.9 q^{74} -62976.2 q^{75} -87401.5 q^{76} +125620. q^{78} +325.479 q^{79} +20152.7 q^{80} -69994.1 q^{81} -166536. q^{82} +92908.3 q^{83} -258334. q^{84} +8176.37 q^{85} +206943. q^{86} -18573.4 q^{87} +23058.0 q^{89} +16480.8 q^{90} -96244.2 q^{91} -102428. q^{92} -8066.65 q^{93} +19455.5 q^{94} -9935.84 q^{95} +197335. q^{96} -5013.44 q^{97} +106437. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 414 q^{10} + 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} - 1380 q^{19} - 3480 q^{20} + 908 q^{21} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} + 12132 q^{26} - 2990 q^{27} - 23712 q^{28} + 3426 q^{29} - 2650 q^{30} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 4756 q^{36} + 17724 q^{37} - 9240 q^{38} + 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 47828 q^{42} + 26208 q^{43} + 18458 q^{45} + 16806 q^{46} - 17232 q^{47} + 61064 q^{48} + 48531 q^{49} - 41070 q^{50} + 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 42312 q^{56} - 20160 q^{57} - 29172 q^{58} - 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} + 35042 q^{69} - 123372 q^{70} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} - 153294 q^{74} - 21916 q^{75} - 103920 q^{76} + 124984 q^{78} - 125148 q^{79} + 118680 q^{80} - 86917 q^{81} - 255372 q^{82} + 143928 q^{83} - 343616 q^{84} + 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 106824 q^{89} - 103424 q^{90} - 109632 q^{91} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} + 22200 q^{95} + 76456 q^{96} + 9684 q^{97} - 3480 q^{98}+O(q^{100}) \) 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 \(-0.129068\pi\)
0.918913 + 0.394460i \(0.129068\pi\)
\(3\) 20.6466 1.32448 0.662241 0.749291i \(-0.269605\pi\)
0.662241 + 0.749291i \(0.269605\pi\)
\(4\) 76.0833 2.37760
\(5\) 8.64919 0.154721 0.0773607 0.997003i \(-0.475351\pi\)
0.0773607 + 0.997003i \(0.475351\pi\)
\(6\) 214.649 2.43417
\(7\) −164.454 −1.26852 −0.634261 0.773119i \(-0.718696\pi\)
−0.634261 + 0.773119i \(0.718696\pi\)
\(8\) 458.304 2.53180
\(9\) 183.283 0.754253
\(10\) 89.9197 0.284351
\(11\) 0 0
\(12\) 1570.86 3.14909
\(13\) 585.236 0.960446 0.480223 0.877147i \(-0.340556\pi\)
0.480223 + 0.877147i \(0.340556\pi\)
\(14\) −1709.71 −2.33132
\(15\) 178.577 0.204926
\(16\) 2330.01 2.27540
\(17\) 945.333 0.793345 0.396673 0.917960i \(-0.370165\pi\)
0.396673 + 0.917960i \(0.370165\pi\)
\(18\) 1905.47 1.38619
\(19\) −1148.76 −0.730037 −0.365019 0.931000i \(-0.618937\pi\)
−0.365019 + 0.931000i \(0.618937\pi\)
\(20\) 658.060 0.367866
\(21\) −3395.41 −1.68014
\(22\) 0 0
\(23\) −1346.27 −0.530654 −0.265327 0.964158i \(-0.585480\pi\)
−0.265327 + 0.964158i \(0.585480\pi\)
\(24\) 9462.44 3.35332
\(25\) −3050.19 −0.976061
\(26\) 6084.30 1.76513
\(27\) −1232.95 −0.325488
\(28\) −12512.2 −3.01604
\(29\) −899.585 −0.198631 −0.0993155 0.995056i \(-0.531665\pi\)
−0.0993155 + 0.995056i \(0.531665\pi\)
\(30\) 1856.54 0.376618
\(31\) −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
\(36\) 13944.8 1.79332
\(37\) −4473.41 −0.537198 −0.268599 0.963252i \(-0.586561\pi\)
−0.268599 + 0.963252i \(0.586561\pi\)
\(38\) −11942.9 −1.34168
\(39\) 12083.2 1.27209
\(40\) 3963.96 0.391723
\(41\) −16018.7 −1.48822 −0.744111 0.668056i \(-0.767127\pi\)
−0.744111 + 0.668056i \(0.767127\pi\)
\(42\) −35299.8 −3.08780
\(43\) 19905.5 1.64173 0.820864 0.571124i \(-0.193492\pi\)
0.820864 + 0.571124i \(0.193492\pi\)
\(44\) 0 0
\(45\) 1585.25 0.116699
\(46\) −13996.2 −0.975250
\(47\) 1871.38 0.123571 0.0617856 0.998089i \(-0.480320\pi\)
0.0617856 + 0.998089i \(0.480320\pi\)
\(48\) 48106.8 3.01372
\(49\) 10238.0 0.609149
\(50\) −31710.7 −1.79383
\(51\) 19517.9 1.05077
\(52\) 44526.7 2.28356
\(53\) 23565.1 1.15234 0.576169 0.817330i \(-0.304547\pi\)
0.576169 + 0.817330i \(0.304547\pi\)
\(54\) −12818.1 −0.598189
\(55\) 0 0
\(56\) −75369.7 −3.21164
\(57\) −23718.0 −0.966922
\(58\) −9352.37 −0.365049
\(59\) −34709.8 −1.29814 −0.649071 0.760727i \(-0.724842\pi\)
−0.649071 + 0.760727i \(0.724842\pi\)
\(60\) 13586.7 0.487233
\(61\) −25776.2 −0.886940 −0.443470 0.896289i \(-0.646253\pi\)
−0.443470 + 0.896289i \(0.646253\pi\)
\(62\) −4061.84 −0.134197
\(63\) −30141.6 −0.956787
\(64\) 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
\(69\) −27795.9 −0.702842
\(70\) −14787.6 −0.360706
\(71\) 56898.4 1.33954 0.669768 0.742571i \(-0.266394\pi\)
0.669768 + 0.742571i \(0.266394\pi\)
\(72\) 83999.6 1.90962
\(73\) 46871.8 1.02945 0.514724 0.857356i \(-0.327894\pi\)
0.514724 + 0.857356i \(0.327894\pi\)
\(74\) −46506.9 −0.987276
\(75\) −62976.2 −1.29278
\(76\) −87401.5 −1.73574
\(77\) 0 0
\(78\) 125620. 2.33789
\(79\) 325.479 0.00586753 0.00293377 0.999996i \(-0.499066\pi\)
0.00293377 + 0.999996i \(0.499066\pi\)
\(80\) 20152.7 0.352053
\(81\) −69994.1 −1.18536
\(82\) −166536. −2.73509
\(83\) 92908.3 1.48033 0.740166 0.672424i \(-0.234747\pi\)
0.740166 + 0.672424i \(0.234747\pi\)
\(84\) −258334. −3.99470
\(85\) 8176.37 0.122748
\(86\) 206943. 3.01721
\(87\) −18573.4 −0.263083
\(88\) 0 0
\(89\) 23058.0 0.308565 0.154283 0.988027i \(-0.450693\pi\)
0.154283 + 0.988027i \(0.450693\pi\)
\(90\) 16480.8 0.214473
\(91\) −96244.2 −1.21835
\(92\) −102428. −1.26169
\(93\) −8066.65 −0.0967132
\(94\) 19455.5 0.227103
\(95\) −9935.84 −0.112952
\(96\) 197335. 2.18538
\(97\) −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 \(-0.559223\pi\)
−0.184984 + 0.982742i \(0.559223\pi\)
\(102\) 202915. 1.93114
\(103\) 180296. 1.67453 0.837265 0.546798i \(-0.184153\pi\)
0.837265 + 0.546798i \(0.184153\pi\)
\(104\) 268216. 2.43165
\(105\) −29367.6 −0.259953
\(106\) 244990. 2.11780
\(107\) −92860.5 −0.784100 −0.392050 0.919944i \(-0.628234\pi\)
−0.392050 + 0.919944i \(0.628234\pi\)
\(108\) −93806.6 −0.773881
\(109\) −180736. −1.45707 −0.728533 0.685011i \(-0.759797\pi\)
−0.728533 + 0.685011i \(0.759797\pi\)
\(110\) 0 0
\(111\) −92360.8 −0.711509
\(112\) −383178. −2.88639
\(113\) −68275.4 −0.503000 −0.251500 0.967857i \(-0.580924\pi\)
−0.251500 + 0.967857i \(0.580924\pi\)
\(114\) −246580. −1.77703
\(115\) −11644.1 −0.0821036
\(116\) −68443.4 −0.472266
\(117\) 107264. 0.724419
\(118\) −360854. −2.38576
\(119\) −155463. −1.00638
\(120\) 81842.5 0.518831
\(121\) 0 0
\(122\) −267977. −1.63004
\(123\) −330732. −1.97112
\(124\) −29725.8 −0.173612
\(125\) −53410.4 −0.305739
\(126\) −313362. −1.75841
\(127\) −27233.1 −0.149826 −0.0749130 0.997190i \(-0.523868\pi\)
−0.0749130 + 0.997190i \(0.523868\pi\)
\(128\) −47966.0 −0.258767
\(129\) 410981. 2.17444
\(130\) 52624.3 0.273104
\(131\) 11887.1 0.0605199 0.0302600 0.999542i \(-0.490366\pi\)
0.0302600 + 0.999542i \(0.490366\pi\)
\(132\) 0 0
\(133\) 188918. 0.926069
\(134\) 575796. 2.77017
\(135\) −10664.0 −0.0503599
\(136\) 433250. 2.00859
\(137\) 35302.2 0.160694 0.0803471 0.996767i \(-0.474397\pi\)
0.0803471 + 0.996767i \(0.474397\pi\)
\(138\) −288975. −1.29170
\(139\) −26248.0 −0.115228 −0.0576141 0.998339i \(-0.518349\pi\)
−0.0576141 + 0.998339i \(0.518349\pi\)
\(140\) −108220. −0.466647
\(141\) 38637.7 0.163668
\(142\) 591533. 2.46183
\(143\) 0 0
\(144\) 427052. 1.71623
\(145\) −7780.68 −0.0307325
\(146\) 487294. 1.89195
\(147\) 211379. 0.806807
\(148\) −340352. −1.27724
\(149\) 226321. 0.835139 0.417570 0.908645i \(-0.362882\pi\)
0.417570 + 0.908645i \(0.362882\pi\)
\(150\) −654720. −2.37590
\(151\) 301067. 1.07453 0.537267 0.843412i \(-0.319457\pi\)
0.537267 + 0.843412i \(0.319457\pi\)
\(152\) −526481. −1.84831
\(153\) 173264. 0.598383
\(154\) 0 0
\(155\) −3379.24 −0.0112977
\(156\) 919327. 3.02453
\(157\) 341482. 1.10565 0.552827 0.833296i \(-0.313549\pi\)
0.552827 + 0.833296i \(0.313549\pi\)
\(158\) 3383.78 0.0107835
\(159\) 486541. 1.52625
\(160\) 82666.8 0.255289
\(161\) 221398. 0.673147
\(162\) −727680. −2.17848
\(163\) 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 \(-0.428830\pi\)
0.221728 + 0.975109i \(0.428830\pi\)
\(168\) −1.55613e6 −4.25376
\(169\) −28791.6 −0.0775442
\(170\) 85004.1 0.225589
\(171\) −210549. −0.550633
\(172\) 1.51447e6 3.90338
\(173\) 499771. 1.26957 0.634783 0.772690i \(-0.281089\pi\)
0.634783 + 0.772690i \(0.281089\pi\)
\(174\) −193095. −0.483501
\(175\) 501615. 1.23816
\(176\) 0 0
\(177\) −716641. −1.71937
\(178\) 239719. 0.567090
\(179\) −626569. −1.46163 −0.730813 0.682578i \(-0.760859\pi\)
−0.730813 + 0.682578i \(0.760859\pi\)
\(180\) 120611. 0.277464
\(181\) 393700. 0.893243 0.446621 0.894723i \(-0.352627\pi\)
0.446621 + 0.894723i \(0.352627\pi\)
\(182\) −1.00058e6 −2.23911
\(183\) −532192. −1.17474
\(184\) −617000. −1.34351
\(185\) −38691.4 −0.0831160
\(186\) −83863.4 −0.177742
\(187\) 0 0
\(188\) 142381. 0.293804
\(189\) 202762. 0.412888
\(190\) −103296. −0.207587
\(191\) 205468. 0.407531 0.203766 0.979020i \(-0.434682\pi\)
0.203766 + 0.979020i \(0.434682\pi\)
\(192\) 512143. 1.00262
\(193\) 349786. 0.675941 0.337971 0.941157i \(-0.390260\pi\)
0.337971 + 0.941157i \(0.390260\pi\)
\(194\) −52121.3 −0.0994285
\(195\) 104510. 0.196820
\(196\) 778938. 1.44831
\(197\) −863902. −1.58598 −0.792992 0.609232i \(-0.791478\pi\)
−0.792992 + 0.609232i \(0.791478\pi\)
\(198\) 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
\(201\) 1.14351e6 1.99640
\(202\) −394318. −0.679936
\(203\) 147940. 0.251968
\(204\) 1.48499e6 2.49832
\(205\) −138549. −0.230260
\(206\) 1.87441e6 3.07749
\(207\) −246749. −0.400248
\(208\) 1.36360e6 2.18540
\(209\) 0 0
\(210\) −305315. −0.477748
\(211\) −166602. −0.257616 −0.128808 0.991670i \(-0.541115\pi\)
−0.128808 + 0.991670i \(0.541115\pi\)
\(212\) 1.79291e6 2.73981
\(213\) 1.17476e6 1.77419
\(214\) −965407. −1.44104
\(215\) 172166. 0.254011
\(216\) −565064. −0.824068
\(217\) 64252.0 0.0926270
\(218\) −1.87899e6 −2.67783
\(219\) 967746. 1.36349
\(220\) 0 0
\(221\) 553243. 0.761965
\(222\) −960212. −1.30763
\(223\) −1.05575e6 −1.42167 −0.710836 0.703358i \(-0.751683\pi\)
−0.710836 + 0.703358i \(0.751683\pi\)
\(224\) −1.57181e6 −2.09305
\(225\) −559050. −0.736197
\(226\) −709812. −0.924427
\(227\) −526562. −0.678242 −0.339121 0.940743i \(-0.610130\pi\)
−0.339121 + 0.940743i \(0.610130\pi\)
\(228\) −1.80455e6 −2.29896
\(229\) 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 \(-0.494380\pi\)
0.0176557 + 0.999844i \(0.494380\pi\)
\(234\) 1.11515e6 1.33136
\(235\) 16185.9 0.0191191
\(236\) −2.64084e6 −3.08647
\(237\) 6720.05 0.00777144
\(238\) −1.61625e6 −1.84954
\(239\) −822476. −0.931384 −0.465692 0.884947i \(-0.654194\pi\)
−0.465692 + 0.884947i \(0.654194\pi\)
\(240\) 416085. 0.466288
\(241\) −762439. −0.845595 −0.422797 0.906224i \(-0.638952\pi\)
−0.422797 + 0.906224i \(0.638952\pi\)
\(242\) 0 0
\(243\) −1.14554e6 −1.24449
\(244\) −1.96114e6 −2.10879
\(245\) 88550.1 0.0942484
\(246\) −3.43840e6 −3.62258
\(247\) −672296. −0.701161
\(248\) −179060. −0.184871
\(249\) 1.91824e6 1.96067
\(250\) −555272. −0.561895
\(251\) 561364. 0.562419 0.281209 0.959646i \(-0.409264\pi\)
0.281209 + 0.959646i \(0.409264\pi\)
\(252\) −2.29327e6 −2.27486
\(253\) 0 0
\(254\) −283123. −0.275354
\(255\) 168814. 0.162577
\(256\) −1.29243e6 −1.23256
\(257\) −764965. −0.722451 −0.361226 0.932478i \(-0.617642\pi\)
−0.361226 + 0.932478i \(0.617642\pi\)
\(258\) 4.27269e6 3.99624
\(259\) 735668. 0.681447
\(260\) 385120. 0.353316
\(261\) −164879. −0.149818
\(262\) 123582. 0.111225
\(263\) 763627. 0.680756 0.340378 0.940289i \(-0.389445\pi\)
0.340378 + 0.940289i \(0.389445\pi\)
\(264\) 0 0
\(265\) 203819. 0.178292
\(266\) 1.96405e6 1.70195
\(267\) 476071. 0.408689
\(268\) 4.21385e6 3.58378
\(269\) 800885. 0.674823 0.337411 0.941357i \(-0.390449\pi\)
0.337411 + 0.941357i \(0.390449\pi\)
\(270\) −110866. −0.0925528
\(271\) −98139.7 −0.0811749 −0.0405874 0.999176i \(-0.512923\pi\)
−0.0405874 + 0.999176i \(0.512923\pi\)
\(272\) 2.20263e6 1.80518
\(273\) −1.98712e6 −1.61368
\(274\) 367013. 0.295328
\(275\) 0 0
\(276\) −2.11480e6 −1.67108
\(277\) 620993. 0.486281 0.243140 0.969991i \(-0.421822\pi\)
0.243140 + 0.969991i \(0.421822\pi\)
\(278\) −272882. −0.211769
\(279\) −71608.9 −0.0550753
\(280\) −651887. −0.496910
\(281\) −1.31191e6 −0.991149 −0.495575 0.868565i \(-0.665043\pi\)
−0.495575 + 0.868565i \(0.665043\pi\)
\(282\) 401690. 0.300793
\(283\) 26897.8 0.0199641 0.00998205 0.999950i \(-0.496823\pi\)
0.00998205 + 0.999950i \(0.496823\pi\)
\(284\) 4.32902e6 3.18488
\(285\) −205142. −0.149604
\(286\) 0 0
\(287\) 2.63433e6 1.88784
\(288\) 1.75178e6 1.24451
\(289\) −526203. −0.370603
\(290\) −80890.4 −0.0564810
\(291\) −103511. −0.0716560
\(292\) 3.56617e6 2.44762
\(293\) −638546. −0.434534 −0.217267 0.976112i \(-0.569714\pi\)
−0.217267 + 0.976112i \(0.569714\pi\)
\(294\) 2.19757e6 1.48277
\(295\) −300212. −0.200851
\(296\) −2.05018e6 −1.36008
\(297\) 0 0
\(298\) 2.35290e6 1.53484
\(299\) −787884. −0.509665
\(300\) −4.79144e6 −3.07371
\(301\) −3.27352e6 −2.08257
\(302\) 3.12998e6 1.97481
\(303\) −783099. −0.490016
\(304\) −2.67662e6 −1.66113
\(305\) −222943. −0.137229
\(306\) 1.80131e6 1.09972
\(307\) −550428. −0.333315 −0.166658 0.986015i \(-0.553297\pi\)
−0.166658 + 0.986015i \(0.553297\pi\)
\(308\) 0 0
\(309\) 3.72250e6 2.21788
\(310\) −35131.7 −0.0207632
\(311\) 186775. 0.109501 0.0547504 0.998500i \(-0.482564\pi\)
0.0547504 + 0.998500i \(0.482564\pi\)
\(312\) 5.53776e6 3.22068
\(313\) −934239. −0.539010 −0.269505 0.962999i \(-0.586860\pi\)
−0.269505 + 0.962999i \(0.586860\pi\)
\(314\) 3.55016e6 2.03200
\(315\) −260701. −0.148035
\(316\) 24763.5 0.0139507
\(317\) −1.88280e6 −1.05234 −0.526170 0.850379i \(-0.676372\pi\)
−0.526170 + 0.850379i \(0.676372\pi\)
\(318\) 5.05823e6 2.80499
\(319\) 0 0
\(320\) 214544. 0.117123
\(321\) −1.91726e6 −1.03853
\(322\) 2.30173e6 1.23713
\(323\) −1.08596e6 −0.579172
\(324\) −5.32538e6 −2.81831
\(325\) −1.78508e6 −0.937454
\(326\) 6.28574e6 3.27576
\(327\) −3.73160e6 −1.92986
\(328\) −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
\(333\) −819901. −0.405183
\(334\) 1.66158e6 0.814993
\(335\) 479033. 0.233213
\(336\) −7.91133e6 −3.82298
\(337\) −387484. −0.185857 −0.0929285 0.995673i \(-0.529623\pi\)
−0.0929285 + 0.995673i \(0.529623\pi\)
\(338\) −299327. −0.142513
\(339\) −1.40966e6 −0.666215
\(340\) 622085. 0.291845
\(341\) 0 0
\(342\) −2.18893e6 −1.01197
\(343\) 1.08030e6 0.495803
\(344\) 9.12276e6 4.15652
\(345\) −240412. −0.108745
\(346\) 5.19577e6 2.33324
\(347\) 2.94793e6 1.31430 0.657148 0.753761i \(-0.271762\pi\)
0.657148 + 0.753761i \(0.271762\pi\)
\(348\) −1.41313e6 −0.625508
\(349\) 924908. 0.406476 0.203238 0.979129i \(-0.434854\pi\)
0.203238 + 0.979129i \(0.434854\pi\)
\(350\) 5.21494e6 2.27551
\(351\) −721564. −0.312613
\(352\) 0 0
\(353\) −4.46816e6 −1.90850 −0.954249 0.299012i \(-0.903343\pi\)
−0.954249 + 0.299012i \(0.903343\pi\)
\(354\) −7.45043e6 −3.15990
\(355\) 492125. 0.207255
\(356\) 1.75433e6 0.733647
\(357\) −3.20979e6 −1.33293
\(358\) −6.51401e6 −2.68621
\(359\) 995937. 0.407846 0.203923 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(360\) 726529. 0.295459
\(361\) −1.15645e6 −0.467045
\(362\) 4.09303e6 1.64162
\(363\) 0 0
\(364\) −7.32258e6 −2.89675
\(365\) 405404. 0.159278
\(366\) −5.53283e6 −2.15896
\(367\) −1.21088e6 −0.469284 −0.234642 0.972082i \(-0.575392\pi\)
−0.234642 + 0.972082i \(0.575392\pi\)
\(368\) −3.13681e6 −1.20745
\(369\) −2.93596e6 −1.12250
\(370\) −402248. −0.152753
\(371\) −3.87537e6 −1.46177
\(372\) −613737. −0.229946
\(373\) −1.82235e6 −0.678203 −0.339102 0.940750i \(-0.610123\pi\)
−0.339102 + 0.940750i \(0.610123\pi\)
\(374\) 0 0
\(375\) −1.10275e6 −0.404946
\(376\) 857662. 0.312857
\(377\) −526470. −0.190774
\(378\) 2.10798e6 0.758817
\(379\) −419357. −0.149964 −0.0749819 0.997185i \(-0.523890\pi\)
−0.0749819 + 0.997185i \(0.523890\pi\)
\(380\) −755952. −0.268556
\(381\) −562271. −0.198442
\(382\) 2.13611e6 0.748972
\(383\) 2.95656e6 1.02989 0.514943 0.857224i \(-0.327813\pi\)
0.514943 + 0.857224i \(0.327813\pi\)
\(384\) −990336. −0.342732
\(385\) 0 0
\(386\) 3.63648e6 1.24226
\(387\) 3.64834e6 1.23828
\(388\) −381439. −0.128631
\(389\) 2.35429e6 0.788834 0.394417 0.918932i \(-0.370947\pi\)
0.394417 + 0.918932i \(0.370947\pi\)
\(390\) 1.08651e6 0.361721
\(391\) −1.27267e6 −0.420992
\(392\) 4.69210e6 1.54224
\(393\) 245429. 0.0801576
\(394\) −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
\(399\) 3.90051e6 1.22656
\(400\) −7.10697e6 −2.22093
\(401\) −5.76535e6 −1.79046 −0.895230 0.445604i \(-0.852989\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(402\) 1.18883e7 3.66904
\(403\) −228652. −0.0701314
\(404\) −2.88574e6 −0.879637
\(405\) −605392. −0.183400
\(406\) 1.53803e6 0.463073
\(407\) 0 0
\(408\) 8.94515e6 2.66034
\(409\) −2.39693e6 −0.708512 −0.354256 0.935148i \(-0.615266\pi\)
−0.354256 + 0.935148i \(0.615266\pi\)
\(410\) −1.44040e6 −0.423178
\(411\) 728872. 0.212837
\(412\) 1.37175e7 3.98137
\(413\) 5.70815e6 1.64672
\(414\) −2.56527e6 −0.735585
\(415\) 803582. 0.229039
\(416\) 5.59354e6 1.58472
\(417\) −541932. −0.152618
\(418\) 0 0
\(419\) −1.41668e6 −0.394220 −0.197110 0.980381i \(-0.563156\pi\)
−0.197110 + 0.980381i \(0.563156\pi\)
\(420\) −2.23438e6 −0.618065
\(421\) −4.80538e6 −1.32136 −0.660682 0.750666i \(-0.729733\pi\)
−0.660682 + 0.750666i \(0.729733\pi\)
\(422\) −1.73204e6 −0.473454
\(423\) 342993. 0.0932040
\(424\) 1.08000e7 2.91749
\(425\) −2.88345e6 −0.774354
\(426\) 1.22132e7 3.26065
\(427\) 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 \(-0.384634\pi\)
0.354551 + 0.935037i \(0.384634\pi\)
\(432\) −2.87277e6 −0.740613
\(433\) 2.71922e6 0.696986 0.348493 0.937311i \(-0.386693\pi\)
0.348493 + 0.937311i \(0.386693\pi\)
\(434\) 667984. 0.170232
\(435\) −160645. −0.0407046
\(436\) −1.37510e7 −3.46433
\(437\) 1.54654e6 0.387397
\(438\) 1.00610e7 2.50585
\(439\) 4.17101e6 1.03295 0.516476 0.856301i \(-0.327243\pi\)
0.516476 + 0.856301i \(0.327243\pi\)
\(440\) 0 0
\(441\) 1.87645e6 0.459452
\(442\) 5.75169e6 1.40036
\(443\) 6.86870e6 1.66290 0.831448 0.555603i \(-0.187513\pi\)
0.831448 + 0.555603i \(0.187513\pi\)
\(444\) −7.02712e6 −1.69169
\(445\) 199433. 0.0477417
\(446\) −1.09759e7 −2.61278
\(447\) 4.67276e6 1.10613
\(448\) −4.07929e6 −0.960262
\(449\) −693812. −0.162415 −0.0812075 0.996697i \(-0.525878\pi\)
−0.0812075 + 0.996697i \(0.525878\pi\)
\(450\) −5.81206e6 −1.35300
\(451\) 0 0
\(452\) −5.19462e6 −1.19594
\(453\) 6.21601e6 1.42320
\(454\) −5.47431e6 −1.24649
\(455\) −832434. −0.188504
\(456\) −1.08701e7 −2.44805
\(457\) −8.12461e6 −1.81975 −0.909876 0.414880i \(-0.863824\pi\)
−0.909876 + 0.414880i \(0.863824\pi\)
\(458\) 1.16121e7 2.58670
\(459\) −1.16554e6 −0.258224
\(460\) −885924. −0.195210
\(461\) 4.48975e6 0.983944 0.491972 0.870611i \(-0.336276\pi\)
0.491972 + 0.870611i \(0.336276\pi\)
\(462\) 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
\(465\) −69770.0 −0.0149636
\(466\) 304217. 0.0648961
\(467\) 7.17275e6 1.52192 0.760962 0.648796i \(-0.224727\pi\)
0.760962 + 0.648796i \(0.224727\pi\)
\(468\) 8.16101e6 1.72238
\(469\) −9.10820e6 −1.91206
\(470\) 168274. 0.0351376
\(471\) 7.05046e6 1.46442
\(472\) −1.59077e7 −3.28663
\(473\) 0 0
\(474\) 69863.7 0.0142826
\(475\) 3.50394e6 0.712561
\(476\) −1.18282e7 −2.39276
\(477\) 4.31910e6 0.869155
\(478\) −8.55072e6 −1.71172
\(479\) 1.51089e6 0.300881 0.150440 0.988619i \(-0.451931\pi\)
0.150440 + 0.988619i \(0.451931\pi\)
\(480\) 1.70679e6 0.338125
\(481\) −2.61800e6 −0.515949
\(482\) −7.92655e6 −1.55406
\(483\) 4.57113e6 0.891571
\(484\) 0 0
\(485\) −43362.2 −0.00837061
\(486\) −1.19093e7 −2.28716
\(487\) 1.63265e6 0.311940 0.155970 0.987762i \(-0.450150\pi\)
0.155970 + 0.987762i \(0.450150\pi\)
\(488\) −1.18133e7 −2.24555
\(489\) 1.24832e7 2.36077
\(490\) 920595. 0.173212
\(491\) −1.24459e6 −0.232982 −0.116491 0.993192i \(-0.537165\pi\)
−0.116491 + 0.993192i \(0.537165\pi\)
\(492\) −2.51632e7 −4.68655
\(493\) −850407. −0.157583
\(494\) −6.98940e6 −1.28861
\(495\) 0 0
\(496\) −910334. −0.166149
\(497\) −9.35714e6 −1.69923
\(498\) 1.99427e7 3.60338
\(499\) −7.66211e6 −1.37752 −0.688759 0.724991i \(-0.741844\pi\)
−0.688759 + 0.724991i \(0.741844\pi\)
\(500\) −4.06364e6 −0.726927
\(501\) 3.29982e6 0.587348
\(502\) 5.83611e6 1.03363
\(503\) 1.07030e7 1.88619 0.943097 0.332518i \(-0.107898\pi\)
0.943097 + 0.332518i \(0.107898\pi\)
\(504\) −1.38140e7 −2.42239
\(505\) −328052. −0.0572420
\(506\) 0 0
\(507\) −594450. −0.102706
\(508\) −2.07198e6 −0.356227
\(509\) 6.27494e6 1.07353 0.536766 0.843731i \(-0.319646\pi\)
0.536766 + 0.843731i \(0.319646\pi\)
\(510\) 1.75505e6 0.298788
\(511\) −7.70824e6 −1.30588
\(512\) −1.19016e7 −2.00647
\(513\) 1.41636e6 0.237618
\(514\) −7.95281e6 −1.32774
\(515\) 1.55941e6 0.259086
\(516\) 3.12688e7 5.16996
\(517\) 0 0
\(518\) 7.64823e6 1.25238
\(519\) 1.03186e7 1.68152
\(520\) 2.31985e6 0.376229
\(521\) 3.60326e6 0.581570 0.290785 0.956788i \(-0.406084\pi\)
0.290785 + 0.956788i \(0.406084\pi\)
\(522\) −1.71413e6 −0.275340
\(523\) −3.56925e6 −0.570589 −0.285294 0.958440i \(-0.592091\pi\)
−0.285294 + 0.958440i \(0.592091\pi\)
\(524\) 904412. 0.143892
\(525\) 1.03567e7 1.63991
\(526\) 7.93890e6 1.25111
\(527\) −369342. −0.0579298
\(528\) 0 0
\(529\) −4.62391e6 −0.718406
\(530\) 2.11897e6 0.327669
\(531\) −6.36174e6 −0.979128
\(532\) 1.43735e7 2.20183
\(533\) −9.37473e6 −1.42936
\(534\) 4.94938e6 0.751100
\(535\) −803169. −0.121317
\(536\) 2.53830e7 3.81620
\(537\) −1.29365e7 −1.93590
\(538\) 8.32625e6 1.24021
\(539\) 0 0
\(540\) −811351. −0.119736
\(541\) −1.16842e7 −1.71635 −0.858176 0.513355i \(-0.828402\pi\)
−0.858176 + 0.513355i \(0.828402\pi\)
\(542\) −1.02029e6 −0.149185
\(543\) 8.12859e6 1.18308
\(544\) 9.03525e6 1.30901
\(545\) −1.56322e6 −0.225439
\(546\) −2.06587e7 −2.96566
\(547\) 4.05598e6 0.579600 0.289800 0.957087i \(-0.406411\pi\)
0.289800 + 0.957087i \(0.406411\pi\)
\(548\) 2.68591e6 0.382067
\(549\) −4.72435e6 −0.668977
\(550\) 0 0
\(551\) 1.03341e6 0.145008
\(552\) −1.27390e7 −1.77945
\(553\) −53526.2 −0.00744309
\(554\) 6.45603e6 0.893699
\(555\) −798846. −0.110086
\(556\) −1.99703e6 −0.273967
\(557\) 284385. 0.0388390 0.0194195 0.999811i \(-0.493818\pi\)
0.0194195 + 0.999811i \(0.493818\pi\)
\(558\) −744469. −0.101219
\(559\) 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 \(-0.525545\pi\)
−0.0801646 + 0.996782i \(0.525545\pi\)
\(564\) 2.93969e6 0.389138
\(565\) −590527. −0.0778249
\(566\) 279637. 0.0366906
\(567\) 1.15108e7 1.50365
\(568\) 2.60768e7 3.39143
\(569\) −3.94580e6 −0.510922 −0.255461 0.966819i \(-0.582227\pi\)
−0.255461 + 0.966819i \(0.582227\pi\)
\(570\) −2.13272e6 −0.274945
\(571\) −5.88346e6 −0.755166 −0.377583 0.925976i \(-0.623245\pi\)
−0.377583 + 0.925976i \(0.623245\pi\)
\(572\) 0 0
\(573\) 4.24222e6 0.539768
\(574\) 2.73874e7 3.46953
\(575\) 4.10637e6 0.517951
\(576\) 4.54637e6 0.570964
\(577\) 3.61005e6 0.451413 0.225706 0.974195i \(-0.427531\pi\)
0.225706 + 0.974195i \(0.427531\pi\)
\(578\) −5.47057e6 −0.681104
\(579\) 7.22190e6 0.895272
\(580\) −591980. −0.0730697
\(581\) −1.52791e7 −1.87783
\(582\) −1.07613e6 −0.131691
\(583\) 0 0
\(584\) 2.14816e7 2.60636
\(585\) 927748. 0.112083
\(586\) −6.63853e6 −0.798597
\(587\) 5.01469e6 0.600688 0.300344 0.953831i \(-0.402899\pi\)
0.300344 + 0.953831i \(0.402899\pi\)
\(588\) 1.60825e7 1.91827
\(589\) 448821. 0.0533070
\(590\) −3.12110e6 −0.369128
\(591\) −1.78367e7 −2.10061
\(592\) −1.04231e7 −1.22234
\(593\) −1.64451e7 −1.92044 −0.960220 0.279244i \(-0.909916\pi\)
−0.960220 + 0.279244i \(0.909916\pi\)
\(594\) 0 0
\(595\) −1.34463e6 −0.155708
\(596\) 1.72192e7 1.98563
\(597\) −1.25973e7 −1.44658
\(598\) −8.19109e6 −0.936675
\(599\) 6.45089e6 0.734603 0.367302 0.930102i \(-0.380282\pi\)
0.367302 + 0.930102i \(0.380282\pi\)
\(600\) −2.88622e7 −3.27305
\(601\) 8.32443e6 0.940087 0.470044 0.882643i \(-0.344238\pi\)
0.470044 + 0.882643i \(0.344238\pi\)
\(602\) −3.40326e7 −3.82740
\(603\) 1.01511e7 1.13689
\(604\) 2.29062e7 2.55482
\(605\) 0 0
\(606\) −8.14134e6 −0.900563
\(607\) −1.47290e7 −1.62256 −0.811279 0.584659i \(-0.801228\pi\)
−0.811279 + 0.584659i \(0.801228\pi\)
\(608\) −1.09796e7 −1.20455
\(609\) 3.05446e6 0.333727
\(610\) −2.31779e6 −0.252202
\(611\) 1.09520e6 0.118683
\(612\) 1.31825e7 1.42272
\(613\) 1.14564e7 1.23139 0.615697 0.787983i \(-0.288874\pi\)
0.615697 + 0.787983i \(0.288874\pi\)
\(614\) −5.72243e6 −0.612575
\(615\) −2.86057e6 −0.304975
\(616\) 0 0
\(617\) −413797. −0.0437597 −0.0218799 0.999761i \(-0.506965\pi\)
−0.0218799 + 0.999761i \(0.506965\pi\)
\(618\) 3.87003e7 4.07609
\(619\) −1.25898e7 −1.32067 −0.660333 0.750973i \(-0.729585\pi\)
−0.660333 + 0.750973i \(0.729585\pi\)
\(620\) −257104. −0.0268615
\(621\) 1.65987e6 0.172721
\(622\) 1.94177e6 0.201243
\(623\) −3.79198e6 −0.391422
\(624\) 2.81538e7 2.89452
\(625\) 9.06989e6 0.928757
\(626\) −9.71264e6 −0.990607
\(627\) 0 0
\(628\) 2.59811e7 2.62881
\(629\) −4.22886e6 −0.426183
\(630\) −2.71033e6 −0.272063
\(631\) −1.55648e7 −1.55621 −0.778107 0.628132i \(-0.783820\pi\)
−0.778107 + 0.628132i \(0.783820\pi\)
\(632\) 149168. 0.0148554
\(633\) −3.43976e6 −0.341208
\(634\) −1.95742e7 −1.93402
\(635\) −235544. −0.0231813
\(636\) 3.70176e7 3.62882
\(637\) 5.99163e6 0.585054
\(638\) 0 0
\(639\) 1.04285e7 1.01035
\(640\) −414867. −0.0400368
\(641\) −1.23075e7 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(642\) −1.99324e7 −1.90863
\(643\) −1.44400e6 −0.137733 −0.0688667 0.997626i \(-0.521938\pi\)
−0.0688667 + 0.997626i \(0.521938\pi\)
\(644\) 1.68447e7 1.60048
\(645\) 3.55465e6 0.336433
\(646\) −1.12900e7 −1.06442
\(647\) −7.35610e6 −0.690855 −0.345427 0.938445i \(-0.612266\pi\)
−0.345427 + 0.938445i \(0.612266\pi\)
\(648\) −3.20786e7 −3.00108
\(649\) 0 0
\(650\) −1.85583e7 −1.72288
\(651\) 1.32659e6 0.122683
\(652\) 4.60009e7 4.23787
\(653\) 3.83734e6 0.352166 0.176083 0.984375i \(-0.443657\pi\)
0.176083 + 0.984375i \(0.443657\pi\)
\(654\) −3.87948e7 −3.54674
\(655\) 102814. 0.00936373
\(656\) −3.73237e7 −3.38630
\(657\) 8.59083e6 0.776465
\(658\) −3.19952e6 −0.288085
\(659\) 1.98049e7 1.77648 0.888239 0.459382i \(-0.151929\pi\)
0.888239 + 0.459382i \(0.151929\pi\)
\(660\) 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
\(663\) 1.14226e7 1.00921
\(664\) 4.25803e7 3.74790
\(665\) 1.63398e6 0.143283
\(666\) −8.52395e6 −0.744656
\(667\) 1.21108e6 0.105404
\(668\) 1.21599e7 1.05436
\(669\) −2.17977e7 −1.88298
\(670\) 4.98017e6 0.428605
\(671\) 0 0
\(672\) −3.24525e7 −2.77220
\(673\) −4.88569e6 −0.415804 −0.207902 0.978150i \(-0.566663\pi\)
−0.207902 + 0.978150i \(0.566663\pi\)
\(674\) −4.02840e6 −0.341573
\(675\) 3.76072e6 0.317696
\(676\) −2.19056e6 −0.184369
\(677\) 1.56799e7 1.31483 0.657416 0.753528i \(-0.271649\pi\)
0.657416 + 0.753528i \(0.271649\pi\)
\(678\) −1.46552e7 −1.22439
\(679\) 824478. 0.0686285
\(680\) 3.74726e6 0.310772
\(681\) −1.08717e7 −0.898320
\(682\) 0 0
\(683\) −1.94703e6 −0.159705 −0.0798527 0.996807i \(-0.525445\pi\)
−0.0798527 + 0.996807i \(0.525445\pi\)
\(684\) −1.60192e7 −1.30919
\(685\) 305336. 0.0248629
\(686\) 1.12311e7 0.911200
\(687\) 2.30611e7 1.86418
\(688\) 4.63799e7 3.73558
\(689\) 1.37912e7 1.10676
\(690\) −2.49940e6 −0.199854
\(691\) −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
\(696\) −8.51227e6 −0.666073
\(697\) −1.51430e7 −1.18067
\(698\) 9.61563e6 0.747032
\(699\) 604162. 0.0467692
\(700\) 3.81645e7 2.94384
\(701\) −5.48228e6 −0.421373 −0.210686 0.977554i \(-0.567570\pi\)
−0.210686 + 0.977554i \(0.567570\pi\)
\(702\) −7.50161e6 −0.574528
\(703\) 5.13887e6 0.392174
\(704\) 0 0
\(705\) 334185. 0.0253230
\(706\) −4.64524e7 −3.50749
\(707\) 6.23750e6 0.469312
\(708\) −5.45244e7 −4.08797
\(709\) −1.30530e7 −0.975200 −0.487600 0.873067i \(-0.662128\pi\)
−0.487600 + 0.873067i \(0.662128\pi\)
\(710\) 5.11629e6 0.380898
\(711\) 59654.9 0.00442560
\(712\) 1.05676e7 0.781225
\(713\) 525987. 0.0387482
\(714\) −3.33700e7 −2.44969
\(715\) 0 0
\(716\) −4.76714e7 −3.47517
\(717\) −1.69814e7 −1.23360
\(718\) 1.03541e7 0.749550
\(719\) −2.17045e7 −1.56577 −0.782885 0.622167i \(-0.786252\pi\)
−0.782885 + 0.622167i \(0.786252\pi\)
\(720\) 3.69365e6 0.265537
\(721\) −2.96503e7 −2.12418
\(722\) −1.20228e7 −0.858348
\(723\) −1.57418e7 −1.11998
\(724\) 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
\(729\) −6.64290e6 −0.462955
\(730\) 4.21470e6 0.292725
\(731\) 1.88173e7 1.30246
\(732\) −4.04909e7 −2.79306
\(733\) −7.35742e6 −0.505785 −0.252892 0.967494i \(-0.581382\pi\)
−0.252892 + 0.967494i \(0.581382\pi\)
\(734\) −1.25887e7 −0.862463
\(735\) 1.82826e6 0.124830
\(736\) −1.28673e7 −0.875573
\(737\) 0 0
\(738\) −3.05232e7 −2.06295
\(739\) 6.81140e6 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(740\) −2.94377e6 −0.197617
\(741\) −1.38806e7 −0.928676
\(742\) −4.02895e7 −2.68647
\(743\) 1.01182e7 0.672405 0.336203 0.941790i \(-0.390857\pi\)
0.336203 + 0.941790i \(0.390857\pi\)
\(744\) −3.69698e6 −0.244858
\(745\) 1.95749e6 0.129214
\(746\) −1.89457e7 −1.24642
\(747\) 1.70286e7 1.11655
\(748\) 0 0
\(749\) 1.52712e7 0.994649
\(750\) −1.14645e7 −0.744220
\(751\) −1.91140e7 −1.23667 −0.618333 0.785916i \(-0.712192\pi\)
−0.618333 + 0.785916i \(0.712192\pi\)
\(752\) 4.36033e6 0.281174
\(753\) 1.15903e7 0.744914
\(754\) −5.47334e6 −0.350610
\(755\) 2.60398e6 0.166254
\(756\) 1.54268e7 0.981685
\(757\) 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 \(-0.266290\pi\)
0.670009 + 0.742353i \(0.266290\pi\)
\(762\) −5.84554e6 −0.364702
\(763\) 2.97227e7 1.84832
\(764\) 1.56327e7 0.968948
\(765\) 1.49859e6 0.0925827
\(766\) 3.07373e7 1.89275
\(767\) −2.03134e7 −1.24680
\(768\) −2.66844e7 −1.63251
\(769\) −1.45027e7 −0.884369 −0.442184 0.896924i \(-0.645796\pi\)
−0.442184 + 0.896924i \(0.645796\pi\)
\(770\) 0 0
\(771\) −1.57939e7 −0.956874
\(772\) 2.66129e7 1.60712
\(773\) 3.18546e7 1.91745 0.958723 0.284342i \(-0.0917750\pi\)
0.958723 + 0.284342i \(0.0917750\pi\)
\(774\) 3.79293e7 2.27574
\(775\) 1.19171e6 0.0712716
\(776\) −2.29768e6 −0.136973
\(777\) 1.51891e7 0.902565
\(778\) 2.44759e7 1.44974
\(779\) 1.84016e7 1.08646
\(780\) 7.95144e6 0.467960
\(781\) 0 0
\(782\) −1.32311e7 −0.773710
\(783\) 1.10914e6 0.0646519
\(784\) 2.38545e7 1.38606
\(785\) 2.95355e6 0.171068
\(786\) 2.55156e6 0.147316
\(787\) 1.68239e7 0.968253 0.484126 0.874998i \(-0.339138\pi\)
0.484126 + 0.874998i \(0.339138\pi\)
\(788\) −6.57285e7 −3.77084
\(789\) 1.57663e7 0.901650
\(790\) 29267.0 0.00166844
\(791\) 1.12281e7 0.638067
\(792\) 0 0
\(793\) −1.50852e7 −0.851858
\(794\) 4.10455e7 2.31055
\(795\) 4.20818e6 0.236144
\(796\) −4.64215e7 −2.59679
\(797\) −2.00376e7 −1.11738 −0.558690 0.829377i \(-0.688696\pi\)
−0.558690 + 0.829377i \(0.688696\pi\)
\(798\) 4.05509e7 2.25421
\(799\) 1.76908e6 0.0980347
\(800\) −2.91530e7 −1.61049
\(801\) 4.22616e6 0.232736
\(802\) −5.99384e7 −3.29055
\(803\) 0 0
\(804\) 8.70018e7 4.74666
\(805\) 1.91492e6 0.104150
\(806\) −2.37714e6 −0.128889
\(807\) 1.65356e7 0.893790
\(808\) −1.73829e7 −0.936683
\(809\) 1.59711e7 0.857954 0.428977 0.903315i \(-0.358874\pi\)
0.428977 + 0.903315i \(0.358874\pi\)
\(810\) −6.29385e6 −0.337057
\(811\) −1.37309e7 −0.733074 −0.366537 0.930403i \(-0.619457\pi\)
−0.366537 + 0.930403i \(0.619457\pi\)
\(812\) 1.12558e7 0.599080
\(813\) −2.02625e6 −0.107515
\(814\) 0 0
\(815\) 5.22941e6 0.275777
\(816\) 4.54769e7 2.39092
\(817\) −2.28666e7 −1.19852
\(818\) −2.49192e7 −1.30212
\(819\) −1.76400e7 −0.918942
\(820\) −1.05413e7 −0.547467
\(821\) −2.26402e7 −1.17226 −0.586128 0.810218i \(-0.699348\pi\)
−0.586128 + 0.810218i \(0.699348\pi\)
\(822\) 7.57758e6 0.391157
\(823\) 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 \(-0.544561\pi\)
−0.139536 + 0.990217i \(0.544561\pi\)
\(828\) −1.87734e7 −0.951630
\(829\) 1.81680e7 0.918164 0.459082 0.888394i \(-0.348178\pi\)
0.459082 + 0.888394i \(0.348178\pi\)
\(830\) 8.35429e6 0.420934
\(831\) 1.28214e7 0.644070
\(832\) 1.45169e7 0.727050
\(833\) 9.67828e6 0.483265
\(834\) −5.63410e6 −0.280485
\(835\) 1.38234e6 0.0686120
\(836\) 0 0
\(837\) 481712. 0.0237670
\(838\) −1.47283e7 −0.724507
\(839\) −1.83237e7 −0.898689 −0.449344 0.893359i \(-0.648342\pi\)
−0.449344 + 0.893359i \(0.648342\pi\)
\(840\) −1.34593e7 −0.658148
\(841\) −1.97019e7 −0.960546
\(842\) −4.99582e7 −2.42844
\(843\) −2.70866e7 −1.31276
\(844\) −1.26756e7 −0.612509
\(845\) −249024. −0.0119978
\(846\) 3.56586e6 0.171293
\(847\) 0 0
\(848\) 5.49069e7 2.62203
\(849\) 555348. 0.0264421
\(850\) −2.99772e7 −1.42313
\(851\) 6.02240e6 0.285066
\(852\) 8.93797e7 4.21832
\(853\) −3.66987e7 −1.72694 −0.863471 0.504398i \(-0.831715\pi\)
−0.863471 + 0.504398i \(0.831715\pi\)
\(854\) 4.40698e7 2.06774
\(855\) −1.82108e6 −0.0851948
\(856\) −4.25584e7 −1.98518
\(857\) −3.80021e7 −1.76749 −0.883743 0.467972i \(-0.844985\pi\)
−0.883743 + 0.467972i \(0.844985\pi\)
\(858\) 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
\(861\) 5.43901e7 2.50042
\(862\) 2.84303e7 1.30321
\(863\) −9.79342e6 −0.447618 −0.223809 0.974633i \(-0.571849\pi\)
−0.223809 + 0.974633i \(0.571849\pi\)
\(864\) −1.17842e7 −0.537050
\(865\) 4.32261e6 0.196429
\(866\) 2.82698e7 1.28094
\(867\) −1.08643e7 −0.490857
\(868\) 4.88851e6 0.220230
\(869\) 0 0
\(870\) −1.67012e6 −0.0748080
\(871\) 3.24131e7 1.44769
\(872\) −8.28322e7 −3.68899
\(873\) −918880. −0.0408059
\(874\) 1.60783e7 0.711969
\(875\) 8.78353e6 0.387837
\(876\) 7.36293e7 3.24183
\(877\) 3.57898e7 1.57130 0.785652 0.618669i \(-0.212328\pi\)
0.785652 + 0.618669i \(0.212328\pi\)
\(878\) 4.33632e7 1.89839
\(879\) −1.31838e7 −0.575532
\(880\) 0 0
\(881\) 7.14038e6 0.309943 0.154971 0.987919i \(-0.450471\pi\)
0.154971 + 0.987919i \(0.450471\pi\)
\(882\) 1.95082e7 0.844393
\(883\) −2.77023e7 −1.19568 −0.597838 0.801617i \(-0.703973\pi\)
−0.597838 + 0.801617i \(0.703973\pi\)
\(884\) 4.20926e7 1.81165
\(885\) −6.19837e6 −0.266023
\(886\) 7.14091e7 3.05611
\(887\) −387870. −0.0165530 −0.00827651 0.999966i \(-0.502635\pi\)
−0.00827651 + 0.999966i \(0.502635\pi\)
\(888\) −4.23293e7 −1.80140
\(889\) 4.47857e6 0.190058
\(890\) 2.07337e6 0.0877410
\(891\) 0 0
\(892\) −8.03250e7 −3.38017
\(893\) −2.14977e6 −0.0902117
\(894\) 4.85795e7 2.03287
\(895\) −5.41932e6 −0.226145
\(896\) 7.88818e6 0.328251
\(897\) −1.62672e7 −0.675042
\(898\) −7.21309e6 −0.298491
\(899\) 351468. 0.0145040
\(900\) −4.25344e7 −1.75039
\(901\) 2.22769e7 0.914203
\(902\) 0 0
\(903\) −6.75873e7 −2.75833
\(904\) −3.12909e7 −1.27349
\(905\) 3.40519e6 0.138204
\(906\) 6.46236e7 2.61560
\(907\) 5.05051e6 0.203853 0.101926 0.994792i \(-0.467499\pi\)
0.101926 + 0.994792i \(0.467499\pi\)
\(908\) −4.00626e7 −1.61259
\(909\) −6.95169e6 −0.279049
\(910\) −8.65425e6 −0.346438
\(911\) −1.72283e7 −0.687774 −0.343887 0.939011i \(-0.611744\pi\)
−0.343887 + 0.939011i \(0.611744\pi\)
\(912\) −5.52631e7 −2.20013
\(913\) 0 0
\(914\) −8.44660e7 −3.34439
\(915\) −4.60303e6 −0.181757
\(916\) 8.49807e7 3.34643
\(917\) −1.95488e6 −0.0767709
\(918\) −1.21174e7 −0.474571
\(919\) −3.89056e7 −1.51958 −0.759789 0.650170i \(-0.774698\pi\)
−0.759789 + 0.650170i \(0.774698\pi\)
\(920\) −5.33655e6 −0.207870
\(921\) −1.13645e7 −0.441470
\(922\) 4.66769e7 1.80832
\(923\) 3.32990e7 1.28655
\(924\) 0 0
\(925\) 1.36447e7 0.524338
\(926\) −9.40340e7 −3.60377
\(927\) 3.30453e7 1.26302
\(928\) −8.59801e6 −0.327739
\(929\) 3.18602e7 1.21118 0.605590 0.795777i \(-0.292937\pi\)
0.605590 + 0.795777i \(0.292937\pi\)
\(930\) −725351. −0.0275005
\(931\) −1.17610e7 −0.444701
\(932\) 2.22635e6 0.0839564
\(933\) 3.85627e6 0.145032
\(934\) 7.45701e7 2.79703
\(935\) 0 0
\(936\) 4.91596e7 1.83408
\(937\) 4.09748e7 1.52464 0.762322 0.647198i \(-0.224059\pi\)
0.762322 + 0.647198i \(0.224059\pi\)
\(938\) −9.46917e7 −3.51402
\(939\) −1.92889e7 −0.713909
\(940\) 1.23148e6 0.0454577
\(941\) 1.76369e7 0.649304 0.324652 0.945833i \(-0.394753\pi\)
0.324652 + 0.945833i \(0.394753\pi\)
\(942\) 7.32988e7 2.69135
\(943\) 2.15655e7 0.789732
\(944\) −8.08741e7 −2.95379
\(945\) 1.75373e6 0.0638827
\(946\) 0 0
\(947\) 4.07232e7 1.47559 0.737797 0.675022i \(-0.235866\pi\)
0.737797 + 0.675022i \(0.235866\pi\)
\(948\) 511284. 0.0184774
\(949\) 2.74311e7 0.988730
\(950\) 3.64280e7 1.30956
\(951\) −3.88735e7 −1.39381
\(952\) −7.12495e7 −2.54794
\(953\) 1.02210e7 0.364553 0.182276 0.983247i \(-0.441653\pi\)
0.182276 + 0.983247i \(0.441653\pi\)
\(954\) 4.49027e7 1.59736
\(955\) 1.77713e6 0.0630538
\(956\) −6.25767e7 −2.21446
\(957\) 0 0
\(958\) 1.57077e7 0.552967
\(959\) −5.80557e6 −0.203844
\(960\) 4.42962e6 0.155127
\(961\) −2.84765e7 −0.994668
\(962\) −2.72175e7 −0.948225
\(963\) −1.70198e7 −0.591410
\(964\) −5.80089e7 −2.01049
\(965\) 3.02536e6 0.104583
\(966\) 4.75229e7 1.63855
\(967\) 7.46133e6 0.256596 0.128298 0.991736i \(-0.459049\pi\)
0.128298 + 0.991736i \(0.459049\pi\)
\(968\) 0 0
\(969\) −2.24214e7 −0.767103
\(970\) −450807. −0.0153837
\(971\) 4.34819e7 1.47999 0.739997 0.672610i \(-0.234827\pi\)
0.739997 + 0.672610i \(0.234827\pi\)
\(972\) −8.71562e7 −2.95892
\(973\) 4.31657e6 0.146170
\(974\) 1.69736e7 0.573292
\(975\) −3.68559e7 −1.24164
\(976\) −6.00587e7 −2.01814
\(977\) −1.52214e7 −0.510173 −0.255087 0.966918i \(-0.582104\pi\)
−0.255087 + 0.966918i \(0.582104\pi\)
\(978\) 1.29779e8 4.33869
\(979\) 0 0
\(980\) 6.73719e6 0.224085
\(981\) −3.31260e7 −1.09900
\(982\) −1.29392e7 −0.428181
\(983\) 2.90192e7 0.957858 0.478929 0.877854i \(-0.341025\pi\)
0.478929 + 0.877854i \(0.341025\pi\)
\(984\) −1.51576e8 −4.99049
\(985\) −7.47205e6 −0.245386
\(986\) −8.84110e6 −0.289610
\(987\) −6.35411e6 −0.207616
\(988\) −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
\(993\) 4.06853e6 0.130938
\(994\) −9.72798e7 −3.12289
\(995\) −5.27722e6 −0.168985
\(996\) 1.45946e8 4.66171
\(997\) −1.05268e7 −0.335397 −0.167699 0.985838i \(-0.553634\pi\)
−0.167699 + 0.985838i \(0.553634\pi\)
\(998\) −7.96577e7 −2.53164
\(999\) 5.51546e6 0.174851
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 121.6.a.d.1.3 3
3.2 odd 2 1089.6.a.r.1.1 3
11.10 odd 2 11.6.a.b.1.1 3
33.32 even 2 99.6.a.g.1.3 3
44.43 even 2 176.6.a.i.1.1 3
55.32 even 4 275.6.b.b.199.1 6
55.43 even 4 275.6.b.b.199.6 6
55.54 odd 2 275.6.a.b.1.3 3
77.76 even 2 539.6.a.e.1.1 3
88.21 odd 2 704.6.a.q.1.1 3
88.43 even 2 704.6.a.t.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.1 3 11.10 odd 2
99.6.a.g.1.3 3 33.32 even 2
121.6.a.d.1.3 3 1.1 even 1 trivial
176.6.a.i.1.1 3 44.43 even 2
275.6.a.b.1.3 3 55.54 odd 2
275.6.b.b.199.1 6 55.32 even 4
275.6.b.b.199.6 6 55.43 even 4
539.6.a.e.1.1 3 77.76 even 2
704.6.a.q.1.1 3 88.21 odd 2
704.6.a.t.1.3 3 88.43 even 2
1089.6.a.r.1.1 3 3.2 odd 2