Properties

Label 275.6.a.b.1.3
Level $275$
Weight $6$
Character 275.1
Self dual yes
Analytic conductor $44.106$
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] = [275,6,Mod(1,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, 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("275.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(44.1055504486\)
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 \(\chi\) \(=\) 275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.3963 q^{2} -20.6466 q^{3} +76.0833 q^{4} -214.649 q^{6} -164.454 q^{7} +458.304 q^{8} +183.283 q^{9} +O(q^{10})\) \(q+10.3963 q^{2} -20.6466 q^{3} +76.0833 q^{4} -214.649 q^{6} -164.454 q^{7} +458.304 q^{8} +183.283 q^{9} +121.000 q^{11} -1570.86 q^{12} +585.236 q^{13} -1709.71 q^{14} +2330.01 q^{16} +945.333 q^{17} +1905.47 q^{18} +1148.76 q^{19} +3395.41 q^{21} +1257.95 q^{22} +1346.27 q^{23} -9462.44 q^{24} +6084.30 q^{26} +1232.95 q^{27} -12512.2 q^{28} +899.585 q^{29} -390.700 q^{31} +9557.75 q^{32} -2498.24 q^{33} +9827.97 q^{34} +13944.8 q^{36} +4473.41 q^{37} +11942.9 q^{38} -12083.2 q^{39} +16018.7 q^{41} +35299.8 q^{42} +19905.5 q^{43} +9206.08 q^{44} +13996.2 q^{46} -1871.38 q^{47} -48106.8 q^{48} +10238.0 q^{49} -19517.9 q^{51} +44526.7 q^{52} -23565.1 q^{53} +12818.1 q^{54} -75369.7 q^{56} -23718.0 q^{57} +9352.37 q^{58} -34709.8 q^{59} +25776.2 q^{61} -4061.84 q^{62} -30141.6 q^{63} +24805.1 q^{64} -25972.5 q^{66} -55384.6 q^{67} +71924.1 q^{68} -27795.9 q^{69} +56898.4 q^{71} +83999.6 q^{72} +46871.8 q^{73} +46506.9 q^{74} +87401.5 q^{76} -19898.9 q^{77} -125620. q^{78} -325.479 q^{79} -69994.1 q^{81} +166536. q^{82} +92908.3 q^{83} +258334. q^{84} +206943. q^{86} -18573.4 q^{87} +55454.8 q^{88} +23058.0 q^{89} -96244.2 q^{91} +102428. q^{92} +8066.65 q^{93} -19455.5 q^{94} -197335. q^{96} +5013.44 q^{97} +106437. q^{98} +22177.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 363 q^{11} - 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} + 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 11748 q^{24} + 12132 q^{26} + 2990 q^{27} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} + 12408 q^{32} - 4114 q^{33} + 25320 q^{34} + 4756 q^{36} - 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 5994 q^{41} + 47828 q^{42} + 26208 q^{43} + 10164 q^{44} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 48531 q^{49} - 22724 q^{51} + 35304 q^{52} - 50586 q^{53} + 18814 q^{54} - 42312 q^{56} - 20160 q^{57} + 29172 q^{58} - 3738 q^{59} + 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} - 24926 q^{66} + 47754 q^{67} + 12600 q^{68} + 35042 q^{69} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 103920 q^{76} - 10164 q^{77} - 124984 q^{78} + 125148 q^{79} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} + 343616 q^{84} + 243060 q^{86} + 19368 q^{87} + 68244 q^{88} - 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 16622 q^{93} - 74928 q^{94} - 76456 q^{96} - 9684 q^{97} - 3480 q^{98} - 847 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.730395\pi\)
−0.662241 + 0.749291i \(0.730395\pi\)
\(4\) 76.0833 2.37760
\(5\) 0 0
\(6\) −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
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −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
\(15\) 0 0
\(16\) 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 \(-0.381063\pi\)
0.365019 + 0.931000i \(0.381063\pi\)
\(20\) 0 0
\(21\) 3395.41 1.68014
\(22\) 1257.95 0.554125
\(23\) 1346.27 0.530654 0.265327 0.964158i \(-0.414520\pi\)
0.265327 + 0.964158i \(0.414520\pi\)
\(24\) −9462.44 −3.35332
\(25\) 0 0
\(26\) 6084.30 1.76513
\(27\) 1232.95 0.325488
\(28\) −12512.2 −3.01604
\(29\) 899.585 0.198631 0.0993155 0.995056i \(-0.468335\pi\)
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
\(33\) −2498.24 −0.399346
\(34\) 9827.97 1.45803
\(35\) 0 0
\(36\) 13944.8 1.79332
\(37\) 4473.41 0.537198 0.268599 0.963252i \(-0.413439\pi\)
0.268599 + 0.963252i \(0.413439\pi\)
\(38\) 11942.9 1.34168
\(39\) −12083.2 −1.27209
\(40\) 0 0
\(41\) 16018.7 1.48822 0.744111 0.668056i \(-0.232873\pi\)
0.744111 + 0.668056i \(0.232873\pi\)
\(42\) 35299.8 3.08780
\(43\) 19905.5 1.64173 0.820864 0.571124i \(-0.193492\pi\)
0.820864 + 0.571124i \(0.193492\pi\)
\(44\) 9206.08 0.716875
\(45\) 0 0
\(46\) 13996.2 0.975250
\(47\) −1871.38 −0.123571 −0.0617856 0.998089i \(-0.519680\pi\)
−0.0617856 + 0.998089i \(0.519680\pi\)
\(48\) −48106.8 −3.01372
\(49\) 10238.0 0.609149
\(50\) 0 0
\(51\) −19517.9 −1.05077
\(52\) 44526.7 2.28356
\(53\) −23565.1 −1.15234 −0.576169 0.817330i \(-0.695453\pi\)
−0.576169 + 0.817330i \(0.695453\pi\)
\(54\) 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 \(0.724842\pi\)
\(60\) 0 0
\(61\) 25776.2 0.886940 0.443470 0.896289i \(-0.353747\pi\)
0.443470 + 0.896289i \(0.353747\pi\)
\(62\) −4061.84 −0.134197
\(63\) −30141.6 −0.956787
\(64\) 24805.1 0.756993
\(65\) 0 0
\(66\) −25972.5 −0.733929
\(67\) −55384.6 −1.50731 −0.753655 0.657271i \(-0.771711\pi\)
−0.753655 + 0.657271i \(0.771711\pi\)
\(68\) 71924.1 1.88626
\(69\) −27795.9 −0.702842
\(70\) 0 0
\(71\) 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
\(75\) 0 0
\(76\) 87401.5 1.73574
\(77\) −19898.9 −0.382474
\(78\) −125620. −2.33789
\(79\) −325.479 −0.00586753 −0.00293377 0.999996i \(-0.500934\pi\)
−0.00293377 + 0.999996i \(0.500934\pi\)
\(80\) 0 0
\(81\) −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
\(85\) 0 0
\(86\) 206943. 3.01721
\(87\) −18573.4 −0.263083
\(88\) 55454.8 0.763365
\(89\) 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
\(92\) 102428. 1.26169
\(93\) 8066.65 0.0967132
\(94\) −19455.5 −0.227103
\(95\) 0 0
\(96\) −197335. −2.18538
\(97\) 5013.44 0.0541011 0.0270506 0.999634i \(-0.491388\pi\)
0.0270506 + 0.999634i \(0.491388\pi\)
\(98\) 106437. 1.11951
\(99\) 22177.3 0.227416
\(100\) 0 0
\(101\) 37928.6 0.369968 0.184984 0.982742i \(-0.440777\pi\)
0.184984 + 0.982742i \(0.440777\pi\)
\(102\) −202915. −1.93114
\(103\) −180296. −1.67453 −0.837265 0.546798i \(-0.815847\pi\)
−0.837265 + 0.546798i \(0.815847\pi\)
\(104\) 268216. 2.43165
\(105\) 0 0
\(106\) −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 \(-0.240203\pi\)
0.728533 + 0.685011i \(0.240203\pi\)
\(110\) 0 0
\(111\) −92360.8 −0.711509
\(112\) −383178. −2.88639
\(113\) 68275.4 0.503000 0.251500 0.967857i \(-0.419076\pi\)
0.251500 + 0.967857i \(0.419076\pi\)
\(114\) −246580. −1.77703
\(115\) 0 0
\(116\) 68443.4 0.472266
\(117\) 107264. 0.724419
\(118\) −360854. −2.38576
\(119\) −155463. −1.00638
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 267977. 1.63004
\(123\) −330732. −1.97112
\(124\) −29725.8 −0.173612
\(125\) 0 0
\(126\) −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
\(130\) 0 0
\(131\) −11887.1 −0.0605199 −0.0302600 0.999542i \(-0.509634\pi\)
−0.0302600 + 0.999542i \(0.509634\pi\)
\(132\) −190075. −0.949488
\(133\) −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 \(0.525603\pi\)
\(138\) −288975. −1.29170
\(139\) 26248.0 0.115228 0.0576141 0.998339i \(-0.481651\pi\)
0.0576141 + 0.998339i \(0.481651\pi\)
\(140\) 0 0
\(141\) 38637.7 0.163668
\(142\) 591533. 2.46183
\(143\) 70813.6 0.289585
\(144\) 427052. 1.71623
\(145\) 0 0
\(146\) 487294. 1.89195
\(147\) −211379. −0.806807
\(148\) 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 \(-0.680543\pi\)
−0.537267 + 0.843412i \(0.680543\pi\)
\(152\) 526481. 1.84831
\(153\) 173264. 0.598383
\(154\) −206875. −0.702920
\(155\) 0 0
\(156\) −919327. −3.02453
\(157\) −341482. −1.10565 −0.552827 0.833296i \(-0.686451\pi\)
−0.552827 + 0.833296i \(0.686451\pi\)
\(158\) −3383.78 −0.0107835
\(159\) 486541. 1.52625
\(160\) 0 0
\(161\) −221398. −0.673147
\(162\) −727680. −2.17848
\(163\) −604612. −1.78241 −0.891205 0.453600i \(-0.850139\pi\)
−0.891205 + 0.453600i \(0.850139\pi\)
\(164\) 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
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) 281931. 0.686058
\(177\) 716641. 1.71937
\(178\) 239719. 0.567090
\(179\) −626569. −1.46163 −0.730813 0.682578i \(-0.760859\pi\)
−0.730813 + 0.682578i \(0.760859\pi\)
\(180\) 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
\(183\) −532192. −1.17474
\(184\) 617000. 1.34351
\(185\) 0 0
\(186\) 83863.4 0.177742
\(187\) 114385. 0.239203
\(188\) −142381. −0.293804
\(189\) −202762. −0.412888
\(190\) 0 0
\(191\) 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
\(195\) 0 0
\(196\) 778938. 1.44831
\(197\) −863902. −1.58598 −0.792992 0.609232i \(-0.791478\pi\)
−0.792992 + 0.609232i \(0.791478\pi\)
\(198\) 230562. 0.417951
\(199\) −610140. −1.09219 −0.546093 0.837725i \(-0.683885\pi\)
−0.546093 + 0.837725i \(0.683885\pi\)
\(200\) 0 0
\(201\) 1.14351e6 1.99640
\(202\) 394318. 0.679936
\(203\) −147940. −0.251968
\(204\) −1.48499e6 −2.49832
\(205\) 0 0
\(206\) −1.87441e6 −3.07749
\(207\) 246749. 0.400248
\(208\) 1.36360e6 2.18540
\(209\) 139000. 0.220115
\(210\) 0 0
\(211\) 166602. 0.257616 0.128808 0.991670i \(-0.458885\pi\)
0.128808 + 0.991670i \(0.458885\pi\)
\(212\) −1.79291e6 −2.73981
\(213\) −1.17476e6 −1.77419
\(214\) −965407. −1.44104
\(215\) 0 0
\(216\) 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 \(-0.248317\pi\)
0.710836 + 0.703358i \(0.248317\pi\)
\(224\) −1.57181e6 −2.09305
\(225\) 0 0
\(226\) 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 \(0.251512\pi\)
\(230\) 0 0
\(231\) 410845. 0.506580
\(232\) 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
\(235\) 0 0
\(236\) −2.64084e6 −3.08647
\(237\) 6720.05 0.00777144
\(238\) −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 \(-0.361048\pi\)
0.422797 + 0.906224i \(0.361048\pi\)
\(242\) 152212. 0.167075
\(243\) 1.14554e6 1.24449
\(244\) 1.96114e6 2.10879
\(245\) 0 0
\(246\) −3.43840e6 −3.62258
\(247\) 672296. 0.701161
\(248\) −179060. −0.184871
\(249\) −1.91824e6 −1.96067
\(250\) 0 0
\(251\) 561364. 0.562419 0.281209 0.959646i \(-0.409264\pi\)
0.281209 + 0.959646i \(0.409264\pi\)
\(252\) −2.29327e6 −2.27486
\(253\) 162898. 0.159998
\(254\) −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 \(0.382358\pi\)
\(258\) −4.27269e6 −3.99624
\(259\) −735668. −0.681447
\(260\) 0 0
\(261\) 164879. 0.149818
\(262\) −123582. −0.111225
\(263\) 763627. 0.680756 0.340378 0.940289i \(-0.389445\pi\)
0.340378 + 0.940289i \(0.389445\pi\)
\(264\) −1.14495e6 −1.01106
\(265\) 0 0
\(266\) −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 \(0.390449\pi\)
\(270\) 0 0
\(271\) 98139.7 0.0811749 0.0405874 0.999176i \(-0.487077\pi\)
0.0405874 + 0.999176i \(0.487077\pi\)
\(272\) 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
\(280\) 0 0
\(281\) 1.31191e6 0.991149 0.495575 0.868565i \(-0.334957\pi\)
0.495575 + 0.868565i \(0.334957\pi\)
\(282\) 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
\(285\) 0 0
\(286\) 736200. 0.532207
\(287\) −2.63433e6 −1.88784
\(288\) 1.75178e6 1.24451
\(289\) −526203. −0.370603
\(290\) 0 0
\(291\) −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
\(295\) 0 0
\(296\) 2.05018e6 1.36008
\(297\) 149186. 0.0981382
\(298\) −2.35290e6 −1.53484
\(299\) 787884. 0.509665
\(300\) 0 0
\(301\) −3.27352e6 −2.08257
\(302\) −3.12998e6 −1.97481
\(303\) −783099. −0.490016
\(304\) 2.67662e6 1.66113
\(305\) 0 0
\(306\) 1.80131e6 1.09972
\(307\) −550428. −0.333315 −0.166658 0.986015i \(-0.553297\pi\)
−0.166658 + 0.986015i \(0.553297\pi\)
\(308\) −1.51397e6 −0.909371
\(309\) 3.72250e6 2.21788
\(310\) 0 0
\(311\) 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 \(-0.413140\pi\)
0.269505 + 0.962999i \(0.413140\pi\)
\(314\) −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 \(0.323628\pi\)
\(318\) 5.05823e6 2.80499
\(319\) 108850. 0.0598895
\(320\) 0 0
\(321\) 1.91726e6 1.03853
\(322\) −2.30173e6 −1.23713
\(323\) 1.08596e6 0.579172
\(324\) −5.32538e6 −2.81831
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 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
\(340\) 0 0
\(341\) −47274.7 −0.0220162
\(342\) 2.18893e6 1.01197
\(343\) 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 \(-0.271762\pi\)
0.657148 + 0.753761i \(0.271762\pi\)
\(348\) −1.41313e6 −0.625508
\(349\) −924908. −0.406476 −0.203238 0.979129i \(-0.565146\pi\)
−0.203238 + 0.979129i \(0.565146\pi\)
\(350\) 0 0
\(351\) 721564. 0.312613
\(352\) 1.15649e6 0.497490
\(353\) 4.46816e6 1.90850 0.954249 0.299012i \(-0.0966570\pi\)
0.954249 + 0.299012i \(0.0966570\pi\)
\(354\) 7.45043e6 3.15990
\(355\) 0 0
\(356\) 1.75433e6 0.733647
\(357\) 3.20979e6 1.33293
\(358\) −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
\(363\) −302287. −0.120407
\(364\) −7.32258e6 −2.89675
\(365\) 0 0
\(366\) −5.53283e6 −2.15896
\(367\) 1.21088e6 0.469284 0.234642 0.972082i \(-0.424608\pi\)
0.234642 + 0.972082i \(0.424608\pi\)
\(368\) 3.13681e6 1.20745
\(369\) 2.93596e6 1.12250
\(370\) 0 0
\(371\) 3.87537e6 1.46177
\(372\) 613737. 0.229946
\(373\) −1.82235e6 −0.678203 −0.339102 0.940750i \(-0.610123\pi\)
−0.339102 + 0.940750i \(0.610123\pi\)
\(374\) 1.18918e6 0.439613
\(375\) 0 0
\(376\) −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 \(0.523890\pi\)
\(380\) 0 0
\(381\) 562271. 0.198442
\(382\) 2.13611e6 0.748972
\(383\) −2.95656e6 −1.02989 −0.514943 0.857224i \(-0.672187\pi\)
−0.514943 + 0.857224i \(0.672187\pi\)
\(384\) 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 \(0.370947\pi\)
\(390\) 0 0
\(391\) 1.27267e6 0.420992
\(392\) 4.69210e6 1.54224
\(393\) 245429. 0.0801576
\(394\) −8.98139e6 −2.91476
\(395\) 0 0
\(396\) 1.68732e6 0.540705
\(397\) −3.94809e6 −1.25722 −0.628609 0.777722i \(-0.716375\pi\)
−0.628609 + 0.777722i \(0.716375\pi\)
\(398\) −6.34320e6 −2.00725
\(399\) 3.90051e6 1.22656
\(400\) 0 0
\(401\) −5.76535e6 −1.79046 −0.895230 0.445604i \(-0.852989\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(402\) 1.18883e7 3.66904
\(403\) −228652. −0.0701314
\(404\) 2.88574e6 0.879637
\(405\) 0 0
\(406\) −1.53803e6 −0.463073
\(407\) 541282. 0.161971
\(408\) −8.94515e6 −2.66034
\(409\) 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
\(412\) −1.37175e7 −3.98137
\(413\) 5.70815e6 1.64672
\(414\) 2.56527e6 0.735585
\(415\) 0 0
\(416\) 5.59354e6 1.58472
\(417\) −541932. −0.152618
\(418\) 1.44509e6 0.404532
\(419\) −1.41668e6 −0.394220 −0.197110 0.980381i \(-0.563156\pi\)
−0.197110 + 0.980381i \(0.563156\pi\)
\(420\) 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
\(423\) −342993. −0.0932040
\(424\) −1.08000e7 −2.91749
\(425\) 0 0
\(426\) −1.22132e7 −3.26065
\(427\) −4.23899e6 −1.12510
\(428\) −7.06514e6 −1.86428
\(429\) −1.46206e6 −0.383551
\(430\) 0 0
\(431\) −2.73465e6 −0.709103 −0.354551 0.935037i \(-0.615366\pi\)
−0.354551 + 0.935037i \(0.615366\pi\)
\(432\) 2.87277e6 0.740613
\(433\) −2.71922e6 −0.696986 −0.348493 0.937311i \(-0.613307\pi\)
−0.348493 + 0.937311i \(0.613307\pi\)
\(434\) 667984. 0.170232
\(435\) 0 0
\(436\) 1.37510e7 3.46433
\(437\) 1.54654e6 0.387397
\(438\) −1.00610e7 −2.50585
\(439\) −4.17101e6 −1.03295 −0.516476 0.856301i \(-0.672757\pi\)
−0.516476 + 0.856301i \(0.672757\pi\)
\(440\) 0 0
\(441\) 1.87645e6 0.459452
\(442\) 5.75169e6 1.40036
\(443\) −6.86870e6 −1.66290 −0.831448 0.555603i \(-0.812487\pi\)
−0.831448 + 0.555603i \(0.812487\pi\)
\(444\) −7.02712e6 −1.69169
\(445\) 0 0
\(446\) 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 \(0.525878\pi\)
\(450\) 0 0
\(451\) 1.93826e6 0.448716
\(452\) 5.19462e6 1.19594
\(453\) 6.21601e6 1.42320
\(454\) −5.47431e6 −1.24649
\(455\) 0 0
\(456\) −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
\(460\) 0 0
\(461\) −4.48975e6 −0.983944 −0.491972 0.870611i \(-0.663724\pi\)
−0.491972 + 0.870611i \(0.663724\pi\)
\(462\) 4.27127e6 0.931006
\(463\) 9.04494e6 1.96089 0.980445 0.196793i \(-0.0630528\pi\)
0.980445 + 0.196793i \(0.0630528\pi\)
\(464\) 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 \(0.775273\pi\)
\(468\) 8.16101e6 1.72238
\(469\) 9.10820e6 1.91206
\(470\) 0 0
\(471\) 7.05046e6 1.46442
\(472\) −1.59077e7 −3.28663
\(473\) 2.40856e6 0.495000
\(474\) 69863.7 0.0142826
\(475\) 0 0
\(476\) −1.18282e7 −2.39276
\(477\) −4.31910e6 −0.869155
\(478\) 8.55072e6 1.71172
\(479\) −1.51089e6 −0.300881 −0.150440 0.988619i \(-0.548069\pi\)
−0.150440 + 0.988619i \(0.548069\pi\)
\(480\) 0 0
\(481\) 2.61800e6 0.515949
\(482\) 7.92655e6 1.55406
\(483\) 4.57113e6 0.891571
\(484\) 1.11394e6 0.216146
\(485\) 0 0
\(486\) 1.19093e7 2.28716
\(487\) −1.63265e6 −0.311940 −0.155970 0.987762i \(-0.549850\pi\)
−0.155970 + 0.987762i \(0.549850\pi\)
\(488\) 1.18133e7 2.24555
\(489\) 1.24832e7 2.36077
\(490\) 0 0
\(491\) 1.24459e6 0.232982 0.116491 0.993192i \(-0.462835\pi\)
0.116491 + 0.993192i \(0.462835\pi\)
\(492\) −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 \(0.741844\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) 1.69354e6 0.294049
\(507\) 594450. 0.102706
\(508\) −2.07198e6 −0.356227
\(509\) 6.27494e6 1.07353 0.536766 0.843731i \(-0.319646\pi\)
0.536766 + 0.843731i \(0.319646\pi\)
\(510\) 0 0
\(511\) −7.70824e6 −1.30588
\(512\) −1.19016e7 −2.00647
\(513\) 1.41636e6 0.237618
\(514\) 7.95281e6 1.32774
\(515\) 0 0
\(516\) −3.12688e7 −5.16996
\(517\) −226437. −0.0372581
\(518\) −7.64823e6 −1.25238
\(519\) −1.03186e7 −1.68152
\(520\) 0 0
\(521\) 3.60326e6 0.581570 0.290785 0.956788i \(-0.406084\pi\)
0.290785 + 0.956788i \(0.406084\pi\)
\(522\) 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
\(525\) 0 0
\(526\) 7.93890e6 1.25111
\(527\) −369342. −0.0579298
\(528\) −5.82092e6 −0.908672
\(529\) −4.62391e6 −0.718406
\(530\) 0 0
\(531\) −6.36174e6 −0.979128
\(532\) −1.43735e7 −2.20183
\(533\) 9.37473e6 1.42936
\(534\) −4.94938e6 −0.751100
\(535\) 0 0
\(536\) −2.53830e7 −3.81620
\(537\) 1.29365e7 1.93590
\(538\) 8.32625e6 1.24021
\(539\) 1.23879e6 0.183665
\(540\) 0 0
\(541\) 1.16842e7 1.71635 0.858176 0.513355i \(-0.171598\pi\)
0.858176 + 0.513355i \(0.171598\pi\)
\(542\) 1.02029e6 0.149185
\(543\) −8.12859e6 −1.18308
\(544\) 9.03525e6 1.30901
\(545\) 0 0
\(546\) 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
\(555\) 0 0
\(556\) 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
\(560\) 0 0
\(561\) −2.36167e6 −0.316820
\(562\) 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
\(565\) 0 0
\(566\) 279637. 0.0366906
\(567\) 1.15108e7 1.50365
\(568\) 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 \(-0.376755\pi\)
0.377583 + 0.925976i \(0.376755\pi\)
\(572\) 5.38773e6 0.688519
\(573\) −4.24222e6 −0.539768
\(574\) −2.73874e7 −3.46953
\(575\) 0 0
\(576\) 4.54637e6 0.570964
\(577\) −3.61005e6 −0.451413 −0.225706 0.974195i \(-0.572469\pi\)
−0.225706 + 0.974195i \(0.572469\pi\)
\(578\) −5.47057e6 −0.681104
\(579\) −7.22190e6 −0.895272
\(580\) 0 0
\(581\) −1.52791e7 −1.87783
\(582\) −1.07613e6 −0.131691
\(583\) −2.85138e6 −0.347443
\(584\) 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 \(0.597101\pi\)
\(588\) −1.60825e7 −1.91827
\(589\) −448821. −0.0533070
\(590\) 0 0
\(591\) 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 \(0.909916\pi\)
\(594\) 1.55099e6 0.180361
\(595\) 0 0
\(596\) −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 \(0.380282\pi\)
\(600\) 0 0
\(601\) −8.32443e6 −0.940087 −0.470044 0.882643i \(-0.655762\pi\)
−0.470044 + 0.882643i \(0.655762\pi\)
\(602\) −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
\(610\) 0 0
\(611\) −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
\(615\) 0 0
\(616\) −9.11974e6 −0.968346
\(617\) 413797. 0.0437597 0.0218799 0.999761i \(-0.493035\pi\)
0.0218799 + 0.999761i \(0.493035\pi\)
\(618\) 3.87003e7 4.07609
\(619\) −1.25898e7 −1.32067 −0.660333 0.750973i \(-0.729585\pi\)
−0.660333 + 0.750973i \(0.729585\pi\)
\(620\) 0 0
\(621\) 1.65987e6 0.172721
\(622\) 1.94177e6 0.201243
\(623\) −3.79198e6 −0.391422
\(624\) −2.81538e7 −2.89452
\(625\) 0 0
\(626\) 9.71264e6 0.990607
\(627\) −2.86988e6 −0.291538
\(628\) −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
\(633\) −3.43976e6 −0.341208
\(634\) 1.95742e7 1.93402
\(635\) 0 0
\(636\) 3.70176e7 3.62882
\(637\) 5.99163e6 0.585054
\(638\) 1.13164e6 0.110067
\(639\) 1.04285e7 1.01035
\(640\) 0 0
\(641\) −1.23075e7 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(642\) 1.99324e7 1.90863
\(643\) 1.44400e6 0.137733 0.0688667 0.997626i \(-0.478062\pi\)
0.0688667 + 0.997626i \(0.478062\pi\)
\(644\) −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 \(0.387734\pi\)
\(648\) −3.20786e7 −3.00108
\(649\) −4.19989e6 −0.391405
\(650\) 0 0
\(651\) −1.32659e6 −0.122683
\(652\) −4.60009e7 −4.23787
\(653\) −3.83734e6 −0.352166 −0.176083 0.984375i \(-0.556343\pi\)
−0.176083 + 0.984375i \(0.556343\pi\)
\(654\) −3.87948e7 −3.54674
\(655\) 0 0
\(656\) 3.73237e7 3.38630
\(657\) 8.59083e6 0.776465
\(658\) 3.19952e6 0.288085
\(659\) −1.98049e7 −1.77648 −0.888239 0.459382i \(-0.848071\pi\)
−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
\(663\) −1.14226e7 −1.00921
\(664\) 4.25803e7 3.74790
\(665\) 0 0
\(666\) 8.52395e6 0.744656
\(667\) 1.21108e6 0.105404
\(668\) 1.21599e7 1.05436
\(669\) −2.17977e7 −1.88298
\(670\) 0 0
\(671\) 3.11892e6 0.267423
\(672\) 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
\(675\) 0 0
\(676\) −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
\(680\) 0 0
\(681\) 1.08717e7 0.898320
\(682\) −491483. −0.0404620
\(683\) 1.94703e6 0.159705 0.0798527 0.996807i \(-0.474555\pi\)
0.0798527 + 0.996807i \(0.474555\pi\)
\(684\) 1.60192e7 1.30919
\(685\) 0 0
\(686\) 1.12311e7 0.911200
\(687\) −2.30611e7 −1.86418
\(688\) 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
\(693\) −3.64714e6 −0.288482
\(694\) 3.06476e7 2.41545
\(695\) 0 0
\(696\) −8.51227e6 −0.666073
\(697\) 1.51430e7 1.18067
\(698\) −9.61563e6 −0.747032
\(699\) −604162. −0.0467692
\(700\) 0 0
\(701\) 5.48228e6 0.421373 0.210686 0.977554i \(-0.432430\pi\)
0.210686 + 0.977554i \(0.432430\pi\)
\(702\) 7.50161e6 0.574528
\(703\) 5.13887e6 0.392174
\(704\) 3.00142e6 0.228242
\(705\) 0 0
\(706\) 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 \(0.662128\pi\)
\(710\) 0 0
\(711\) −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 \(0.786252\pi\)
\(720\) 0 0
\(721\) 2.96503e7 2.12418
\(722\) −1.20228e7 −0.858348
\(723\) −1.57418e7 −1.11998
\(724\) 2.99540e7 2.12378
\(725\) 0 0
\(726\) −3.14267e6 −0.221288
\(727\) 1.07681e7 0.755619 0.377809 0.925883i \(-0.376677\pi\)
0.377809 + 0.925883i \(0.376677\pi\)
\(728\) −4.41091e7 −3.08461
\(729\) −6.64290e6 −0.462955
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) 1.28673e7 0.875573
\(737\) −6.70154e6 −0.454471
\(738\) 3.05232e7 2.06295
\(739\) −6.81140e6 −0.458802 −0.229401 0.973332i \(-0.573677\pi\)
−0.229401 + 0.973332i \(0.573677\pi\)
\(740\) 0 0
\(741\) −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
\(745\) 0 0
\(746\) −1.89457e7 −1.24642
\(747\) 1.70286e7 1.11655
\(748\) 8.70281e6 0.568729
\(749\) 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
\(753\) −1.15903e7 −0.744914
\(754\) 5.47334e6 0.350610
\(755\) 0 0
\(756\) −1.54268e7 −0.981685
\(757\) −1.48895e7 −0.944366 −0.472183 0.881501i \(-0.656534\pi\)
−0.472183 + 0.881501i \(0.656534\pi\)
\(758\) −4.35977e6 −0.275607
\(759\) −3.36330e6 −0.211915
\(760\) 0 0
\(761\) −2.14078e7 −1.34002 −0.670009 0.742353i \(-0.733710\pi\)
−0.670009 + 0.742353i \(0.733710\pi\)
\(762\) 5.84554e6 0.364702
\(763\) −2.97227e7 −1.84832
\(764\) 1.56327e7 0.968948
\(765\) 0 0
\(766\) −3.07373e7 −1.89275
\(767\) −2.03134e7 −1.24680
\(768\) 2.66844e7 1.63251
\(769\) 1.45027e7 0.884369 0.442184 0.896924i \(-0.354204\pi\)
0.442184 + 0.896924i \(0.354204\pi\)
\(770\) 0 0
\(771\) −1.57939e7 −0.956874
\(772\) 2.66129e7 1.60712
\(773\) −3.18546e7 −1.91745 −0.958723 0.284342i \(-0.908225\pi\)
−0.958723 + 0.284342i \(0.908225\pi\)
\(774\) 3.79293e7 2.27574
\(775\) 0 0
\(776\) 2.29768e6 0.136973
\(777\) 1.51891e7 0.902565
\(778\) 2.44759e7 1.44974
\(779\) 1.84016e7 1.08646
\(780\) 0 0
\(781\) 6.88470e6 0.403885
\(782\) 1.32311e7 0.773710
\(783\) 1.10914e6 0.0646519
\(784\) 2.38545e7 1.38606
\(785\) 0 0
\(786\) 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
\(790\) 0 0
\(791\) −1.12281e7 −0.638067
\(792\) 1.01639e7 0.575771
\(793\) 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 \(0.311304\pi\)
\(798\) 4.05509e7 2.25421
\(799\) −1.76908e6 −0.0980347
\(800\) 0 0
\(801\) 4.22616e6 0.232736
\(802\) −5.99384e7 −3.29055
\(803\) 5.67149e6 0.310391
\(804\) 8.70018e7 4.74666
\(805\) 0 0
\(806\) −2.37714e6 −0.128889
\(807\) −1.65356e7 −0.893790
\(808\) 1.73829e7 0.936683
\(809\) −1.59711e7 −0.857954 −0.428977 0.903315i \(-0.641126\pi\)
−0.428977 + 0.903315i \(0.641126\pi\)
\(810\) 0 0
\(811\) 1.37309e7 0.733074 0.366537 0.930403i \(-0.380543\pi\)
0.366537 + 0.930403i \(0.380543\pi\)
\(812\) −1.12558e7 −0.599080
\(813\) −2.02625e6 −0.107515
\(814\) 5.62734e6 0.297675
\(815\) 0 0
\(816\) −4.54769e7 −2.39092
\(817\) 2.28666e7 1.19852
\(818\) 2.49192e7 1.30212
\(819\) −1.76400e7 −0.918942
\(820\) 0 0
\(821\) 2.26402e7 1.17226 0.586128 0.810218i \(-0.300652\pi\)
0.586128 + 0.810218i \(0.300652\pi\)
\(822\) 7.57758e6 0.391157
\(823\) −1.16510e7 −0.599603 −0.299802 0.954002i \(-0.596921\pi\)
−0.299802 + 0.954002i \(0.596921\pi\)
\(824\) −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 \(0.348178\pi\)
\(830\) 0 0
\(831\) −1.28214e7 −0.644070
\(832\) 1.45169e7 0.727050
\(833\) 9.67828e6 0.483265
\(834\) −5.63410e6 −0.280485
\(835\) 0 0
\(836\) 1.05756e7 0.523345
\(837\) −481712. −0.0237670
\(838\) −1.47283e7 −0.724507
\(839\) −1.83237e7 −0.898689 −0.449344 0.893359i \(-0.648342\pi\)
−0.449344 + 0.893359i \(0.648342\pi\)
\(840\) 0 0
\(841\) −1.97019e7 −0.960546
\(842\) −4.99582e7 −2.42844
\(843\) −2.70866e7 −1.31276
\(844\) 1.26756e7 0.612509
\(845\) 0 0
\(846\) −3.56586e6 −0.171293
\(847\) −2.40776e6 −0.115320
\(848\) −5.49069e7 −2.62203
\(849\) −555348. −0.0264421
\(850\) 0 0
\(851\) 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
\(855\) 0 0
\(856\) −4.25584e7 −1.98518
\(857\) −3.80021e7 −1.76749 −0.883743 0.467972i \(-0.844985\pi\)
−0.883743 + 0.467972i \(0.844985\pi\)
\(858\) −1.52001e7 −0.704899
\(859\) −7.40664e6 −0.342482 −0.171241 0.985229i \(-0.554778\pi\)
−0.171241 + 0.985229i \(0.554778\pi\)
\(860\) 0 0
\(861\) 5.43901e7 2.50042
\(862\) −2.84303e7 −1.30321
\(863\) 9.79342e6 0.447618 0.223809 0.974633i \(-0.428151\pi\)
0.223809 + 0.974633i \(0.428151\pi\)
\(864\) 1.17842e7 0.537050
\(865\) 0 0
\(866\) −2.82698e7 −1.28094
\(867\) 1.08643e7 0.490857
\(868\) 4.88851e6 0.220230
\(869\) −39383.0 −0.00176913
\(870\) 0 0
\(871\) −3.24131e7 −1.44769
\(872\) 8.28322e7 3.68899
\(873\) 918880. 0.0408059
\(874\) 1.60783e7 0.711969
\(875\) 0 0
\(876\) −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 \(-0.296027\pi\)
0.597838 + 0.801617i \(0.296027\pi\)
\(884\) 4.20926e7 1.81165
\(885\) 0 0
\(886\) −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
\(890\) 0 0
\(891\) −8.46928e6 −0.357398
\(892\) 8.03250e7 3.38017
\(893\) −2.14977e6 −0.0902117
\(894\) 4.85795e7 2.03287
\(895\) 0 0
\(896\) 7.88818e6 0.328251
\(897\) −1.62672e7 −0.675042
\(898\) −7.21309e6 −0.298491
\(899\) −351468. −0.0145040
\(900\) 0 0
\(901\) −2.22769e7 −0.914203
\(902\) 2.01508e7 0.824662
\(903\) 6.75873e7 2.75833
\(904\) 3.12909e7 1.27349
\(905\) 0 0
\(906\) 6.46236e7 2.61560
\(907\) −5.05051e6 −0.203853 −0.101926 0.994792i \(-0.532501\pi\)
−0.101926 + 0.994792i \(0.532501\pi\)
\(908\) −4.00626e7 −1.61259
\(909\) 6.95169e6 0.279049
\(910\) 0 0
\(911\) −1.72283e7 −0.687774 −0.343887 0.939011i \(-0.611744\pi\)
−0.343887 + 0.939011i \(0.611744\pi\)
\(912\) −5.52631e7 −2.20013
\(913\) 1.12419e7 0.446337
\(914\) −8.44660e7 −3.34439
\(915\) 0 0
\(916\) 8.49807e7 3.34643
\(917\) 1.95488e6 0.0767709
\(918\) 1.21174e7 0.474571
\(919\) 3.89056e7 1.51958 0.759789 0.650170i \(-0.225302\pi\)
0.759789 + 0.650170i \(0.225302\pi\)
\(920\) 0 0
\(921\) 1.13645e7 0.441470
\(922\) −4.66769e7 −1.80832
\(923\) 3.32990e7 1.28655
\(924\) 3.12584e7 1.20445
\(925\) 0 0
\(926\) 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 \(0.292937\pi\)
\(930\) 0 0
\(931\) 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
\(940\) 0 0
\(941\) −1.76369e7 −0.649304 −0.324652 0.945833i \(-0.605247\pi\)
−0.324652 + 0.945833i \(0.605247\pi\)
\(942\) 7.32988e7 2.69135
\(943\) 2.15655e7 0.789732
\(944\) −8.08741e7 −2.95379
\(945\) 0 0
\(946\) 2.50402e7 0.909723
\(947\) −4.07232e7 −1.47559 −0.737797 0.675022i \(-0.764134\pi\)
−0.737797 + 0.675022i \(0.764134\pi\)
\(948\) 511284. 0.0184774
\(949\) 2.74311e7 0.988730
\(950\) 0 0
\(951\) −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
\(955\) 0 0
\(956\) 6.25767e7 2.21446
\(957\) −2.24738e6 −0.0793226
\(958\) −1.57077e7 −0.552967
\(959\) 5.80557e6 0.203844
\(960\) 0 0
\(961\) −2.84765e7 −0.994668
\(962\) 2.72175e7 0.948225
\(963\) −1.70198e7 −0.591410
\(964\) 5.80089e7 2.01049
\(965\) 0 0
\(966\) 4.75229e7 1.63855
\(967\) 7.46133e6 0.256596 0.128298 0.991736i \(-0.459049\pi\)
0.128298 + 0.991736i \(0.459049\pi\)
\(968\) 6.71003e6 0.230163
\(969\) −2.24214e7 −0.767103
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) 6.00587e7 2.01814
\(977\) 1.52214e7 0.510173 0.255087 0.966918i \(-0.417896\pi\)
0.255087 + 0.966918i \(0.417896\pi\)
\(978\) 1.29779e8 4.33869
\(979\) 2.79002e6 0.0930360
\(980\) 0 0
\(981\) 3.31260e7 1.09900
\(982\) 1.29392e7 0.428181
\(983\) −2.90192e7 −0.957858 −0.478929 0.877854i \(-0.658975\pi\)
−0.478929 + 0.877854i \(0.658975\pi\)
\(984\) −1.51576e8 −4.99049
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) −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 275.6.a.b.1.3 3
5.2 odd 4 275.6.b.b.199.6 6
5.3 odd 4 275.6.b.b.199.1 6
5.4 even 2 11.6.a.b.1.1 3
15.14 odd 2 99.6.a.g.1.3 3
20.19 odd 2 176.6.a.i.1.1 3
35.34 odd 2 539.6.a.e.1.1 3
40.19 odd 2 704.6.a.t.1.3 3
40.29 even 2 704.6.a.q.1.1 3
55.54 odd 2 121.6.a.d.1.3 3
165.164 even 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 5.4 even 2
99.6.a.g.1.3 3 15.14 odd 2
121.6.a.d.1.3 3 55.54 odd 2
176.6.a.i.1.1 3 20.19 odd 2
275.6.a.b.1.3 3 1.1 even 1 trivial
275.6.b.b.199.1 6 5.3 odd 4
275.6.b.b.199.6 6 5.2 odd 4
539.6.a.e.1.1 3 35.34 odd 2
704.6.a.q.1.1 3 40.29 even 2
704.6.a.t.1.3 3 40.19 odd 2
1089.6.a.r.1.1 3 165.164 even 2