Properties

Label 722.6.a.q.1.5
Level $722$
Weight $6$
Character 722.1
Self dual yes
Analytic conductor $115.797$
Analytic rank $0$
Dimension $15$
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] = [722,6,Mod(1,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, 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("722.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 722.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: \(115.797117905\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} - 6060225486 x^{8} + 437045334939 x^{7} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 19^{6} \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-12.5784\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -11.0464 q^{3} +16.0000 q^{4} -52.2115 q^{5} +44.1854 q^{6} +215.645 q^{7} -64.0000 q^{8} -120.978 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -11.0464 q^{3} +16.0000 q^{4} -52.2115 q^{5} +44.1854 q^{6} +215.645 q^{7} -64.0000 q^{8} -120.978 q^{9} +208.846 q^{10} +546.889 q^{11} -176.742 q^{12} -257.843 q^{13} -862.581 q^{14} +576.747 q^{15} +256.000 q^{16} +1895.04 q^{17} +483.912 q^{18} -835.384 q^{20} -2382.10 q^{21} -2187.56 q^{22} +2232.65 q^{23} +706.967 q^{24} -398.958 q^{25} +1031.37 q^{26} +4020.63 q^{27} +3450.32 q^{28} +3582.58 q^{29} -2306.99 q^{30} +8443.79 q^{31} -1024.00 q^{32} -6041.13 q^{33} -7580.16 q^{34} -11259.2 q^{35} -1935.65 q^{36} +3258.91 q^{37} +2848.22 q^{39} +3341.54 q^{40} -8960.15 q^{41} +9528.38 q^{42} +18035.6 q^{43} +8750.23 q^{44} +6316.44 q^{45} -8930.60 q^{46} -20284.9 q^{47} -2827.87 q^{48} +29695.9 q^{49} +1595.83 q^{50} -20933.3 q^{51} -4125.48 q^{52} -3592.76 q^{53} -16082.5 q^{54} -28553.9 q^{55} -13801.3 q^{56} -14330.3 q^{58} +3069.98 q^{59} +9227.95 q^{60} -5773.68 q^{61} -33775.2 q^{62} -26088.3 q^{63} +4096.00 q^{64} +13462.4 q^{65} +24164.5 q^{66} -38679.3 q^{67} +30320.6 q^{68} -24662.7 q^{69} +45036.7 q^{70} -42588.1 q^{71} +7742.59 q^{72} +44359.4 q^{73} -13035.6 q^{74} +4407.04 q^{75} +117934. q^{77} -11392.9 q^{78} -11044.5 q^{79} -13366.1 q^{80} -15015.7 q^{81} +35840.6 q^{82} -21521.6 q^{83} -38113.5 q^{84} -98942.9 q^{85} -72142.3 q^{86} -39574.5 q^{87} -35000.9 q^{88} +61451.6 q^{89} -25265.8 q^{90} -55602.6 q^{91} +35722.4 q^{92} -93273.1 q^{93} +81139.5 q^{94} +11311.5 q^{96} -157318. q^{97} -118784. q^{98} -66161.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} - 960 q^{8} + 2127 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} - 960 q^{8} + 2127 q^{9} - 432 q^{10} + 126 q^{11} + 114 q^{13} - 336 q^{14} + 3840 q^{16} + 4119 q^{17} - 8508 q^{18} + 1728 q^{20} - 3408 q^{21} - 504 q^{22} + 3936 q^{23} + 26895 q^{25} - 456 q^{26} + 13017 q^{27} + 1344 q^{28} - 14658 q^{29} - 6840 q^{31} - 15360 q^{32} + 3945 q^{33} - 16476 q^{34} + 12636 q^{35} + 34032 q^{36} + 4278 q^{37} + 4956 q^{39} - 6912 q^{40} - 5112 q^{41} + 13632 q^{42} + 94191 q^{43} + 2016 q^{44} + 31770 q^{45} - 15744 q^{46} + 702 q^{47} + 63777 q^{49} - 107580 q^{50} + 108 q^{51} + 1824 q^{52} - 47544 q^{53} - 52068 q^{54} + 16848 q^{55} - 5376 q^{56} + 58632 q^{58} + 8832 q^{59} + 119196 q^{61} + 27360 q^{62} - 88068 q^{63} + 61440 q^{64} - 80646 q^{65} - 15780 q^{66} - 64248 q^{67} + 65904 q^{68} - 124224 q^{69} - 50544 q^{70} + 53364 q^{71} - 136128 q^{72} - 4908 q^{73} - 17112 q^{74} + 87480 q^{75} + 121218 q^{77} - 19824 q^{78} + 115500 q^{79} + 27648 q^{80} + 481659 q^{81} + 20448 q^{82} + 201630 q^{83} - 54528 q^{84} - 150282 q^{85} - 376764 q^{86} + 376512 q^{87} - 8064 q^{88} + 101505 q^{89} - 127080 q^{90} - 414918 q^{91} + 62976 q^{92} + 165960 q^{93} - 2808 q^{94} - 297114 q^{97} - 255108 q^{98} - 149895 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −11.0464 −0.708624 −0.354312 0.935127i \(-0.615285\pi\)
−0.354312 + 0.935127i \(0.615285\pi\)
\(4\) 16.0000 0.500000
\(5\) −52.2115 −0.933988 −0.466994 0.884261i \(-0.654663\pi\)
−0.466994 + 0.884261i \(0.654663\pi\)
\(6\) 44.1854 0.501073
\(7\) 215.645 1.66339 0.831697 0.555230i \(-0.187370\pi\)
0.831697 + 0.555230i \(0.187370\pi\)
\(8\) −64.0000 −0.353553
\(9\) −120.978 −0.497852
\(10\) 208.846 0.660429
\(11\) 546.889 1.36275 0.681377 0.731932i \(-0.261381\pi\)
0.681377 + 0.731932i \(0.261381\pi\)
\(12\) −176.742 −0.354312
\(13\) −257.843 −0.423152 −0.211576 0.977362i \(-0.567860\pi\)
−0.211576 + 0.977362i \(0.567860\pi\)
\(14\) −862.581 −1.17620
\(15\) 576.747 0.661846
\(16\) 256.000 0.250000
\(17\) 1895.04 1.59036 0.795181 0.606372i \(-0.207376\pi\)
0.795181 + 0.606372i \(0.207376\pi\)
\(18\) 483.912 0.352034
\(19\) 0 0
\(20\) −835.384 −0.466994
\(21\) −2382.10 −1.17872
\(22\) −2187.56 −0.963613
\(23\) 2232.65 0.880038 0.440019 0.897989i \(-0.354972\pi\)
0.440019 + 0.897989i \(0.354972\pi\)
\(24\) 706.967 0.250536
\(25\) −398.958 −0.127667
\(26\) 1031.37 0.299214
\(27\) 4020.63 1.06141
\(28\) 3450.32 0.831697
\(29\) 3582.58 0.791045 0.395522 0.918456i \(-0.370564\pi\)
0.395522 + 0.918456i \(0.370564\pi\)
\(30\) −2306.99 −0.467996
\(31\) 8443.79 1.57809 0.789047 0.614332i \(-0.210575\pi\)
0.789047 + 0.614332i \(0.210575\pi\)
\(32\) −1024.00 −0.176777
\(33\) −6041.13 −0.965681
\(34\) −7580.16 −1.12456
\(35\) −11259.2 −1.55359
\(36\) −1935.65 −0.248926
\(37\) 3258.91 0.391353 0.195676 0.980669i \(-0.437310\pi\)
0.195676 + 0.980669i \(0.437310\pi\)
\(38\) 0 0
\(39\) 2848.22 0.299856
\(40\) 3341.54 0.330215
\(41\) −8960.15 −0.832445 −0.416222 0.909263i \(-0.636646\pi\)
−0.416222 + 0.909263i \(0.636646\pi\)
\(42\) 9528.38 0.833481
\(43\) 18035.6 1.48751 0.743753 0.668455i \(-0.233044\pi\)
0.743753 + 0.668455i \(0.233044\pi\)
\(44\) 8750.23 0.681377
\(45\) 6316.44 0.464988
\(46\) −8930.60 −0.622281
\(47\) −20284.9 −1.33945 −0.669727 0.742607i \(-0.733589\pi\)
−0.669727 + 0.742607i \(0.733589\pi\)
\(48\) −2827.87 −0.177156
\(49\) 29695.9 1.76688
\(50\) 1595.83 0.0902740
\(51\) −20933.3 −1.12697
\(52\) −4125.48 −0.211576
\(53\) −3592.76 −0.175687 −0.0878434 0.996134i \(-0.527998\pi\)
−0.0878434 + 0.996134i \(0.527998\pi\)
\(54\) −16082.5 −0.750533
\(55\) −28553.9 −1.27280
\(56\) −13801.3 −0.588098
\(57\) 0 0
\(58\) −14330.3 −0.559353
\(59\) 3069.98 0.114817 0.0574084 0.998351i \(-0.481716\pi\)
0.0574084 + 0.998351i \(0.481716\pi\)
\(60\) 9227.95 0.330923
\(61\) −5773.68 −0.198668 −0.0993340 0.995054i \(-0.531671\pi\)
−0.0993340 + 0.995054i \(0.531671\pi\)
\(62\) −33775.2 −1.11588
\(63\) −26088.3 −0.828123
\(64\) 4096.00 0.125000
\(65\) 13462.4 0.395219
\(66\) 24164.5 0.682839
\(67\) −38679.3 −1.05267 −0.526334 0.850278i \(-0.676434\pi\)
−0.526334 + 0.850278i \(0.676434\pi\)
\(68\) 30320.6 0.795181
\(69\) −24662.7 −0.623616
\(70\) 45036.7 1.09855
\(71\) −42588.1 −1.00263 −0.501317 0.865264i \(-0.667151\pi\)
−0.501317 + 0.865264i \(0.667151\pi\)
\(72\) 7742.59 0.176017
\(73\) 44359.4 0.974269 0.487134 0.873327i \(-0.338042\pi\)
0.487134 + 0.873327i \(0.338042\pi\)
\(74\) −13035.6 −0.276728
\(75\) 4407.04 0.0904677
\(76\) 0 0
\(77\) 117934. 2.26680
\(78\) −11392.9 −0.212030
\(79\) −11044.5 −0.199104 −0.0995520 0.995032i \(-0.531741\pi\)
−0.0995520 + 0.995032i \(0.531741\pi\)
\(80\) −13366.1 −0.233497
\(81\) −15015.7 −0.254292
\(82\) 35840.6 0.588627
\(83\) −21521.6 −0.342910 −0.171455 0.985192i \(-0.554847\pi\)
−0.171455 + 0.985192i \(0.554847\pi\)
\(84\) −38113.5 −0.589360
\(85\) −98942.9 −1.48538
\(86\) −72142.3 −1.05183
\(87\) −39574.5 −0.560554
\(88\) −35000.9 −0.481807
\(89\) 61451.6 0.822353 0.411176 0.911556i \(-0.365118\pi\)
0.411176 + 0.911556i \(0.365118\pi\)
\(90\) −25265.8 −0.328796
\(91\) −55602.6 −0.703868
\(92\) 35722.4 0.440019
\(93\) −93273.1 −1.11828
\(94\) 81139.5 0.947137
\(95\) 0 0
\(96\) 11311.5 0.125268
\(97\) −157318. −1.69765 −0.848827 0.528671i \(-0.822691\pi\)
−0.848827 + 0.528671i \(0.822691\pi\)
\(98\) −118784. −1.24937
\(99\) −66161.6 −0.678450
\(100\) −6383.33 −0.0638333
\(101\) 77058.5 0.751652 0.375826 0.926690i \(-0.377359\pi\)
0.375826 + 0.926690i \(0.377359\pi\)
\(102\) 83733.1 0.796887
\(103\) −132907. −1.23440 −0.617200 0.786806i \(-0.711733\pi\)
−0.617200 + 0.786806i \(0.711733\pi\)
\(104\) 16501.9 0.149607
\(105\) 124373. 1.10091
\(106\) 14371.1 0.124229
\(107\) −112225. −0.947615 −0.473808 0.880628i \(-0.657121\pi\)
−0.473808 + 0.880628i \(0.657121\pi\)
\(108\) 64330.1 0.530707
\(109\) 60711.5 0.489446 0.244723 0.969593i \(-0.421303\pi\)
0.244723 + 0.969593i \(0.421303\pi\)
\(110\) 114216. 0.900003
\(111\) −35999.1 −0.277322
\(112\) 55205.2 0.415848
\(113\) −44583.9 −0.328459 −0.164230 0.986422i \(-0.552514\pi\)
−0.164230 + 0.986422i \(0.552514\pi\)
\(114\) 0 0
\(115\) −116570. −0.821944
\(116\) 57321.3 0.395522
\(117\) 31193.3 0.210667
\(118\) −12279.9 −0.0811877
\(119\) 408656. 2.64540
\(120\) −36911.8 −0.233998
\(121\) 138037. 0.857100
\(122\) 23094.7 0.140479
\(123\) 98977.0 0.589891
\(124\) 135101. 0.789047
\(125\) 183991. 1.05323
\(126\) 104353. 0.585572
\(127\) 306590. 1.68674 0.843372 0.537331i \(-0.180567\pi\)
0.843372 + 0.537331i \(0.180567\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −199227. −1.05408
\(130\) −53849.4 −0.279462
\(131\) 19963.0 0.101636 0.0508181 0.998708i \(-0.483817\pi\)
0.0508181 + 0.998708i \(0.483817\pi\)
\(132\) −96658.1 −0.482840
\(133\) 0 0
\(134\) 154717. 0.744349
\(135\) −209923. −0.991348
\(136\) −121283. −0.562278
\(137\) 144949. 0.659801 0.329900 0.944016i \(-0.392985\pi\)
0.329900 + 0.944016i \(0.392985\pi\)
\(138\) 98650.6 0.440963
\(139\) 91551.0 0.401907 0.200954 0.979601i \(-0.435596\pi\)
0.200954 + 0.979601i \(0.435596\pi\)
\(140\) −180147. −0.776794
\(141\) 224074. 0.949170
\(142\) 170352. 0.708969
\(143\) −141011. −0.576652
\(144\) −30970.4 −0.124463
\(145\) −187052. −0.738826
\(146\) −177438. −0.688912
\(147\) −328031. −1.25205
\(148\) 52142.6 0.195676
\(149\) −259829. −0.958786 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(150\) −17628.1 −0.0639703
\(151\) 172996. 0.617438 0.308719 0.951153i \(-0.400100\pi\)
0.308719 + 0.951153i \(0.400100\pi\)
\(152\) 0 0
\(153\) −229258. −0.791764
\(154\) −471736. −1.60287
\(155\) −440863. −1.47392
\(156\) 45571.6 0.149928
\(157\) 32495.4 0.105214 0.0526070 0.998615i \(-0.483247\pi\)
0.0526070 + 0.998615i \(0.483247\pi\)
\(158\) 44178.2 0.140788
\(159\) 39687.0 0.124496
\(160\) 53464.6 0.165107
\(161\) 481461. 1.46385
\(162\) 60062.7 0.179811
\(163\) 222554. 0.656094 0.328047 0.944661i \(-0.393610\pi\)
0.328047 + 0.944661i \(0.393610\pi\)
\(164\) −143362. −0.416222
\(165\) 315417. 0.901934
\(166\) 86086.5 0.242474
\(167\) −401806. −1.11487 −0.557436 0.830220i \(-0.688215\pi\)
−0.557436 + 0.830220i \(0.688215\pi\)
\(168\) 152454. 0.416741
\(169\) −304810. −0.820942
\(170\) 395771. 1.05032
\(171\) 0 0
\(172\) 288569. 0.743753
\(173\) 289986. 0.736652 0.368326 0.929697i \(-0.379931\pi\)
0.368326 + 0.929697i \(0.379931\pi\)
\(174\) 158298. 0.396371
\(175\) −86033.5 −0.212360
\(176\) 140004. 0.340689
\(177\) −33912.1 −0.0813619
\(178\) −245806. −0.581491
\(179\) 308157. 0.718852 0.359426 0.933174i \(-0.382973\pi\)
0.359426 + 0.933174i \(0.382973\pi\)
\(180\) 101063. 0.232494
\(181\) 130725. 0.296593 0.148296 0.988943i \(-0.452621\pi\)
0.148296 + 0.988943i \(0.452621\pi\)
\(182\) 222410. 0.497710
\(183\) 63778.1 0.140781
\(184\) −142890. −0.311140
\(185\) −170153. −0.365519
\(186\) 373092. 0.790741
\(187\) 1.03638e6 2.16727
\(188\) −324558. −0.669727
\(189\) 867030. 1.76555
\(190\) 0 0
\(191\) −828301. −1.64288 −0.821438 0.570297i \(-0.806828\pi\)
−0.821438 + 0.570297i \(0.806828\pi\)
\(192\) −45245.9 −0.0885780
\(193\) 747808. 1.44510 0.722548 0.691321i \(-0.242971\pi\)
0.722548 + 0.691321i \(0.242971\pi\)
\(194\) 629272. 1.20042
\(195\) −148710. −0.280062
\(196\) 475134. 0.883438
\(197\) −716653. −1.31566 −0.657830 0.753167i \(-0.728525\pi\)
−0.657830 + 0.753167i \(0.728525\pi\)
\(198\) 264646. 0.479736
\(199\) 256756. 0.459608 0.229804 0.973237i \(-0.426191\pi\)
0.229804 + 0.973237i \(0.426191\pi\)
\(200\) 25533.3 0.0451370
\(201\) 427266. 0.745947
\(202\) −308234. −0.531498
\(203\) 772567. 1.31582
\(204\) −334933. −0.563484
\(205\) 467823. 0.777493
\(206\) 531630. 0.872853
\(207\) −270102. −0.438128
\(208\) −66007.7 −0.105788
\(209\) 0 0
\(210\) −497491. −0.778461
\(211\) −300557. −0.464751 −0.232376 0.972626i \(-0.574650\pi\)
−0.232376 + 0.972626i \(0.574650\pi\)
\(212\) −57484.2 −0.0878434
\(213\) 470443. 0.710490
\(214\) 448902. 0.670065
\(215\) −941664. −1.38931
\(216\) −257320. −0.375267
\(217\) 1.82086e6 2.62499
\(218\) −242846. −0.346090
\(219\) −490010. −0.690390
\(220\) −456863. −0.636398
\(221\) −488622. −0.672965
\(222\) 143996. 0.196096
\(223\) 1.11984e6 1.50797 0.753984 0.656893i \(-0.228130\pi\)
0.753984 + 0.656893i \(0.228130\pi\)
\(224\) −220821. −0.294049
\(225\) 48265.2 0.0635591
\(226\) 178335. 0.232256
\(227\) −1.42711e6 −1.83820 −0.919099 0.394026i \(-0.871082\pi\)
−0.919099 + 0.394026i \(0.871082\pi\)
\(228\) 0 0
\(229\) 843458. 1.06286 0.531428 0.847103i \(-0.321656\pi\)
0.531428 + 0.847103i \(0.321656\pi\)
\(230\) 466280. 0.581202
\(231\) −1.30274e6 −1.60631
\(232\) −229285. −0.279677
\(233\) 482086. 0.581748 0.290874 0.956761i \(-0.406054\pi\)
0.290874 + 0.956761i \(0.406054\pi\)
\(234\) −124773. −0.148964
\(235\) 1.05910e6 1.25103
\(236\) 49119.7 0.0574084
\(237\) 122002. 0.141090
\(238\) −1.63463e6 −1.87058
\(239\) 715888. 0.810682 0.405341 0.914166i \(-0.367153\pi\)
0.405341 + 0.914166i \(0.367153\pi\)
\(240\) 147647. 0.165462
\(241\) 1.58668e6 1.75973 0.879867 0.475220i \(-0.157632\pi\)
0.879867 + 0.475220i \(0.157632\pi\)
\(242\) −552147. −0.606061
\(243\) −811145. −0.881217
\(244\) −92378.8 −0.0993340
\(245\) −1.55047e6 −1.65024
\(246\) −395908. −0.417116
\(247\) 0 0
\(248\) −540402. −0.557941
\(249\) 237736. 0.242994
\(250\) −735965. −0.744744
\(251\) 280916. 0.281444 0.140722 0.990049i \(-0.455058\pi\)
0.140722 + 0.990049i \(0.455058\pi\)
\(252\) −417413. −0.414062
\(253\) 1.22101e6 1.19928
\(254\) −1.22636e6 −1.19271
\(255\) 1.09296e6 1.05258
\(256\) 65536.0 0.0625000
\(257\) 1.89954e6 1.79397 0.896985 0.442061i \(-0.145752\pi\)
0.896985 + 0.442061i \(0.145752\pi\)
\(258\) 796909. 0.745349
\(259\) 702769. 0.650973
\(260\) 215398. 0.197609
\(261\) −433414. −0.393823
\(262\) −79852.2 −0.0718677
\(263\) 1.73196e6 1.54400 0.772000 0.635622i \(-0.219256\pi\)
0.772000 + 0.635622i \(0.219256\pi\)
\(264\) 386633. 0.341420
\(265\) 187584. 0.164089
\(266\) 0 0
\(267\) −678816. −0.582739
\(268\) −618869. −0.526334
\(269\) −575238. −0.484693 −0.242347 0.970190i \(-0.577917\pi\)
−0.242347 + 0.970190i \(0.577917\pi\)
\(270\) 839693. 0.700989
\(271\) 1.71192e6 1.41599 0.707995 0.706218i \(-0.249600\pi\)
0.707995 + 0.706218i \(0.249600\pi\)
\(272\) 485130. 0.397590
\(273\) 614206. 0.498778
\(274\) −579795. −0.466550
\(275\) −218186. −0.173978
\(276\) −394603. −0.311808
\(277\) −2.38150e6 −1.86488 −0.932442 0.361319i \(-0.882326\pi\)
−0.932442 + 0.361319i \(0.882326\pi\)
\(278\) −366204. −0.284191
\(279\) −1.02151e6 −0.785657
\(280\) 720587. 0.549277
\(281\) 929370. 0.702138 0.351069 0.936350i \(-0.385818\pi\)
0.351069 + 0.936350i \(0.385818\pi\)
\(282\) −896296. −0.671164
\(283\) 321844. 0.238880 0.119440 0.992841i \(-0.461890\pi\)
0.119440 + 0.992841i \(0.461890\pi\)
\(284\) −681409. −0.501317
\(285\) 0 0
\(286\) 564046. 0.407755
\(287\) −1.93221e6 −1.38468
\(288\) 123881. 0.0880086
\(289\) 2.17132e6 1.52925
\(290\) 748208. 0.522429
\(291\) 1.73779e6 1.20300
\(292\) 709751. 0.487134
\(293\) 287213. 0.195450 0.0977250 0.995213i \(-0.468843\pi\)
0.0977250 + 0.995213i \(0.468843\pi\)
\(294\) 1.31213e6 0.885334
\(295\) −160288. −0.107237
\(296\) −208570. −0.138364
\(297\) 2.19884e6 1.44645
\(298\) 1.03932e6 0.677964
\(299\) −575673. −0.372390
\(300\) 70512.6 0.0452338
\(301\) 3.88928e6 2.47431
\(302\) −691983. −0.436595
\(303\) −851215. −0.532639
\(304\) 0 0
\(305\) 301452. 0.185554
\(306\) 917032. 0.559862
\(307\) 71384.1 0.0432270 0.0216135 0.999766i \(-0.493120\pi\)
0.0216135 + 0.999766i \(0.493120\pi\)
\(308\) 1.88695e6 1.13340
\(309\) 1.46814e6 0.874726
\(310\) 1.76345e6 1.04222
\(311\) 1.91268e6 1.12135 0.560676 0.828035i \(-0.310541\pi\)
0.560676 + 0.828035i \(0.310541\pi\)
\(312\) −182286. −0.106015
\(313\) 863929. 0.498445 0.249222 0.968446i \(-0.419825\pi\)
0.249222 + 0.968446i \(0.419825\pi\)
\(314\) −129982. −0.0743975
\(315\) 1.36211e6 0.773457
\(316\) −176713. −0.0995520
\(317\) 2.49607e6 1.39511 0.697554 0.716532i \(-0.254272\pi\)
0.697554 + 0.716532i \(0.254272\pi\)
\(318\) −158748. −0.0880319
\(319\) 1.95928e6 1.07800
\(320\) −213858. −0.116748
\(321\) 1.23968e6 0.671503
\(322\) −1.92584e6 −1.03510
\(323\) 0 0
\(324\) −240251. −0.127146
\(325\) 102868. 0.0540224
\(326\) −890215. −0.463928
\(327\) −670641. −0.346833
\(328\) 573449. 0.294314
\(329\) −4.37434e6 −2.22804
\(330\) −1.26167e6 −0.637764
\(331\) −1.22909e6 −0.616617 −0.308308 0.951286i \(-0.599763\pi\)
−0.308308 + 0.951286i \(0.599763\pi\)
\(332\) −344346. −0.171455
\(333\) −394257. −0.194836
\(334\) 1.60722e6 0.788334
\(335\) 2.01951e6 0.983180
\(336\) −609816. −0.294680
\(337\) −905116. −0.434140 −0.217070 0.976156i \(-0.569650\pi\)
−0.217070 + 0.976156i \(0.569650\pi\)
\(338\) 1.21924e6 0.580494
\(339\) 492489. 0.232754
\(340\) −1.58309e6 −0.742689
\(341\) 4.61782e6 2.15056
\(342\) 0 0
\(343\) 2.77943e6 1.27562
\(344\) −1.15428e6 −0.525913
\(345\) 1.28767e6 0.582450
\(346\) −1.15995e6 −0.520891
\(347\) −1.59170e6 −0.709639 −0.354819 0.934935i \(-0.615458\pi\)
−0.354819 + 0.934935i \(0.615458\pi\)
\(348\) −633192. −0.280277
\(349\) −101389. −0.0445582 −0.0222791 0.999752i \(-0.507092\pi\)
−0.0222791 + 0.999752i \(0.507092\pi\)
\(350\) 344134. 0.150161
\(351\) −1.03669e6 −0.449140
\(352\) −560015. −0.240903
\(353\) −1.12463e6 −0.480365 −0.240182 0.970728i \(-0.577207\pi\)
−0.240182 + 0.970728i \(0.577207\pi\)
\(354\) 135648. 0.0575316
\(355\) 2.22359e6 0.936447
\(356\) 983225. 0.411176
\(357\) −4.51416e6 −1.87459
\(358\) −1.23263e6 −0.508305
\(359\) −3.20159e6 −1.31108 −0.655540 0.755160i \(-0.727559\pi\)
−0.655540 + 0.755160i \(0.727559\pi\)
\(360\) −404252. −0.164398
\(361\) 0 0
\(362\) −522898. −0.209723
\(363\) −1.52480e6 −0.607362
\(364\) −889641. −0.351934
\(365\) −2.31607e6 −0.909955
\(366\) −255112. −0.0995472
\(367\) −1.66625e6 −0.645764 −0.322882 0.946439i \(-0.604652\pi\)
−0.322882 + 0.946439i \(0.604652\pi\)
\(368\) 571559. 0.220009
\(369\) 1.08398e6 0.414434
\(370\) 680611. 0.258461
\(371\) −774763. −0.292236
\(372\) −1.49237e6 −0.559138
\(373\) −4.06188e6 −1.51166 −0.755832 0.654765i \(-0.772767\pi\)
−0.755832 + 0.654765i \(0.772767\pi\)
\(374\) −4.14551e6 −1.53249
\(375\) −2.03243e6 −0.746342
\(376\) 1.29823e6 0.473569
\(377\) −923743. −0.334732
\(378\) −3.46812e6 −1.24843
\(379\) −122287. −0.0437302 −0.0218651 0.999761i \(-0.506960\pi\)
−0.0218651 + 0.999761i \(0.506960\pi\)
\(380\) 0 0
\(381\) −3.38671e6 −1.19527
\(382\) 3.31321e6 1.16169
\(383\) 1.42368e6 0.495925 0.247963 0.968770i \(-0.420239\pi\)
0.247963 + 0.968770i \(0.420239\pi\)
\(384\) 180984. 0.0626341
\(385\) −6.15752e6 −2.11716
\(386\) −2.99123e6 −1.02184
\(387\) −2.18191e6 −0.740557
\(388\) −2.51709e6 −0.848827
\(389\) −2.08327e6 −0.698026 −0.349013 0.937118i \(-0.613483\pi\)
−0.349013 + 0.937118i \(0.613483\pi\)
\(390\) 594840. 0.198034
\(391\) 4.23096e6 1.39958
\(392\) −1.90054e6 −0.624685
\(393\) −220519. −0.0720219
\(394\) 2.86661e6 0.930312
\(395\) 576652. 0.185961
\(396\) −1.05858e6 −0.339225
\(397\) 1.84336e6 0.586994 0.293497 0.955960i \(-0.405181\pi\)
0.293497 + 0.955960i \(0.405181\pi\)
\(398\) −1.02702e6 −0.324992
\(399\) 0 0
\(400\) −102133. −0.0319167
\(401\) 44488.5 0.0138161 0.00690807 0.999976i \(-0.497801\pi\)
0.00690807 + 0.999976i \(0.497801\pi\)
\(402\) −1.70906e6 −0.527464
\(403\) −2.17717e6 −0.667774
\(404\) 1.23294e6 0.375826
\(405\) 783991. 0.237505
\(406\) −3.09027e6 −0.930424
\(407\) 1.78226e6 0.533318
\(408\) 1.33973e6 0.398444
\(409\) 702028. 0.207513 0.103757 0.994603i \(-0.466914\pi\)
0.103757 + 0.994603i \(0.466914\pi\)
\(410\) −1.87129e6 −0.549771
\(411\) −1.60115e6 −0.467551
\(412\) −2.12652e6 −0.617200
\(413\) 662026. 0.190985
\(414\) 1.08041e6 0.309803
\(415\) 1.12368e6 0.320274
\(416\) 264031. 0.0748034
\(417\) −1.01131e6 −0.284801
\(418\) 0 0
\(419\) −5.55566e6 −1.54597 −0.772984 0.634425i \(-0.781237\pi\)
−0.772984 + 0.634425i \(0.781237\pi\)
\(420\) 1.98996e6 0.550455
\(421\) 2.11205e6 0.580763 0.290382 0.956911i \(-0.406218\pi\)
0.290382 + 0.956911i \(0.406218\pi\)
\(422\) 1.20223e6 0.328629
\(423\) 2.45402e6 0.666850
\(424\) 229937. 0.0621146
\(425\) −756042. −0.203036
\(426\) −1.88177e6 −0.502392
\(427\) −1.24507e6 −0.330463
\(428\) −1.79561e6 −0.473808
\(429\) 1.55766e6 0.408630
\(430\) 3.76666e6 0.982392
\(431\) −529667. −0.137344 −0.0686720 0.997639i \(-0.521876\pi\)
−0.0686720 + 0.997639i \(0.521876\pi\)
\(432\) 1.02928e6 0.265353
\(433\) −3.40387e6 −0.872475 −0.436238 0.899832i \(-0.643689\pi\)
−0.436238 + 0.899832i \(0.643689\pi\)
\(434\) −7.28345e6 −1.85615
\(435\) 2.06624e6 0.523550
\(436\) 971384. 0.244723
\(437\) 0 0
\(438\) 1.96004e6 0.488180
\(439\) −568141. −0.140700 −0.0703502 0.997522i \(-0.522412\pi\)
−0.0703502 + 0.997522i \(0.522412\pi\)
\(440\) 1.82745e6 0.450001
\(441\) −3.59255e6 −0.879643
\(442\) 1.95449e6 0.475858
\(443\) 3.19364e6 0.773174 0.386587 0.922253i \(-0.373654\pi\)
0.386587 + 0.922253i \(0.373654\pi\)
\(444\) −575986. −0.138661
\(445\) −3.20848e6 −0.768067
\(446\) −4.47934e6 −1.06629
\(447\) 2.87016e6 0.679419
\(448\) 883283. 0.207924
\(449\) −1.27971e6 −0.299567 −0.149784 0.988719i \(-0.547858\pi\)
−0.149784 + 0.988719i \(0.547858\pi\)
\(450\) −193061. −0.0449431
\(451\) −4.90021e6 −1.13442
\(452\) −713342. −0.164230
\(453\) −1.91097e6 −0.437531
\(454\) 5.70843e6 1.29980
\(455\) 2.90309e6 0.657404
\(456\) 0 0
\(457\) 5.15509e6 1.15464 0.577319 0.816519i \(-0.304099\pi\)
0.577319 + 0.816519i \(0.304099\pi\)
\(458\) −3.37383e6 −0.751553
\(459\) 7.61925e6 1.68803
\(460\) −1.86512e6 −0.410972
\(461\) −3.73593e6 −0.818742 −0.409371 0.912368i \(-0.634252\pi\)
−0.409371 + 0.912368i \(0.634252\pi\)
\(462\) 5.21097e6 1.13583
\(463\) 7.57260e6 1.64170 0.820848 0.571147i \(-0.193502\pi\)
0.820848 + 0.571147i \(0.193502\pi\)
\(464\) 917141. 0.197761
\(465\) 4.86993e6 1.04446
\(466\) −1.92834e6 −0.411358
\(467\) −8.04618e6 −1.70725 −0.853626 0.520886i \(-0.825602\pi\)
−0.853626 + 0.520886i \(0.825602\pi\)
\(468\) 499093. 0.105334
\(469\) −8.34101e6 −1.75100
\(470\) −4.23642e6 −0.884615
\(471\) −358956. −0.0745571
\(472\) −196479. −0.0405939
\(473\) 9.86346e6 2.02710
\(474\) −488008. −0.0997656
\(475\) 0 0
\(476\) 6.53850e6 1.32270
\(477\) 434645. 0.0874660
\(478\) −2.86355e6 −0.573238
\(479\) −309721. −0.0616782 −0.0308391 0.999524i \(-0.509818\pi\)
−0.0308391 + 0.999524i \(0.509818\pi\)
\(480\) −590589. −0.116999
\(481\) −840287. −0.165602
\(482\) −6.34672e6 −1.24432
\(483\) −5.31839e6 −1.03732
\(484\) 2.20859e6 0.428550
\(485\) 8.21381e6 1.58559
\(486\) 3.24458e6 0.623114
\(487\) 3.23195e6 0.617508 0.308754 0.951142i \(-0.400088\pi\)
0.308754 + 0.951142i \(0.400088\pi\)
\(488\) 369515. 0.0702397
\(489\) −2.45841e6 −0.464924
\(490\) 6.20187e6 1.16690
\(491\) −7.32124e6 −1.37051 −0.685253 0.728305i \(-0.740309\pi\)
−0.685253 + 0.728305i \(0.740309\pi\)
\(492\) 1.58363e6 0.294945
\(493\) 6.78914e6 1.25805
\(494\) 0 0
\(495\) 3.45439e6 0.633664
\(496\) 2.16161e6 0.394524
\(497\) −9.18392e6 −1.66777
\(498\) −950943. −0.171823
\(499\) −4.31611e6 −0.775963 −0.387981 0.921667i \(-0.626828\pi\)
−0.387981 + 0.921667i \(0.626828\pi\)
\(500\) 2.94386e6 0.526613
\(501\) 4.43849e6 0.790026
\(502\) −1.12366e6 −0.199011
\(503\) −4.64391e6 −0.818396 −0.409198 0.912446i \(-0.634191\pi\)
−0.409198 + 0.912446i \(0.634191\pi\)
\(504\) 1.66965e6 0.292786
\(505\) −4.02334e6 −0.702034
\(506\) −4.88405e6 −0.848016
\(507\) 3.36704e6 0.581740
\(508\) 4.90544e6 0.843372
\(509\) −7.29470e6 −1.24800 −0.623998 0.781426i \(-0.714492\pi\)
−0.623998 + 0.781426i \(0.714492\pi\)
\(510\) −4.37183e6 −0.744283
\(511\) 9.56590e6 1.62059
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) −7.59815e6 −1.26853
\(515\) 6.93930e6 1.15292
\(516\) −3.18764e6 −0.527041
\(517\) −1.10936e7 −1.82535
\(518\) −2.81108e6 −0.460308
\(519\) −3.20329e6 −0.522009
\(520\) −861591. −0.139731
\(521\) 7.11898e6 1.14901 0.574505 0.818501i \(-0.305195\pi\)
0.574505 + 0.818501i \(0.305195\pi\)
\(522\) 1.73365e6 0.278475
\(523\) 1.16755e7 1.86647 0.933237 0.359261i \(-0.116971\pi\)
0.933237 + 0.359261i \(0.116971\pi\)
\(524\) 319409. 0.0508181
\(525\) 950357. 0.150483
\(526\) −6.92782e6 −1.09177
\(527\) 1.60013e7 2.50974
\(528\) −1.54653e6 −0.241420
\(529\) −1.45161e6 −0.225534
\(530\) −750335. −0.116029
\(531\) −371400. −0.0571617
\(532\) 0 0
\(533\) 2.31031e6 0.352251
\(534\) 2.71526e6 0.412059
\(535\) 5.85946e6 0.885061
\(536\) 2.47548e6 0.372175
\(537\) −3.40401e6 −0.509396
\(538\) 2.30095e6 0.342730
\(539\) 1.62404e7 2.40782
\(540\) −3.35877e6 −0.495674
\(541\) −5.42739e6 −0.797256 −0.398628 0.917113i \(-0.630514\pi\)
−0.398628 + 0.917113i \(0.630514\pi\)
\(542\) −6.84768e6 −1.00126
\(543\) −1.44403e6 −0.210173
\(544\) −1.94052e6 −0.281139
\(545\) −3.16984e6 −0.457136
\(546\) −2.45682e6 −0.352689
\(547\) −1.85061e6 −0.264452 −0.132226 0.991220i \(-0.542212\pi\)
−0.132226 + 0.991220i \(0.542212\pi\)
\(548\) 2.31918e6 0.329900
\(549\) 698488. 0.0989072
\(550\) 872744. 0.123021
\(551\) 0 0
\(552\) 1.57841e6 0.220482
\(553\) −2.38170e6 −0.331188
\(554\) 9.52602e6 1.31867
\(555\) 1.87957e6 0.259015
\(556\) 1.46482e6 0.200954
\(557\) −3123.46 −0.000426578 0 −0.000213289 1.00000i \(-0.500068\pi\)
−0.000213289 1.00000i \(0.500068\pi\)
\(558\) 4.08605e6 0.555544
\(559\) −4.65034e6 −0.629441
\(560\) −2.88235e6 −0.388397
\(561\) −1.14482e7 −1.53578
\(562\) −3.71748e6 −0.496487
\(563\) −3.45261e6 −0.459068 −0.229534 0.973301i \(-0.573720\pi\)
−0.229534 + 0.973301i \(0.573720\pi\)
\(564\) 3.58519e6 0.474585
\(565\) 2.32779e6 0.306777
\(566\) −1.28738e6 −0.168914
\(567\) −3.23806e6 −0.422987
\(568\) 2.72564e6 0.354484
\(569\) −9.38291e6 −1.21495 −0.607473 0.794341i \(-0.707817\pi\)
−0.607473 + 0.794341i \(0.707817\pi\)
\(570\) 0 0
\(571\) 8.27080e6 1.06159 0.530796 0.847500i \(-0.321893\pi\)
0.530796 + 0.847500i \(0.321893\pi\)
\(572\) −2.25618e6 −0.288326
\(573\) 9.14971e6 1.16418
\(574\) 7.72885e6 0.979119
\(575\) −890735. −0.112351
\(576\) −495526. −0.0622315
\(577\) −5.47580e6 −0.684713 −0.342356 0.939570i \(-0.611225\pi\)
−0.342356 + 0.939570i \(0.611225\pi\)
\(578\) −8.68527e6 −1.08134
\(579\) −8.26055e6 −1.02403
\(580\) −2.99283e6 −0.369413
\(581\) −4.64104e6 −0.570394
\(582\) −6.95116e6 −0.850648
\(583\) −1.96484e6 −0.239418
\(584\) −2.83900e6 −0.344456
\(585\) −1.62865e6 −0.196760
\(586\) −1.14885e6 −0.138204
\(587\) −1.17010e7 −1.40161 −0.700806 0.713352i \(-0.747176\pi\)
−0.700806 + 0.713352i \(0.747176\pi\)
\(588\) −5.24850e6 −0.626026
\(589\) 0 0
\(590\) 641153. 0.0758283
\(591\) 7.91641e6 0.932308
\(592\) 834282. 0.0978382
\(593\) −7.06987e6 −0.825609 −0.412805 0.910820i \(-0.635451\pi\)
−0.412805 + 0.910820i \(0.635451\pi\)
\(594\) −8.79536e6 −1.02279
\(595\) −2.13366e7 −2.47077
\(596\) −4.15726e6 −0.479393
\(597\) −2.83622e6 −0.325689
\(598\) 2.30269e6 0.263319
\(599\) 5.20555e6 0.592788 0.296394 0.955066i \(-0.404216\pi\)
0.296394 + 0.955066i \(0.404216\pi\)
\(600\) −282050. −0.0319852
\(601\) 8.10399e6 0.915193 0.457597 0.889160i \(-0.348710\pi\)
0.457597 + 0.889160i \(0.348710\pi\)
\(602\) −1.55571e7 −1.74960
\(603\) 4.67935e6 0.524073
\(604\) 2.76793e6 0.308719
\(605\) −7.20711e6 −0.800521
\(606\) 3.40486e6 0.376633
\(607\) 5.55114e6 0.611520 0.305760 0.952109i \(-0.401090\pi\)
0.305760 + 0.952109i \(0.401090\pi\)
\(608\) 0 0
\(609\) −8.53405e6 −0.932421
\(610\) −1.20581e6 −0.131206
\(611\) 5.23031e6 0.566793
\(612\) −3.66813e6 −0.395882
\(613\) 1.00552e7 1.08079 0.540393 0.841413i \(-0.318276\pi\)
0.540393 + 0.841413i \(0.318276\pi\)
\(614\) −285536. −0.0305661
\(615\) −5.16774e6 −0.550951
\(616\) −7.54778e6 −0.801434
\(617\) −1.57668e6 −0.166736 −0.0833681 0.996519i \(-0.526568\pi\)
−0.0833681 + 0.996519i \(0.526568\pi\)
\(618\) −5.87257e6 −0.618525
\(619\) 1.15688e7 1.21356 0.606779 0.794871i \(-0.292461\pi\)
0.606779 + 0.794871i \(0.292461\pi\)
\(620\) −7.05381e6 −0.736961
\(621\) 8.97666e6 0.934084
\(622\) −7.65073e6 −0.792916
\(623\) 1.32517e7 1.36790
\(624\) 729145. 0.0749639
\(625\) −8.35971e6 −0.856035
\(626\) −3.45572e6 −0.352454
\(627\) 0 0
\(628\) 519927. 0.0526070
\(629\) 6.17577e6 0.622392
\(630\) −5.44844e6 −0.546917
\(631\) −4.82735e6 −0.482653 −0.241327 0.970444i \(-0.577583\pi\)
−0.241327 + 0.970444i \(0.577583\pi\)
\(632\) 706851. 0.0703939
\(633\) 3.32006e6 0.329334
\(634\) −9.98427e6 −0.986491
\(635\) −1.60075e7 −1.57540
\(636\) 634991. 0.0622479
\(637\) −7.65687e6 −0.747657
\(638\) −7.83710e6 −0.762261
\(639\) 5.15222e6 0.499163
\(640\) 855433. 0.0825536
\(641\) 2.00328e7 1.92573 0.962865 0.269982i \(-0.0870177\pi\)
0.962865 + 0.269982i \(0.0870177\pi\)
\(642\) −4.95873e6 −0.474824
\(643\) 5.98749e6 0.571107 0.285553 0.958363i \(-0.407823\pi\)
0.285553 + 0.958363i \(0.407823\pi\)
\(644\) 7.70337e6 0.731924
\(645\) 1.04020e7 0.984500
\(646\) 0 0
\(647\) 1.64159e6 0.154172 0.0770859 0.997024i \(-0.475438\pi\)
0.0770859 + 0.997024i \(0.475438\pi\)
\(648\) 961003. 0.0899057
\(649\) 1.67894e6 0.156467
\(650\) −411474. −0.0381996
\(651\) −2.01139e7 −1.86013
\(652\) 3.56086e6 0.328047
\(653\) −6.57658e6 −0.603555 −0.301778 0.953378i \(-0.597580\pi\)
−0.301778 + 0.953378i \(0.597580\pi\)
\(654\) 2.68256e6 0.245248
\(655\) −1.04230e6 −0.0949270
\(656\) −2.29380e6 −0.208111
\(657\) −5.36651e6 −0.485041
\(658\) 1.74974e7 1.57546
\(659\) −1.35882e7 −1.21884 −0.609420 0.792847i \(-0.708598\pi\)
−0.609420 + 0.792847i \(0.708598\pi\)
\(660\) 5.04667e6 0.450967
\(661\) 9.37715e6 0.834771 0.417385 0.908730i \(-0.362947\pi\)
0.417385 + 0.908730i \(0.362947\pi\)
\(662\) 4.91638e6 0.436014
\(663\) 5.39749e6 0.476879
\(664\) 1.37738e6 0.121237
\(665\) 0 0
\(666\) 1.57703e6 0.137770
\(667\) 7.99866e6 0.696149
\(668\) −6.42890e6 −0.557436
\(669\) −1.23701e7 −1.06858
\(670\) −8.07802e6 −0.695213
\(671\) −3.15756e6 −0.270736
\(672\) 2.43927e6 0.208370
\(673\) 2.15238e7 1.83182 0.915908 0.401387i \(-0.131472\pi\)
0.915908 + 0.401387i \(0.131472\pi\)
\(674\) 3.62046e6 0.306983
\(675\) −1.60406e6 −0.135507
\(676\) −4.87696e6 −0.410471
\(677\) 1.44381e7 1.21071 0.605353 0.795957i \(-0.293032\pi\)
0.605353 + 0.795957i \(0.293032\pi\)
\(678\) −1.96996e6 −0.164582
\(679\) −3.39249e7 −2.82387
\(680\) 6.33234e6 0.525161
\(681\) 1.57644e7 1.30259
\(682\) −1.84713e7 −1.52067
\(683\) −1.39407e7 −1.14349 −0.571747 0.820430i \(-0.693734\pi\)
−0.571747 + 0.820430i \(0.693734\pi\)
\(684\) 0 0
\(685\) −7.56799e6 −0.616246
\(686\) −1.11177e7 −0.901997
\(687\) −9.31713e6 −0.753166
\(688\) 4.61710e6 0.371876
\(689\) 926368. 0.0743422
\(690\) −5.15070e6 −0.411854
\(691\) 6.84023e6 0.544974 0.272487 0.962159i \(-0.412154\pi\)
0.272487 + 0.962159i \(0.412154\pi\)
\(692\) 4.63978e6 0.368326
\(693\) −1.42674e7 −1.12853
\(694\) 6.36680e6 0.501790
\(695\) −4.78002e6 −0.375377
\(696\) 2.53277e6 0.198186
\(697\) −1.69798e7 −1.32389
\(698\) 405557. 0.0315074
\(699\) −5.32529e6 −0.412240
\(700\) −1.37654e6 −0.106180
\(701\) −6.35337e6 −0.488325 −0.244162 0.969734i \(-0.578513\pi\)
−0.244162 + 0.969734i \(0.578513\pi\)
\(702\) 4.14676e6 0.317590
\(703\) 0 0
\(704\) 2.24006e6 0.170344
\(705\) −1.16992e7 −0.886513
\(706\) 4.49850e6 0.339669
\(707\) 1.66173e7 1.25029
\(708\) −542593. −0.0406810
\(709\) 1.29660e6 0.0968704 0.0484352 0.998826i \(-0.484577\pi\)
0.0484352 + 0.998826i \(0.484577\pi\)
\(710\) −8.89435e6 −0.662168
\(711\) 1.33615e6 0.0991243
\(712\) −3.93290e6 −0.290746
\(713\) 1.88520e7 1.38878
\(714\) 1.80567e7 1.32554
\(715\) 7.36242e6 0.538586
\(716\) 4.93051e6 0.359426
\(717\) −7.90795e6 −0.574469
\(718\) 1.28064e7 0.927074
\(719\) 3.56565e6 0.257227 0.128614 0.991695i \(-0.458947\pi\)
0.128614 + 0.991695i \(0.458947\pi\)
\(720\) 1.61701e6 0.116247
\(721\) −2.86609e7 −2.05329
\(722\) 0 0
\(723\) −1.75270e7 −1.24699
\(724\) 2.09159e6 0.148296
\(725\) −1.42930e6 −0.100990
\(726\) 6.09922e6 0.429470
\(727\) −4.17549e6 −0.293003 −0.146501 0.989210i \(-0.546801\pi\)
−0.146501 + 0.989210i \(0.546801\pi\)
\(728\) 3.55856e6 0.248855
\(729\) 1.26090e7 0.878743
\(730\) 9.26429e6 0.643435
\(731\) 3.41781e7 2.36567
\(732\) 1.02045e6 0.0703905
\(733\) 1.27641e7 0.877469 0.438734 0.898617i \(-0.355427\pi\)
0.438734 + 0.898617i \(0.355427\pi\)
\(734\) 6.66499e6 0.456624
\(735\) 1.71270e7 1.16940
\(736\) −2.28623e6 −0.155570
\(737\) −2.11533e7 −1.43453
\(738\) −4.33592e6 −0.293049
\(739\) 1.59151e7 1.07201 0.536004 0.844215i \(-0.319933\pi\)
0.536004 + 0.844215i \(0.319933\pi\)
\(740\) −2.72244e6 −0.182759
\(741\) 0 0
\(742\) 3.09905e6 0.206642
\(743\) 9.97067e6 0.662602 0.331301 0.943525i \(-0.392513\pi\)
0.331301 + 0.943525i \(0.392513\pi\)
\(744\) 5.96948e6 0.395370
\(745\) 1.35661e7 0.895495
\(746\) 1.62475e7 1.06891
\(747\) 2.60364e6 0.170718
\(748\) 1.65820e7 1.08364
\(749\) −2.42009e7 −1.57626
\(750\) 8.12973e6 0.527744
\(751\) −550344. −0.0356069 −0.0178034 0.999842i \(-0.505667\pi\)
−0.0178034 + 0.999842i \(0.505667\pi\)
\(752\) −5.19293e6 −0.334864
\(753\) −3.10309e6 −0.199438
\(754\) 3.69497e6 0.236691
\(755\) −9.03237e6 −0.576680
\(756\) 1.38725e7 0.882774
\(757\) −2.51587e7 −1.59569 −0.797843 0.602865i \(-0.794026\pi\)
−0.797843 + 0.602865i \(0.794026\pi\)
\(758\) 489147. 0.0309219
\(759\) −1.34877e7 −0.849835
\(760\) 0 0
\(761\) 9.64716e6 0.603862 0.301931 0.953330i \(-0.402369\pi\)
0.301931 + 0.953330i \(0.402369\pi\)
\(762\) 1.35468e7 0.845182
\(763\) 1.30921e7 0.814141
\(764\) −1.32528e7 −0.821438
\(765\) 1.19699e7 0.739498
\(766\) −5.69473e6 −0.350672
\(767\) −791572. −0.0485849
\(768\) −723934. −0.0442890
\(769\) −2.07985e7 −1.26829 −0.634143 0.773216i \(-0.718647\pi\)
−0.634143 + 0.773216i \(0.718647\pi\)
\(770\) 2.46301e7 1.49706
\(771\) −2.09830e7 −1.27125
\(772\) 1.19649e7 0.722548
\(773\) 2.05987e7 1.23991 0.619955 0.784637i \(-0.287151\pi\)
0.619955 + 0.784637i \(0.287151\pi\)
\(774\) 8.72762e6 0.523653
\(775\) −3.36872e6 −0.201470
\(776\) 1.00684e7 0.600211
\(777\) −7.76304e6 −0.461296
\(778\) 8.33308e6 0.493579
\(779\) 0 0
\(780\) −2.37936e6 −0.140031
\(781\) −2.32910e7 −1.36634
\(782\) −1.69238e7 −0.989651
\(783\) 1.44042e7 0.839626
\(784\) 7.60215e6 0.441719
\(785\) −1.69664e6 −0.0982685
\(786\) 882076. 0.0509272
\(787\) 3.14384e7 1.80935 0.904677 0.426098i \(-0.140112\pi\)
0.904677 + 0.426098i \(0.140112\pi\)
\(788\) −1.14665e7 −0.657830
\(789\) −1.91318e7 −1.09412
\(790\) −2.30661e6 −0.131494
\(791\) −9.61430e6 −0.546357
\(792\) 4.23434e6 0.239868
\(793\) 1.48870e6 0.0840668
\(794\) −7.37343e6 −0.415067
\(795\) −2.07212e6 −0.116278
\(796\) 4.10809e6 0.229804
\(797\) 1.25351e7 0.699006 0.349503 0.936935i \(-0.386350\pi\)
0.349503 + 0.936935i \(0.386350\pi\)
\(798\) 0 0
\(799\) −3.84407e7 −2.13022
\(800\) 408533. 0.0225685
\(801\) −7.43429e6 −0.409410
\(802\) −177954. −0.00976949
\(803\) 2.42597e7 1.32769
\(804\) 6.83625e6 0.372973
\(805\) −2.51378e7 −1.36722
\(806\) 8.70868e6 0.472188
\(807\) 6.35429e6 0.343465
\(808\) −4.93174e6 −0.265749
\(809\) −1.51423e7 −0.813432 −0.406716 0.913555i \(-0.633326\pi\)
−0.406716 + 0.913555i \(0.633326\pi\)
\(810\) −3.13596e6 −0.167942
\(811\) −3.18034e6 −0.169793 −0.0848967 0.996390i \(-0.527056\pi\)
−0.0848967 + 0.996390i \(0.527056\pi\)
\(812\) 1.23611e7 0.657909
\(813\) −1.89105e7 −1.00340
\(814\) −7.12906e6 −0.377113
\(815\) −1.16199e7 −0.612784
\(816\) −5.35892e6 −0.281742
\(817\) 0 0
\(818\) −2.80811e6 −0.146734
\(819\) 6.72669e6 0.350422
\(820\) 7.48516e6 0.388747
\(821\) 1.60665e7 0.831885 0.415942 0.909391i \(-0.363452\pi\)
0.415942 + 0.909391i \(0.363452\pi\)
\(822\) 6.40462e6 0.330608
\(823\) 1.77578e7 0.913883 0.456941 0.889497i \(-0.348945\pi\)
0.456941 + 0.889497i \(0.348945\pi\)
\(824\) 8.50607e6 0.436426
\(825\) 2.41016e6 0.123285
\(826\) −2.64811e6 −0.135047
\(827\) −1.79406e6 −0.0912166 −0.0456083 0.998959i \(-0.514523\pi\)
−0.0456083 + 0.998959i \(0.514523\pi\)
\(828\) −4.32163e6 −0.219064
\(829\) −2.37927e7 −1.20242 −0.601212 0.799090i \(-0.705315\pi\)
−0.601212 + 0.799090i \(0.705315\pi\)
\(830\) −4.49471e6 −0.226468
\(831\) 2.63069e7 1.32150
\(832\) −1.05612e6 −0.0528940
\(833\) 5.62749e7 2.80997
\(834\) 4.04522e6 0.201385
\(835\) 2.09789e7 1.04128
\(836\) 0 0
\(837\) 3.39494e7 1.67501
\(838\) 2.22226e7 1.09316
\(839\) 4.82941e6 0.236859 0.118429 0.992962i \(-0.462214\pi\)
0.118429 + 0.992962i \(0.462214\pi\)
\(840\) −7.95986e6 −0.389231
\(841\) −7.67625e6 −0.374248
\(842\) −8.44820e6 −0.410661
\(843\) −1.02661e7 −0.497552
\(844\) −4.80891e6 −0.232376
\(845\) 1.59146e7 0.766750
\(846\) −9.81610e6 −0.471534
\(847\) 2.97670e7 1.42569
\(848\) −919748. −0.0439217
\(849\) −3.55521e6 −0.169276
\(850\) 3.02417e6 0.143568
\(851\) 7.27601e6 0.344405
\(852\) 7.52709e6 0.355245
\(853\) 2.91328e6 0.137091 0.0685456 0.997648i \(-0.478164\pi\)
0.0685456 + 0.997648i \(0.478164\pi\)
\(854\) 4.98027e6 0.233673
\(855\) 0 0
\(856\) 7.18243e6 0.335033
\(857\) −5.81789e6 −0.270591 −0.135295 0.990805i \(-0.543198\pi\)
−0.135295 + 0.990805i \(0.543198\pi\)
\(858\) −6.23065e6 −0.288945
\(859\) 1.53027e7 0.707597 0.353799 0.935322i \(-0.384890\pi\)
0.353799 + 0.935322i \(0.384890\pi\)
\(860\) −1.50666e7 −0.694656
\(861\) 2.13439e7 0.981220
\(862\) 2.11867e6 0.0971169
\(863\) 2.58064e7 1.17951 0.589754 0.807583i \(-0.299225\pi\)
0.589754 + 0.807583i \(0.299225\pi\)
\(864\) −4.11713e6 −0.187633
\(865\) −1.51406e7 −0.688024
\(866\) 1.36155e7 0.616933
\(867\) −2.39852e7 −1.08366
\(868\) 2.91338e7 1.31250
\(869\) −6.04014e6 −0.271330
\(870\) −8.26497e6 −0.370206
\(871\) 9.97318e6 0.445439
\(872\) −3.88553e6 −0.173045
\(873\) 1.90320e7 0.845180
\(874\) 0 0
\(875\) 3.96768e7 1.75193
\(876\) −7.84016e6 −0.345195
\(877\) −4.01238e7 −1.76158 −0.880792 0.473503i \(-0.842989\pi\)
−0.880792 + 0.473503i \(0.842989\pi\)
\(878\) 2.27257e6 0.0994902
\(879\) −3.17266e6 −0.138501
\(880\) −7.30980e6 −0.318199
\(881\) −1.71504e7 −0.744448 −0.372224 0.928143i \(-0.621405\pi\)
−0.372224 + 0.928143i \(0.621405\pi\)
\(882\) 1.43702e7 0.622001
\(883\) 1.09709e7 0.473523 0.236762 0.971568i \(-0.423914\pi\)
0.236762 + 0.971568i \(0.423914\pi\)
\(884\) −7.81795e6 −0.336482
\(885\) 1.77060e6 0.0759911
\(886\) −1.27746e7 −0.546717
\(887\) 1.75319e7 0.748204 0.374102 0.927388i \(-0.377951\pi\)
0.374102 + 0.927388i \(0.377951\pi\)
\(888\) 2.30394e6 0.0980481
\(889\) 6.61147e7 2.80572
\(890\) 1.28339e7 0.543106
\(891\) −8.21191e6 −0.346537
\(892\) 1.79174e7 0.753984
\(893\) 0 0
\(894\) −1.14807e7 −0.480422
\(895\) −1.60893e7 −0.671399
\(896\) −3.53313e6 −0.147025
\(897\) 6.35909e6 0.263884
\(898\) 5.11882e6 0.211826
\(899\) 3.02506e7 1.24834
\(900\) 772243. 0.0317795
\(901\) −6.80843e6 −0.279406
\(902\) 1.96008e7 0.802155
\(903\) −4.29624e7 −1.75335
\(904\) 2.85337e6 0.116128
\(905\) −6.82533e6 −0.277014
\(906\) 7.64390e6 0.309381
\(907\) 4.35347e7 1.75718 0.878592 0.477573i \(-0.158483\pi\)
0.878592 + 0.477573i \(0.158483\pi\)
\(908\) −2.28337e7 −0.919099
\(909\) −9.32238e6 −0.374211
\(910\) −1.16124e7 −0.464855
\(911\) 6.91498e6 0.276055 0.138027 0.990428i \(-0.455924\pi\)
0.138027 + 0.990428i \(0.455924\pi\)
\(912\) 0 0
\(913\) −1.17699e7 −0.467302
\(914\) −2.06204e7 −0.816453
\(915\) −3.32995e6 −0.131488
\(916\) 1.34953e7 0.531428
\(917\) 4.30494e6 0.169061
\(918\) −3.04770e7 −1.19362
\(919\) 1.42108e7 0.555047 0.277523 0.960719i \(-0.410486\pi\)
0.277523 + 0.960719i \(0.410486\pi\)
\(920\) 7.46048e6 0.290601
\(921\) −788534. −0.0306317
\(922\) 1.49437e7 0.578938
\(923\) 1.09810e7 0.424266
\(924\) −2.08439e7 −0.803153
\(925\) −1.30017e6 −0.0499627
\(926\) −3.02904e7 −1.16085
\(927\) 1.60789e7 0.614549
\(928\) −3.66856e6 −0.139838
\(929\) −4.49843e6 −0.171010 −0.0855051 0.996338i \(-0.527250\pi\)
−0.0855051 + 0.996338i \(0.527250\pi\)
\(930\) −1.94797e7 −0.738542
\(931\) 0 0
\(932\) 7.71338e6 0.290874
\(933\) −2.11282e7 −0.794618
\(934\) 3.21847e7 1.20721
\(935\) −5.41108e7 −2.02421
\(936\) −1.99637e6 −0.0744820
\(937\) −1.97821e7 −0.736078 −0.368039 0.929810i \(-0.619971\pi\)
−0.368039 + 0.929810i \(0.619971\pi\)
\(938\) 3.33640e7 1.23815
\(939\) −9.54327e6 −0.353210
\(940\) 1.69457e7 0.625517
\(941\) 4.01748e6 0.147904 0.0739519 0.997262i \(-0.476439\pi\)
0.0739519 + 0.997262i \(0.476439\pi\)
\(942\) 1.43583e6 0.0527199
\(943\) −2.00049e7 −0.732583
\(944\) 785914. 0.0287042
\(945\) −4.52690e7 −1.64900
\(946\) −3.94538e7 −1.43338
\(947\) −1.27746e7 −0.462884 −0.231442 0.972849i \(-0.574344\pi\)
−0.231442 + 0.972849i \(0.574344\pi\)
\(948\) 1.95203e6 0.0705449
\(949\) −1.14378e7 −0.412264
\(950\) 0 0
\(951\) −2.75725e7 −0.988608
\(952\) −2.61540e7 −0.935289
\(953\) 2.59500e7 0.925561 0.462781 0.886473i \(-0.346852\pi\)
0.462781 + 0.886473i \(0.346852\pi\)
\(954\) −1.73858e6 −0.0618478
\(955\) 4.32469e7 1.53443
\(956\) 1.14542e7 0.405341
\(957\) −2.16429e7 −0.763897
\(958\) 1.23888e6 0.0436131
\(959\) 3.12575e7 1.09751
\(960\) 2.36236e6 0.0827308
\(961\) 4.26684e7 1.49038
\(962\) 3.36115e6 0.117098
\(963\) 1.35768e7 0.471772
\(964\) 2.53869e7 0.879867
\(965\) −3.90442e7 −1.34970
\(966\) 2.12735e7 0.733495
\(967\) 3.62654e7 1.24717 0.623586 0.781755i \(-0.285675\pi\)
0.623586 + 0.781755i \(0.285675\pi\)
\(968\) −8.83436e6 −0.303031
\(969\) 0 0
\(970\) −3.28552e7 −1.12118
\(971\) −7.02568e6 −0.239133 −0.119567 0.992826i \(-0.538151\pi\)
−0.119567 + 0.992826i \(0.538151\pi\)
\(972\) −1.29783e7 −0.440608
\(973\) 1.97425e7 0.668530
\(974\) −1.29278e7 −0.436644
\(975\) −1.13632e6 −0.0382816
\(976\) −1.47806e6 −0.0496670
\(977\) 3.44151e7 1.15349 0.576744 0.816925i \(-0.304323\pi\)
0.576744 + 0.816925i \(0.304323\pi\)
\(978\) 9.83363e6 0.328751
\(979\) 3.36072e7 1.12066
\(980\) −2.48075e7 −0.825121
\(981\) −7.34475e6 −0.243671
\(982\) 2.92850e7 0.969095
\(983\) −9.27370e6 −0.306104 −0.153052 0.988218i \(-0.548910\pi\)
−0.153052 + 0.988218i \(0.548910\pi\)
\(984\) −6.33453e6 −0.208558
\(985\) 3.74176e7 1.22881
\(986\) −2.71565e7 −0.889574
\(987\) 4.83205e7 1.57884
\(988\) 0 0
\(989\) 4.02671e7 1.30906
\(990\) −1.38176e7 −0.448068
\(991\) −3.75908e7 −1.21590 −0.607950 0.793975i \(-0.708008\pi\)
−0.607950 + 0.793975i \(0.708008\pi\)
\(992\) −8.64644e6 −0.278970
\(993\) 1.35770e7 0.436949
\(994\) 3.67357e7 1.17929
\(995\) −1.34056e7 −0.429268
\(996\) 3.80377e6 0.121497
\(997\) −7.17501e6 −0.228605 −0.114302 0.993446i \(-0.536463\pi\)
−0.114302 + 0.993446i \(0.536463\pi\)
\(998\) 1.72644e7 0.548689
\(999\) 1.31029e7 0.415387
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 722.6.a.q.1.5 15
19.14 odd 18 38.6.e.b.25.2 30
19.15 odd 18 38.6.e.b.35.2 yes 30
19.18 odd 2 722.6.a.r.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.6.e.b.25.2 30 19.14 odd 18
38.6.e.b.35.2 yes 30 19.15 odd 18
722.6.a.q.1.5 15 1.1 even 1 trivial
722.6.a.r.1.11 15 19.18 odd 2