Properties

Label 176.6.a.i.1.1
Level $176$
Weight $6$
Character 176.1
Self dual yes
Analytic conductor $28.228$
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] = [176,6,Mod(1,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("176.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 176.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: \(28.2275522871\)
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: \( 2^{3} \)
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 \(\chi\) \(=\) 176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-20.6466 q^{3} +8.64919 q^{5} -164.454 q^{7} +183.283 q^{9} +O(q^{10})\) \(q-20.6466 q^{3} +8.64919 q^{5} -164.454 q^{7} +183.283 q^{9} -121.000 q^{11} -585.236 q^{13} -178.577 q^{15} -945.333 q^{17} -1148.76 q^{19} +3395.41 q^{21} +1346.27 q^{23} -3050.19 q^{25} +1232.95 q^{27} +899.585 q^{29} +390.700 q^{31} +2498.24 q^{33} -1422.39 q^{35} -4473.41 q^{37} +12083.2 q^{39} +16018.7 q^{41} +19905.5 q^{43} +1585.25 q^{45} -1871.38 q^{47} +10238.0 q^{49} +19517.9 q^{51} +23565.1 q^{53} -1046.55 q^{55} +23718.0 q^{57} +34709.8 q^{59} +25776.2 q^{61} -30141.6 q^{63} -5061.82 q^{65} -55384.6 q^{67} -27795.9 q^{69} -56898.4 q^{71} -46871.8 q^{73} +62976.2 q^{75} +19898.9 q^{77} +325.479 q^{79} -69994.1 q^{81} +92908.3 q^{83} -8176.37 q^{85} -18573.4 q^{87} +23058.0 q^{89} +96244.2 q^{91} -8066.65 q^{93} -9935.84 q^{95} -5013.44 q^{97} -22177.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 24 q^{5} - 84 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 24 q^{5} - 84 q^{7} - 7 q^{9} - 363 q^{11} + 486 q^{13} - 1654 q^{15} + 1086 q^{17} - 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 57 q^{25} + 2990 q^{27} - 3426 q^{29} + 4098 q^{31} + 4114 q^{33} + 24228 q^{35} + 17724 q^{37} + 6560 q^{39} + 5994 q^{41} + 26208 q^{43} + 18458 q^{45} + 17232 q^{47} + 48531 q^{49} + 22724 q^{51} + 50586 q^{53} - 2904 q^{55} + 20160 q^{57} + 3738 q^{59} + 18486 q^{61} + 12496 q^{63} - 7668 q^{65} + 47754 q^{67} + 35042 q^{69} - 39282 q^{71} + 15426 q^{73} + 21916 q^{75} + 10164 q^{77} - 125148 q^{79} - 86917 q^{81} + 143928 q^{83} - 104040 q^{85} + 19368 q^{87} - 106824 q^{89} + 109632 q^{91} - 16622 q^{93} + 22200 q^{95} + 9684 q^{97} + 847 q^{99}+O(q^{100}) \) 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 0
\(3\) −20.6466 −1.32448 −0.662241 0.749291i \(-0.730395\pi\)
−0.662241 + 0.749291i \(0.730395\pi\)
\(4\) 0 0
\(5\) 8.64919 0.154721 0.0773607 0.997003i \(-0.475351\pi\)
0.0773607 + 0.997003i \(0.475351\pi\)
\(6\) 0 0
\(7\) −164.454 −1.26852 −0.634261 0.773119i \(-0.718696\pi\)
−0.634261 + 0.773119i \(0.718696\pi\)
\(8\) 0 0
\(9\) 183.283 0.754253
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −585.236 −0.960446 −0.480223 0.877147i \(-0.659444\pi\)
−0.480223 + 0.877147i \(0.659444\pi\)
\(14\) 0 0
\(15\) −178.577 −0.204926
\(16\) 0 0
\(17\) −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 \(0.618937\pi\)
\(20\) 0 0
\(21\) 3395.41 1.68014
\(22\) 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
\(26\) 0 0
\(27\) 1232.95 0.325488
\(28\) 0 0
\(29\) 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 \(-0.488376\pi\)
0.0365098 + 0.999333i \(0.488376\pi\)
\(32\) 0 0
\(33\) 2498.24 0.399346
\(34\) 0 0
\(35\) −1422.39 −0.196268
\(36\) 0 0
\(37\) −4473.41 −0.537198 −0.268599 0.963252i \(-0.586561\pi\)
−0.268599 + 0.963252i \(0.586561\pi\)
\(38\) 0 0
\(39\) 12083.2 1.27209
\(40\) 0 0
\(41\) 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
\(45\) 1585.25 0.116699
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) 19517.9 1.05077
\(52\) 0 0
\(53\) 23565.1 1.15234 0.576169 0.817330i \(-0.304547\pi\)
0.576169 + 0.817330i \(0.304547\pi\)
\(54\) 0 0
\(55\) −1046.55 −0.0466503
\(56\) 0 0
\(57\) 23718.0 0.966922
\(58\) 0 0
\(59\) 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 \(-0.353747\pi\)
0.443470 + 0.896289i \(0.353747\pi\)
\(62\) 0 0
\(63\) −30141.6 −0.956787
\(64\) 0 0
\(65\) −5061.82 −0.148602
\(66\) 0 0
\(67\) −55384.6 −1.50731 −0.753655 0.657271i \(-0.771711\pi\)
−0.753655 + 0.657271i \(0.771711\pi\)
\(68\) 0 0
\(69\) −27795.9 −0.702842
\(70\) 0 0
\(71\) −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 \(-0.672106\pi\)
−0.514724 + 0.857356i \(0.672106\pi\)
\(74\) 0 0
\(75\) 62976.2 1.29278
\(76\) 0 0
\(77\) 19898.9 0.382474
\(78\) 0 0
\(79\) 325.479 0.00586753 0.00293377 0.999996i \(-0.499066\pi\)
0.00293377 + 0.999996i \(0.499066\pi\)
\(80\) 0 0
\(81\) −69994.1 −1.18536
\(82\) 0 0
\(83\) 92908.3 1.48033 0.740166 0.672424i \(-0.234747\pi\)
0.740166 + 0.672424i \(0.234747\pi\)
\(84\) 0 0
\(85\) −8176.37 −0.122748
\(86\) 0 0
\(87\) −18573.4 −0.263083
\(88\) 0 0
\(89\) 23058.0 0.308565 0.154283 0.988027i \(-0.450693\pi\)
0.154283 + 0.988027i \(0.450693\pi\)
\(90\) 0 0
\(91\) 96244.2 1.21835
\(92\) 0 0
\(93\) −8066.65 −0.0967132
\(94\) 0 0
\(95\) −9935.84 −0.112952
\(96\) 0 0
\(97\) −5013.44 −0.0541011 −0.0270506 0.999634i \(-0.508612\pi\)
−0.0270506 + 0.999634i \(0.508612\pi\)
\(98\) 0 0
\(99\) −22177.3 −0.227416
\(100\) 0 0
\(101\) 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 \(-0.815847\pi\)
−0.837265 + 0.546798i \(0.815847\pi\)
\(104\) 0 0
\(105\) 29367.6 0.259953
\(106\) 0 0
\(107\) −92860.5 −0.784100 −0.392050 0.919944i \(-0.628234\pi\)
−0.392050 + 0.919944i \(0.628234\pi\)
\(108\) 0 0
\(109\) 180736. 1.45707 0.728533 0.685011i \(-0.240203\pi\)
0.728533 + 0.685011i \(0.240203\pi\)
\(110\) 0 0
\(111\) 92360.8 0.711509
\(112\) 0 0
\(113\) −68275.4 −0.503000 −0.251500 0.967857i \(-0.580924\pi\)
−0.251500 + 0.967857i \(0.580924\pi\)
\(114\) 0 0
\(115\) 11644.1 0.0821036
\(116\) 0 0
\(117\) −107264. −0.724419
\(118\) 0 0
\(119\) 155463. 1.00638
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 0 0
\(123\) −330732. −1.97112
\(124\) 0 0
\(125\) −53410.4 −0.305739
\(126\) 0 0
\(127\) −27233.1 −0.149826 −0.0749130 0.997190i \(-0.523868\pi\)
−0.0749130 + 0.997190i \(0.523868\pi\)
\(128\) 0 0
\(129\) −410981. −2.17444
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) 10664.0 0.0503599
\(136\) 0 0
\(137\) 35302.2 0.160694 0.0803471 0.996767i \(-0.474397\pi\)
0.0803471 + 0.996767i \(0.474397\pi\)
\(138\) 0 0
\(139\) −26248.0 −0.115228 −0.0576141 0.998339i \(-0.518349\pi\)
−0.0576141 + 0.998339i \(0.518349\pi\)
\(140\) 0 0
\(141\) 38637.7 0.163668
\(142\) 0 0
\(143\) 70813.6 0.289585
\(144\) 0 0
\(145\) 7780.68 0.0307325
\(146\) 0 0
\(147\) −211379. −0.806807
\(148\) 0 0
\(149\) −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 \(0.319457\pi\)
\(152\) 0 0
\(153\) −173264. −0.598383
\(154\) 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 \(0.313549\pi\)
\(158\) 0 0
\(159\) −486541. −1.52625
\(160\) 0 0
\(161\) −221398. −0.673147
\(162\) 0 0
\(163\) −604612. −1.78241 −0.891205 0.453600i \(-0.850139\pi\)
−0.891205 + 0.453600i \(0.850139\pi\)
\(164\) 0 0
\(165\) 21607.8 0.0617875
\(166\) 0 0
\(167\) 159824. 0.443455 0.221728 0.975109i \(-0.428830\pi\)
0.221728 + 0.975109i \(0.428830\pi\)
\(168\) 0 0
\(169\) −28791.6 −0.0775442
\(170\) 0 0
\(171\) −210549. −0.550633
\(172\) 0 0
\(173\) −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
\(177\) −716641. −1.71937
\(178\) 0 0
\(179\) 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 \(0.352627\pi\)
\(182\) 0 0
\(183\) −532192. −1.17474
\(184\) 0 0
\(185\) −38691.4 −0.0831160
\(186\) 0 0
\(187\) 114385. 0.239203
\(188\) 0 0
\(189\) −202762. −0.412888
\(190\) 0 0
\(191\) −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 \(-0.609740\pi\)
−0.337971 + 0.941157i \(0.609740\pi\)
\(194\) 0 0
\(195\) 104510. 0.196820
\(196\) 0 0
\(197\) 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 \(-0.316115\pi\)
0.546093 + 0.837725i \(0.316115\pi\)
\(200\) 0 0
\(201\) 1.14351e6 1.99640
\(202\) 0 0
\(203\) −147940. −0.251968
\(204\) 0 0
\(205\) 138549. 0.230260
\(206\) 0 0
\(207\) 246749. 0.400248
\(208\) 0 0
\(209\) 139000. 0.220115
\(210\) 0 0
\(211\) −166602. −0.257616 −0.128808 0.991670i \(-0.541115\pi\)
−0.128808 + 0.991670i \(0.541115\pi\)
\(212\) 0 0
\(213\) 1.17476e6 1.77419
\(214\) 0 0
\(215\) 172166. 0.254011
\(216\) 0 0
\(217\) −64252.0 −0.0926270
\(218\) 0 0
\(219\) 967746. 1.36349
\(220\) 0 0
\(221\) 553243. 0.761965
\(222\) 0 0
\(223\) 1.05575e6 1.42167 0.710836 0.703358i \(-0.248317\pi\)
0.710836 + 0.703358i \(0.248317\pi\)
\(224\) 0 0
\(225\) −559050. −0.736197
\(226\) 0 0
\(227\) −526562. −0.678242 −0.339121 0.940743i \(-0.610130\pi\)
−0.339121 + 0.940743i \(0.610130\pi\)
\(228\) 0 0
\(229\) 1.11694e6 1.40748 0.703740 0.710458i \(-0.251512\pi\)
0.703740 + 0.710458i \(0.251512\pi\)
\(230\) 0 0
\(231\) −410845. −0.506580
\(232\) 0 0
\(233\) −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
\(236\) 0 0
\(237\) −6720.05 −0.00777144
\(238\) 0 0
\(239\) −822476. −0.931384 −0.465692 0.884947i \(-0.654194\pi\)
−0.465692 + 0.884947i \(0.654194\pi\)
\(240\) 0 0
\(241\) 762439. 0.845595 0.422797 0.906224i \(-0.361048\pi\)
0.422797 + 0.906224i \(0.361048\pi\)
\(242\) 0 0
\(243\) 1.14554e6 1.24449
\(244\) 0 0
\(245\) 88550.1 0.0942484
\(246\) 0 0
\(247\) 672296. 0.701161
\(248\) 0 0
\(249\) −1.91824e6 −1.96067
\(250\) 0 0
\(251\) −561364. −0.562419 −0.281209 0.959646i \(-0.590736\pi\)
−0.281209 + 0.959646i \(0.590736\pi\)
\(252\) 0 0
\(253\) −162898. −0.159998
\(254\) 0 0
\(255\) 168814. 0.162577
\(256\) 0 0
\(257\) −764965. −0.722451 −0.361226 0.932478i \(-0.617642\pi\)
−0.361226 + 0.932478i \(0.617642\pi\)
\(258\) 0 0
\(259\) 735668. 0.681447
\(260\) 0 0
\(261\) 164879. 0.149818
\(262\) 0 0
\(263\) 763627. 0.680756 0.340378 0.940289i \(-0.389445\pi\)
0.340378 + 0.940289i \(0.389445\pi\)
\(264\) 0 0
\(265\) 203819. 0.178292
\(266\) 0 0
\(267\) −476071. −0.408689
\(268\) 0 0
\(269\) 800885. 0.674823 0.337411 0.941357i \(-0.390449\pi\)
0.337411 + 0.941357i \(0.390449\pi\)
\(270\) 0 0
\(271\) −98139.7 −0.0811749 −0.0405874 0.999176i \(-0.512923\pi\)
−0.0405874 + 0.999176i \(0.512923\pi\)
\(272\) 0 0
\(273\) −1.98712e6 −1.61368
\(274\) 0 0
\(275\) 369073. 0.294294
\(276\) 0 0
\(277\) −620993. −0.486281 −0.243140 0.969991i \(-0.578178\pi\)
−0.243140 + 0.969991i \(0.578178\pi\)
\(278\) 0 0
\(279\) 71608.9 0.0550753
\(280\) 0 0
\(281\) 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 \(0.496823\pi\)
\(284\) 0 0
\(285\) 205142. 0.149604
\(286\) 0 0
\(287\) −2.63433e6 −1.88784
\(288\) 0 0
\(289\) −526203. −0.370603
\(290\) 0 0
\(291\) 103511. 0.0716560
\(292\) 0 0
\(293\) 638546. 0.434534 0.217267 0.976112i \(-0.430286\pi\)
0.217267 + 0.976112i \(0.430286\pi\)
\(294\) 0 0
\(295\) 300212. 0.200851
\(296\) 0 0
\(297\) −149186. −0.0981382
\(298\) 0 0
\(299\) −787884. −0.509665
\(300\) 0 0
\(301\) −3.27352e6 −2.08257
\(302\) 0 0
\(303\) −783099. −0.490016
\(304\) 0 0
\(305\) 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
\(309\) 3.72250e6 2.21788
\(310\) 0 0
\(311\) −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 \(0.586860\pi\)
\(314\) 0 0
\(315\) −260701. −0.148035
\(316\) 0 0
\(317\) −1.88280e6 −1.05234 −0.526170 0.850379i \(-0.676372\pi\)
−0.526170 + 0.850379i \(0.676372\pi\)
\(318\) 0 0
\(319\) −108850. −0.0598895
\(320\) 0 0
\(321\) 1.91726e6 1.03853
\(322\) 0 0
\(323\) 1.08596e6 0.579172
\(324\) 0 0
\(325\) 1.78508e6 0.937454
\(326\) 0 0
\(327\) −3.73160e6 −1.92986
\(328\) 0 0
\(329\) 307755. 0.156753
\(330\) 0 0
\(331\) −197056. −0.0988596 −0.0494298 0.998778i \(-0.515740\pi\)
−0.0494298 + 0.998778i \(0.515740\pi\)
\(332\) 0 0
\(333\) −819901. −0.405183
\(334\) 0 0
\(335\) −479033. −0.233213
\(336\) 0 0
\(337\) 387484. 0.185857 0.0929285 0.995673i \(-0.470377\pi\)
0.0929285 + 0.995673i \(0.470377\pi\)
\(338\) 0 0
\(339\) 1.40966e6 0.666215
\(340\) 0 0
\(341\) −47274.7 −0.0220162
\(342\) 0 0
\(343\) 1.08030e6 0.495803
\(344\) 0 0
\(345\) −240412. −0.108745
\(346\) 0 0
\(347\) 2.94793e6 1.31430 0.657148 0.753761i \(-0.271762\pi\)
0.657148 + 0.753761i \(0.271762\pi\)
\(348\) 0 0
\(349\) −924908. −0.406476 −0.203238 0.979129i \(-0.565146\pi\)
−0.203238 + 0.979129i \(0.565146\pi\)
\(350\) 0 0
\(351\) −721564. −0.312613
\(352\) 0 0
\(353\) −4.46816e6 −1.90850 −0.954249 0.299012i \(-0.903343\pi\)
−0.954249 + 0.299012i \(0.903343\pi\)
\(354\) 0 0
\(355\) −492125. −0.207255
\(356\) 0 0
\(357\) −3.20979e6 −1.33293
\(358\) 0 0
\(359\) 995937. 0.407846 0.203923 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(360\) 0 0
\(361\) −1.15645e6 −0.467045
\(362\) 0 0
\(363\) −302287. −0.120407
\(364\) 0 0
\(365\) −405404. −0.159278
\(366\) 0 0
\(367\) 1.21088e6 0.469284 0.234642 0.972082i \(-0.424608\pi\)
0.234642 + 0.972082i \(0.424608\pi\)
\(368\) 0 0
\(369\) 2.93596e6 1.12250
\(370\) 0 0
\(371\) −3.87537e6 −1.46177
\(372\) 0 0
\(373\) 1.82235e6 0.678203 0.339102 0.940750i \(-0.389877\pi\)
0.339102 + 0.940750i \(0.389877\pi\)
\(374\) 0 0
\(375\) 1.10275e6 0.404946
\(376\) 0 0
\(377\) −526470. −0.190774
\(378\) 0 0
\(379\) 419357. 0.149964 0.0749819 0.997185i \(-0.476110\pi\)
0.0749819 + 0.997185i \(0.476110\pi\)
\(380\) 0 0
\(381\) 562271. 0.198442
\(382\) 0 0
\(383\) −2.95656e6 −1.02989 −0.514943 0.857224i \(-0.672187\pi\)
−0.514943 + 0.857224i \(0.672187\pi\)
\(384\) 0 0
\(385\) 172109. 0.0591769
\(386\) 0 0
\(387\) 3.64834e6 1.23828
\(388\) 0 0
\(389\) 2.35429e6 0.788834 0.394417 0.918932i \(-0.370947\pi\)
0.394417 + 0.918932i \(0.370947\pi\)
\(390\) 0 0
\(391\) −1.27267e6 −0.420992
\(392\) 0 0
\(393\) −245429. −0.0801576
\(394\) 0 0
\(395\) 2815.13 0.000907833 0
\(396\) 0 0
\(397\) 3.94809e6 1.25722 0.628609 0.777722i \(-0.283625\pi\)
0.628609 + 0.777722i \(0.283625\pi\)
\(398\) 0 0
\(399\) −3.90051e6 −1.22656
\(400\) 0 0
\(401\) −5.76535e6 −1.79046 −0.895230 0.445604i \(-0.852989\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(402\) 0 0
\(403\) −228652. −0.0701314
\(404\) 0 0
\(405\) −605392. −0.183400
\(406\) 0 0
\(407\) 541282. 0.161971
\(408\) 0 0
\(409\) 2.39693e6 0.708512 0.354256 0.935148i \(-0.384734\pi\)
0.354256 + 0.935148i \(0.384734\pi\)
\(410\) 0 0
\(411\) −728872. −0.212837
\(412\) 0 0
\(413\) −5.70815e6 −1.64672
\(414\) 0 0
\(415\) 803582. 0.229039
\(416\) 0 0
\(417\) 541932. 0.152618
\(418\) 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 \(0.729733\pi\)
\(422\) 0 0
\(423\) −342993. −0.0932040
\(424\) 0 0
\(425\) 2.88345e6 0.774354
\(426\) 0 0
\(427\) −4.23899e6 −1.12510
\(428\) 0 0
\(429\) −1.46206e6 −0.383551
\(430\) 0 0
\(431\) 2.73465e6 0.709103 0.354551 0.935037i \(-0.384634\pi\)
0.354551 + 0.935037i \(0.384634\pi\)
\(432\) 0 0
\(433\) 2.71922e6 0.696986 0.348493 0.937311i \(-0.386693\pi\)
0.348493 + 0.937311i \(0.386693\pi\)
\(434\) 0 0
\(435\) −160645. −0.0407046
\(436\) 0 0
\(437\) −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
\(441\) 1.87645e6 0.459452
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 4.67276e6 1.10613
\(448\) 0 0
\(449\) −693812. −0.162415 −0.0812075 0.996697i \(-0.525878\pi\)
−0.0812075 + 0.996697i \(0.525878\pi\)
\(450\) 0 0
\(451\) −1.93826e6 −0.448716
\(452\) 0 0
\(453\) −6.21601e6 −1.42320
\(454\) 0 0
\(455\) 832434. 0.188504
\(456\) 0 0
\(457\) 8.12461e6 1.81975 0.909876 0.414880i \(-0.136176\pi\)
0.909876 + 0.414880i \(0.136176\pi\)
\(458\) 0 0
\(459\) −1.16554e6 −0.258224
\(460\) 0 0
\(461\) −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 \(-0.0630528\pi\)
0.980445 + 0.196793i \(0.0630528\pi\)
\(464\) 0 0
\(465\) −69770.0 −0.0149636
\(466\) 0 0
\(467\) −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
\(470\) 0 0
\(471\) −7.05046e6 −1.46442
\(472\) 0 0
\(473\) −2.40856e6 −0.495000
\(474\) 0 0
\(475\) 3.50394e6 0.712561
\(476\) 0 0
\(477\) 4.31910e6 0.869155
\(478\) 0 0
\(479\) 1.51089e6 0.300881 0.150440 0.988619i \(-0.451931\pi\)
0.150440 + 0.988619i \(0.451931\pi\)
\(480\) 0 0
\(481\) 2.61800e6 0.515949
\(482\) 0 0
\(483\) 4.57113e6 0.891571
\(484\) 0 0
\(485\) −43362.2 −0.00837061
\(486\) 0 0
\(487\) −1.63265e6 −0.311940 −0.155970 0.987762i \(-0.549850\pi\)
−0.155970 + 0.987762i \(0.549850\pi\)
\(488\) 0 0
\(489\) 1.24832e7 2.36077
\(490\) 0 0
\(491\) −1.24459e6 −0.232982 −0.116491 0.993192i \(-0.537165\pi\)
−0.116491 + 0.993192i \(0.537165\pi\)
\(492\) 0 0
\(493\) −850407. −0.157583
\(494\) 0 0
\(495\) −191816. −0.0351861
\(496\) 0 0
\(497\) 9.35714e6 1.69923
\(498\) 0 0
\(499\) 7.66211e6 1.37752 0.688759 0.724991i \(-0.258156\pi\)
0.688759 + 0.724991i \(0.258156\pi\)
\(500\) 0 0
\(501\) −3.29982e6 −0.587348
\(502\) 0 0
\(503\) 1.07030e7 1.88619 0.943097 0.332518i \(-0.107898\pi\)
0.943097 + 0.332518i \(0.107898\pi\)
\(504\) 0 0
\(505\) 328052. 0.0572420
\(506\) 0 0
\(507\) 594450. 0.102706
\(508\) 0 0
\(509\) 6.27494e6 1.07353 0.536766 0.843731i \(-0.319646\pi\)
0.536766 + 0.843731i \(0.319646\pi\)
\(510\) 0 0
\(511\) 7.70824e6 1.30588
\(512\) 0 0
\(513\) −1.41636e6 −0.237618
\(514\) 0 0
\(515\) −1.55941e6 −0.259086
\(516\) 0 0
\(517\) 226437. 0.0372581
\(518\) 0 0
\(519\) 1.03186e7 1.68152
\(520\) 0 0
\(521\) 3.60326e6 0.581570 0.290785 0.956788i \(-0.406084\pi\)
0.290785 + 0.956788i \(0.406084\pi\)
\(522\) 0 0
\(523\) −3.56925e6 −0.570589 −0.285294 0.958440i \(-0.592091\pi\)
−0.285294 + 0.958440i \(0.592091\pi\)
\(524\) 0 0
\(525\) −1.03567e7 −1.63991
\(526\) 0 0
\(527\) −369342. −0.0579298
\(528\) 0 0
\(529\) −4.62391e6 −0.718406
\(530\) 0 0
\(531\) 6.36174e6 0.979128
\(532\) 0 0
\(533\) −9.37473e6 −1.42936
\(534\) 0 0
\(535\) −803169. −0.121317
\(536\) 0 0
\(537\) −1.29365e7 −1.93590
\(538\) 0 0
\(539\) −1.23879e6 −0.183665
\(540\) 0 0
\(541\) 1.16842e7 1.71635 0.858176 0.513355i \(-0.171598\pi\)
0.858176 + 0.513355i \(0.171598\pi\)
\(542\) 0 0
\(543\) −8.12859e6 −1.18308
\(544\) 0 0
\(545\) 1.56322e6 0.225439
\(546\) 0 0
\(547\) 4.05598e6 0.579600 0.289800 0.957087i \(-0.406411\pi\)
0.289800 + 0.957087i \(0.406411\pi\)
\(548\) 0 0
\(549\) 4.72435e6 0.668977
\(550\) 0 0
\(551\) −1.03341e6 −0.145008
\(552\) 0 0
\(553\) −53526.2 −0.00744309
\(554\) 0 0
\(555\) 798846. 0.110086
\(556\) 0 0
\(557\) −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
\(560\) 0 0
\(561\) −2.36167e6 −0.316820
\(562\) 0 0
\(563\) −1.20582e6 −0.160329 −0.0801646 0.996782i \(-0.525545\pi\)
−0.0801646 + 0.996782i \(0.525545\pi\)
\(564\) 0 0
\(565\) −590527. −0.0778249
\(566\) 0 0
\(567\) 1.15108e7 1.50365
\(568\) 0 0
\(569\) 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
\(573\) 4.24222e6 0.539768
\(574\) 0 0
\(575\) −4.10637e6 −0.517951
\(576\) 0 0
\(577\) 3.61005e6 0.451413 0.225706 0.974195i \(-0.427531\pi\)
0.225706 + 0.974195i \(0.427531\pi\)
\(578\) 0 0
\(579\) 7.22190e6 0.895272
\(580\) 0 0
\(581\) −1.52791e7 −1.87783
\(582\) 0 0
\(583\) −2.85138e6 −0.347443
\(584\) 0 0
\(585\) −927748. −0.112083
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) −1.78367e7 −2.10061
\(592\) 0 0
\(593\) 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
\(596\) 0 0
\(597\) −1.25973e7 −1.44658
\(598\) 0 0
\(599\) −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 \(-0.655762\pi\)
−0.470044 + 0.882643i \(0.655762\pi\)
\(602\) 0 0
\(603\) −1.01511e7 −1.13689
\(604\) 0 0
\(605\) 126633. 0.0140656
\(606\) 0 0
\(607\) −1.47290e7 −1.62256 −0.811279 0.584659i \(-0.801228\pi\)
−0.811279 + 0.584659i \(0.801228\pi\)
\(608\) 0 0
\(609\) 3.05446e6 0.333727
\(610\) 0 0
\(611\) 1.09520e6 0.118683
\(612\) 0 0
\(613\) −1.14564e7 −1.23139 −0.615697 0.787983i \(-0.711126\pi\)
−0.615697 + 0.787983i \(0.711126\pi\)
\(614\) 0 0
\(615\) −2.86057e6 −0.304975
\(616\) 0 0
\(617\) −413797. −0.0437597 −0.0218799 0.999761i \(-0.506965\pi\)
−0.0218799 + 0.999761i \(0.506965\pi\)
\(618\) 0 0
\(619\) 1.25898e7 1.32067 0.660333 0.750973i \(-0.270415\pi\)
0.660333 + 0.750973i \(0.270415\pi\)
\(620\) 0 0
\(621\) 1.65987e6 0.172721
\(622\) 0 0
\(623\) −3.79198e6 −0.391422
\(624\) 0 0
\(625\) 9.06989e6 0.928757
\(626\) 0 0
\(627\) −2.86988e6 −0.291538
\(628\) 0 0
\(629\) 4.22886e6 0.426183
\(630\) 0 0
\(631\) 1.55648e7 1.55621 0.778107 0.628132i \(-0.216180\pi\)
0.778107 + 0.628132i \(0.216180\pi\)
\(632\) 0 0
\(633\) 3.43976e6 0.341208
\(634\) 0 0
\(635\) −235544. −0.0231813
\(636\) 0 0
\(637\) −5.99163e6 −0.585054
\(638\) 0 0
\(639\) −1.04285e7 −1.01035
\(640\) 0 0
\(641\) −1.23075e7 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(642\) 0 0
\(643\) 1.44400e6 0.137733 0.0688667 0.997626i \(-0.478062\pi\)
0.0688667 + 0.997626i \(0.478062\pi\)
\(644\) 0 0
\(645\) −3.55465e6 −0.336433
\(646\) 0 0
\(647\) 7.35610e6 0.690855 0.345427 0.938445i \(-0.387734\pi\)
0.345427 + 0.938445i \(0.387734\pi\)
\(648\) 0 0
\(649\) −4.19989e6 −0.391405
\(650\) 0 0
\(651\) 1.32659e6 0.122683
\(652\) 0 0
\(653\) 3.83734e6 0.352166 0.176083 0.984375i \(-0.443657\pi\)
0.176083 + 0.984375i \(0.443657\pi\)
\(654\) 0 0
\(655\) 102814. 0.00936373
\(656\) 0 0
\(657\) −8.59083e6 −0.776465
\(658\) 0 0
\(659\) 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 \(0.214116\pi\)
\(662\) 0 0
\(663\) −1.14226e7 −1.00921
\(664\) 0 0
\(665\) 1.63398e6 0.143283
\(666\) 0 0
\(667\) 1.21108e6 0.105404
\(668\) 0 0
\(669\) −2.17977e7 −1.88298
\(670\) 0 0
\(671\) −3.11892e6 −0.267423
\(672\) 0 0
\(673\) 4.88569e6 0.415804 0.207902 0.978150i \(-0.433337\pi\)
0.207902 + 0.978150i \(0.433337\pi\)
\(674\) 0 0
\(675\) −3.76072e6 −0.317696
\(676\) 0 0
\(677\) −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
\(680\) 0 0
\(681\) 1.08717e7 0.898320
\(682\) 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
\(686\) 0 0
\(687\) −2.30611e7 −1.86418
\(688\) 0 0
\(689\) −1.37912e7 −1.10676
\(690\) 0 0
\(691\) 5.39805e6 0.430073 0.215036 0.976606i \(-0.431013\pi\)
0.215036 + 0.976606i \(0.431013\pi\)
\(692\) 0 0
\(693\) 3.64714e6 0.288482
\(694\) 0 0
\(695\) −227024. −0.0178283
\(696\) 0 0
\(697\) −1.51430e7 −1.18067
\(698\) 0 0
\(699\) 604162. 0.0467692
\(700\) 0 0
\(701\) 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
\(705\) 334185. 0.0253230
\(706\) 0 0
\(707\) −6.23750e6 −0.469312
\(708\) 0 0
\(709\) −1.30530e7 −0.975200 −0.487600 0.873067i \(-0.662128\pi\)
−0.487600 + 0.873067i \(0.662128\pi\)
\(710\) 0 0
\(711\) 59654.9 0.00442560
\(712\) 0 0
\(713\) 525987. 0.0387482
\(714\) 0 0
\(715\) 612480. 0.0448051
\(716\) 0 0
\(717\) 1.69814e7 1.23360
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) −1.57418e7 −1.11998
\(724\) 0 0
\(725\) −2.74391e6 −0.193876
\(726\) 0 0
\(727\) 1.07681e7 0.755619 0.377809 0.925883i \(-0.376677\pi\)
0.377809 + 0.925883i \(0.376677\pi\)
\(728\) 0 0
\(729\) −6.64290e6 −0.462955
\(730\) 0 0
\(731\) −1.88173e7 −1.30246
\(732\) 0 0
\(733\) 7.35742e6 0.505785 0.252892 0.967494i \(-0.418618\pi\)
0.252892 + 0.967494i \(0.418618\pi\)
\(734\) 0 0
\(735\) −1.82826e6 −0.124830
\(736\) 0 0
\(737\) 6.70154e6 0.454471
\(738\) 0 0
\(739\) 6.81140e6 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(740\) 0 0
\(741\) −1.38806e7 −0.928676
\(742\) 0 0
\(743\) 1.01182e7 0.672405 0.336203 0.941790i \(-0.390857\pi\)
0.336203 + 0.941790i \(0.390857\pi\)
\(744\) 0 0
\(745\) −1.95749e6 −0.129214
\(746\) 0 0
\(747\) 1.70286e7 1.11655
\(748\) 0 0
\(749\) 1.52712e7 0.994649
\(750\) 0 0
\(751\) 1.91140e7 1.23667 0.618333 0.785916i \(-0.287808\pi\)
0.618333 + 0.785916i \(0.287808\pi\)
\(752\) 0 0
\(753\) 1.15903e7 0.744914
\(754\) 0 0
\(755\) 2.60398e6 0.166254
\(756\) 0 0
\(757\) 1.48895e7 0.944366 0.472183 0.881501i \(-0.343466\pi\)
0.472183 + 0.881501i \(0.343466\pi\)
\(758\) 0 0
\(759\) 3.36330e6 0.211915
\(760\) 0 0
\(761\) −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
\(764\) 0 0
\(765\) −1.49859e6 −0.0925827
\(766\) 0 0
\(767\) −2.03134e7 −1.24680
\(768\) 0 0
\(769\) 1.45027e7 0.884369 0.442184 0.896924i \(-0.354204\pi\)
0.442184 + 0.896924i \(0.354204\pi\)
\(770\) 0 0
\(771\) 1.57939e7 0.956874
\(772\) 0 0
\(773\) 3.18546e7 1.91745 0.958723 0.284342i \(-0.0917750\pi\)
0.958723 + 0.284342i \(0.0917750\pi\)
\(774\) 0 0
\(775\) −1.19171e6 −0.0712716
\(776\) 0 0
\(777\) −1.51891e7 −0.902565
\(778\) 0 0
\(779\) −1.84016e7 −1.08646
\(780\) 0 0
\(781\) 6.88470e6 0.403885
\(782\) 0 0
\(783\) 1.10914e6 0.0646519
\(784\) 0 0
\(785\) 2.95355e6 0.171068
\(786\) 0 0
\(787\) 1.68239e7 0.968253 0.484126 0.874998i \(-0.339138\pi\)
0.484126 + 0.874998i \(0.339138\pi\)
\(788\) 0 0
\(789\) −1.57663e7 −0.901650
\(790\) 0 0
\(791\) 1.12281e7 0.638067
\(792\) 0 0
\(793\) −1.50852e7 −0.851858
\(794\) 0 0
\(795\) −4.20818e6 −0.236144
\(796\) 0 0
\(797\) −2.00376e7 −1.11738 −0.558690 0.829377i \(-0.688696\pi\)
−0.558690 + 0.829377i \(0.688696\pi\)
\(798\) 0 0
\(799\) 1.76908e6 0.0980347
\(800\) 0 0
\(801\) 4.22616e6 0.232736
\(802\) 0 0
\(803\) 5.67149e6 0.310391
\(804\) 0 0
\(805\) −1.91492e6 −0.104150
\(806\) 0 0
\(807\) −1.65356e7 −0.893790
\(808\) 0 0
\(809\) −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 \(0.619457\pi\)
\(812\) 0 0
\(813\) 2.02625e6 0.107515
\(814\) 0 0
\(815\) −5.22941e6 −0.275777
\(816\) 0 0
\(817\) −2.28666e7 −1.19852
\(818\) 0 0
\(819\) 1.76400e7 0.918942
\(820\) 0 0
\(821\) 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 \(-0.596921\pi\)
−0.299802 + 0.954002i \(0.596921\pi\)
\(824\) 0 0
\(825\) −7.62012e6 −0.389787
\(826\) 0 0
\(827\) −5.48883e6 −0.279072 −0.139536 0.990217i \(-0.544561\pi\)
−0.139536 + 0.990217i \(0.544561\pi\)
\(828\) 0 0
\(829\) 1.81680e7 0.918164 0.459082 0.888394i \(-0.348178\pi\)
0.459082 + 0.888394i \(0.348178\pi\)
\(830\) 0 0
\(831\) 1.28214e7 0.644070
\(832\) 0 0
\(833\) −9.67828e6 −0.483265
\(834\) 0 0
\(835\) 1.38234e6 0.0686120
\(836\) 0 0
\(837\) 481712. 0.0237670
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) −2.70866e7 −1.31276
\(844\) 0 0
\(845\) −249024. −0.0119978
\(846\) 0 0
\(847\) −2.40776e6 −0.115320
\(848\) 0 0
\(849\) −555348. −0.0264421
\(850\) 0 0
\(851\) −6.02240e6 −0.285066
\(852\) 0 0
\(853\) 3.66987e7 1.72694 0.863471 0.504398i \(-0.168285\pi\)
0.863471 + 0.504398i \(0.168285\pi\)
\(854\) 0 0
\(855\) −1.82108e6 −0.0851948
\(856\) 0 0
\(857\) 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 \(-0.445222\pi\)
0.171241 + 0.985229i \(0.445222\pi\)
\(860\) 0 0
\(861\) 5.43901e7 2.50042
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 1.08643e7 0.490857
\(868\) 0 0
\(869\) −39383.0 −0.00176913
\(870\) 0 0
\(871\) 3.24131e7 1.44769
\(872\) 0 0
\(873\) −918880. −0.0408059
\(874\) 0 0
\(875\) 8.78353e6 0.387837
\(876\) 0 0
\(877\) −3.57898e7 −1.57130 −0.785652 0.618669i \(-0.787672\pi\)
−0.785652 + 0.618669i \(0.787672\pi\)
\(878\) 0 0
\(879\) −1.31838e7 −0.575532
\(880\) 0 0
\(881\) 7.14038e6 0.309943 0.154971 0.987919i \(-0.450471\pi\)
0.154971 + 0.987919i \(0.450471\pi\)
\(882\) 0 0
\(883\) 2.77023e7 1.19568 0.597838 0.801617i \(-0.296027\pi\)
0.597838 + 0.801617i \(0.296027\pi\)
\(884\) 0 0
\(885\) −6.19837e6 −0.266023
\(886\) 0 0
\(887\) −387870. −0.0165530 −0.00827651 0.999966i \(-0.502635\pi\)
−0.00827651 + 0.999966i \(0.502635\pi\)
\(888\) 0 0
\(889\) 4.47857e6 0.190058
\(890\) 0 0
\(891\) 8.46928e6 0.357398
\(892\) 0 0
\(893\) 2.14977e6 0.0902117
\(894\) 0 0
\(895\) 5.41932e6 0.226145
\(896\) 0 0
\(897\) 1.62672e7 0.675042
\(898\) 0 0
\(899\) 351468. 0.0145040
\(900\) 0 0
\(901\) −2.22769e7 −0.914203
\(902\) 0 0
\(903\) 6.75873e7 2.75833
\(904\) 0 0
\(905\) 3.40519e6 0.138204
\(906\) 0 0
\(907\) −5.05051e6 −0.203853 −0.101926 0.994792i \(-0.532501\pi\)
−0.101926 + 0.994792i \(0.532501\pi\)
\(908\) 0 0
\(909\) 6.95169e6 0.279049
\(910\) 0 0
\(911\) 1.72283e7 0.687774 0.343887 0.939011i \(-0.388256\pi\)
0.343887 + 0.939011i \(0.388256\pi\)
\(912\) 0 0
\(913\) −1.12419e7 −0.446337
\(914\) 0 0
\(915\) −4.60303e6 −0.181757
\(916\) 0 0
\(917\) −1.95488e6 −0.0767709
\(918\) 0 0
\(919\) −3.89056e7 −1.51958 −0.759789 0.650170i \(-0.774698\pi\)
−0.759789 + 0.650170i \(0.774698\pi\)
\(920\) 0 0
\(921\) 1.13645e7 0.441470
\(922\) 0 0
\(923\) 3.32990e7 1.28655
\(924\) 0 0
\(925\) 1.36447e7 0.524338
\(926\) 0 0
\(927\) −3.30453e7 −1.26302
\(928\) 0 0
\(929\) 3.18602e7 1.21118 0.605590 0.795777i \(-0.292937\pi\)
0.605590 + 0.795777i \(0.292937\pi\)
\(930\) 0 0
\(931\) −1.17610e7 −0.444701
\(932\) 0 0
\(933\) 3.85627e6 0.145032
\(934\) 0 0
\(935\) 989340. 0.0370098
\(936\) 0 0
\(937\) −4.09748e7 −1.52464 −0.762322 0.647198i \(-0.775941\pi\)
−0.762322 + 0.647198i \(0.775941\pi\)
\(938\) 0 0
\(939\) 1.92889e7 0.713909
\(940\) 0 0
\(941\) −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
\(944\) 0 0
\(945\) −1.75373e6 −0.0638827
\(946\) 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
\(950\) 0 0
\(951\) 3.88735e7 1.39381
\(952\) 0 0
\(953\) −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
\(956\) 0 0
\(957\) 2.24738e6 0.0793226
\(958\) 0 0
\(959\) −5.80557e6 −0.203844
\(960\) 0 0
\(961\) −2.84765e7 −0.994668
\(962\) 0 0
\(963\) −1.70198e7 −0.591410
\(964\) 0 0
\(965\) −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
\(969\) −2.24214e7 −0.767103
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) −3.68559e7 −1.24164
\(976\) 0 0
\(977\) −1.52214e7 −0.510173 −0.255087 0.966918i \(-0.582104\pi\)
−0.255087 + 0.966918i \(0.582104\pi\)
\(978\) 0 0
\(979\) −2.79002e6 −0.0930360
\(980\) 0 0
\(981\) 3.31260e7 1.09900
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) −6.35411e6 −0.207616
\(988\) 0 0
\(989\) 2.67981e7 0.871190
\(990\) 0 0
\(991\) −3.46744e7 −1.12156 −0.560782 0.827963i \(-0.689500\pi\)
−0.560782 + 0.827963i \(0.689500\pi\)
\(992\) 0 0
\(993\) 4.06853e6 0.130938
\(994\) 0 0
\(995\) 5.27722e6 0.168985
\(996\) 0 0
\(997\) 1.05268e7 0.335397 0.167699 0.985838i \(-0.446366\pi\)
0.167699 + 0.985838i \(0.446366\pi\)
\(998\) 0 0
\(999\) −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 176.6.a.i.1.1 3
4.3 odd 2 11.6.a.b.1.1 3
8.3 odd 2 704.6.a.q.1.1 3
8.5 even 2 704.6.a.t.1.3 3
12.11 even 2 99.6.a.g.1.3 3
20.3 even 4 275.6.b.b.199.6 6
20.7 even 4 275.6.b.b.199.1 6
20.19 odd 2 275.6.a.b.1.3 3
28.27 even 2 539.6.a.e.1.1 3
44.43 even 2 121.6.a.d.1.3 3
132.131 odd 2 1089.6.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.1 3 4.3 odd 2
99.6.a.g.1.3 3 12.11 even 2
121.6.a.d.1.3 3 44.43 even 2
176.6.a.i.1.1 3 1.1 even 1 trivial
275.6.a.b.1.3 3 20.19 odd 2
275.6.b.b.199.1 6 20.7 even 4
275.6.b.b.199.6 6 20.3 even 4
539.6.a.e.1.1 3 28.27 even 2
704.6.a.q.1.1 3 8.3 odd 2
704.6.a.t.1.3 3 8.5 even 2
1089.6.a.r.1.1 3 132.131 odd 2