Properties

Label 165.6.a.g.1.2
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
Dimension $5$
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] = [165,6,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.93734\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.93734 q^{2} -9.00000 q^{3} +3.25206 q^{4} -25.0000 q^{5} +53.4361 q^{6} -105.553 q^{7} +170.686 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.93734 q^{2} -9.00000 q^{3} +3.25206 q^{4} -25.0000 q^{5} +53.4361 q^{6} -105.553 q^{7} +170.686 q^{8} +81.0000 q^{9} +148.434 q^{10} -121.000 q^{11} -29.2685 q^{12} +124.247 q^{13} +626.706 q^{14} +225.000 q^{15} -1117.49 q^{16} +1766.45 q^{17} -480.925 q^{18} +1998.70 q^{19} -81.3014 q^{20} +949.979 q^{21} +718.419 q^{22} -2546.65 q^{23} -1536.18 q^{24} +625.000 q^{25} -737.700 q^{26} -729.000 q^{27} -343.265 q^{28} -320.493 q^{29} -1335.90 q^{30} +8554.39 q^{31} +1172.96 q^{32} +1089.00 q^{33} -10488.0 q^{34} +2638.83 q^{35} +263.417 q^{36} +4126.28 q^{37} -11867.0 q^{38} -1118.23 q^{39} -4267.16 q^{40} +4531.80 q^{41} -5640.35 q^{42} -13412.4 q^{43} -393.499 q^{44} -2025.00 q^{45} +15120.3 q^{46} -24103.2 q^{47} +10057.4 q^{48} -5665.52 q^{49} -3710.84 q^{50} -15898.1 q^{51} +404.060 q^{52} -28493.8 q^{53} +4328.32 q^{54} +3025.00 q^{55} -18016.5 q^{56} -17988.3 q^{57} +1902.88 q^{58} -18203.6 q^{59} +731.713 q^{60} +4696.58 q^{61} -50790.3 q^{62} -8549.81 q^{63} +28795.4 q^{64} -3106.19 q^{65} -6465.77 q^{66} +5374.30 q^{67} +5744.60 q^{68} +22919.9 q^{69} -15667.6 q^{70} -16054.9 q^{71} +13825.6 q^{72} +26176.0 q^{73} -24499.2 q^{74} -5625.00 q^{75} +6499.88 q^{76} +12771.9 q^{77} +6639.30 q^{78} -21818.8 q^{79} +27937.2 q^{80} +6561.00 q^{81} -26906.8 q^{82} -65176.7 q^{83} +3089.39 q^{84} -44161.3 q^{85} +79633.8 q^{86} +2884.44 q^{87} -20653.1 q^{88} +28370.0 q^{89} +12023.1 q^{90} -13114.7 q^{91} -8281.86 q^{92} -76989.5 q^{93} +143109. q^{94} -49967.5 q^{95} -10556.6 q^{96} -51795.2 q^{97} +33638.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 45 q^{3} + 127 q^{4} - 125 q^{5} + 9 q^{6} + 116 q^{7} - 153 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} - 45 q^{3} + 127 q^{4} - 125 q^{5} + 9 q^{6} + 116 q^{7} - 153 q^{8} + 405 q^{9} + 25 q^{10} - 605 q^{11} - 1143 q^{12} - 926 q^{13} + 368 q^{14} + 1125 q^{15} + 1891 q^{16} - 246 q^{17} - 81 q^{18} + 3420 q^{19} - 3175 q^{20} - 1044 q^{21} + 121 q^{22} - 4244 q^{23} + 1377 q^{24} + 3125 q^{25} - 8862 q^{26} - 3645 q^{27} - 4904 q^{28} - 2922 q^{29} - 225 q^{30} - 6112 q^{31} - 24757 q^{32} + 5445 q^{33} + 10866 q^{34} - 2900 q^{35} + 10287 q^{36} + 6654 q^{37} - 45692 q^{38} + 8334 q^{39} + 3825 q^{40} - 14934 q^{41} - 3312 q^{42} + 10804 q^{43} - 15367 q^{44} - 10125 q^{45} - 101500 q^{46} - 41460 q^{47} - 17019 q^{48} - 12099 q^{49} - 625 q^{50} + 2214 q^{51} - 97742 q^{52} - 62398 q^{53} + 729 q^{54} + 15125 q^{55} - 74368 q^{56} - 30780 q^{57} - 27822 q^{58} + 8524 q^{59} + 28575 q^{60} + 59010 q^{61} - 142624 q^{62} + 9396 q^{63} + 13799 q^{64} + 23150 q^{65} - 1089 q^{66} - 15772 q^{67} - 83686 q^{68} + 38196 q^{69} - 9200 q^{70} + 88124 q^{71} - 12393 q^{72} - 118358 q^{73} + 67194 q^{74} - 28125 q^{75} + 100668 q^{76} - 14036 q^{77} + 79758 q^{78} + 57324 q^{79} - 47275 q^{80} + 32805 q^{81} + 29102 q^{82} - 7268 q^{83} + 44136 q^{84} + 6150 q^{85} - 35288 q^{86} + 26298 q^{87} + 18513 q^{88} + 72978 q^{89} + 2025 q^{90} - 1464 q^{91} + 62148 q^{92} + 55008 q^{93} + 344836 q^{94} - 85500 q^{95} + 222813 q^{96} - 59174 q^{97} + 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.93734 −1.04958 −0.524792 0.851230i \(-0.675857\pi\)
−0.524792 + 0.851230i \(0.675857\pi\)
\(3\) −9.00000 −0.577350
\(4\) 3.25206 0.101627
\(5\) −25.0000 −0.447214
\(6\) 53.4361 0.605978
\(7\) −105.553 −0.814191 −0.407096 0.913386i \(-0.633458\pi\)
−0.407096 + 0.913386i \(0.633458\pi\)
\(8\) 170.686 0.942918
\(9\) 81.0000 0.333333
\(10\) 148.434 0.469388
\(11\) −121.000 −0.301511
\(12\) −29.2685 −0.0586742
\(13\) 124.247 0.203906 0.101953 0.994789i \(-0.467491\pi\)
0.101953 + 0.994789i \(0.467491\pi\)
\(14\) 626.706 0.854562
\(15\) 225.000 0.258199
\(16\) −1117.49 −1.09130
\(17\) 1766.45 1.48245 0.741224 0.671257i \(-0.234245\pi\)
0.741224 + 0.671257i \(0.234245\pi\)
\(18\) −480.925 −0.349861
\(19\) 1998.70 1.27017 0.635087 0.772441i \(-0.280964\pi\)
0.635087 + 0.772441i \(0.280964\pi\)
\(20\) −81.3014 −0.0454489
\(21\) 949.979 0.470073
\(22\) 718.419 0.316462
\(23\) −2546.65 −1.00381 −0.501903 0.864924i \(-0.667367\pi\)
−0.501903 + 0.864924i \(0.667367\pi\)
\(24\) −1536.18 −0.544394
\(25\) 625.000 0.200000
\(26\) −737.700 −0.214016
\(27\) −729.000 −0.192450
\(28\) −343.265 −0.0827436
\(29\) −320.493 −0.0707659 −0.0353829 0.999374i \(-0.511265\pi\)
−0.0353829 + 0.999374i \(0.511265\pi\)
\(30\) −1335.90 −0.271001
\(31\) 8554.39 1.59876 0.799382 0.600823i \(-0.205160\pi\)
0.799382 + 0.600823i \(0.205160\pi\)
\(32\) 1172.96 0.202492
\(33\) 1089.00 0.174078
\(34\) −10488.0 −1.55595
\(35\) 2638.83 0.364117
\(36\) 263.417 0.0338756
\(37\) 4126.28 0.495513 0.247756 0.968822i \(-0.420307\pi\)
0.247756 + 0.968822i \(0.420307\pi\)
\(38\) −11867.0 −1.33315
\(39\) −1118.23 −0.117725
\(40\) −4267.16 −0.421686
\(41\) 4531.80 0.421028 0.210514 0.977591i \(-0.432486\pi\)
0.210514 + 0.977591i \(0.432486\pi\)
\(42\) −5640.35 −0.493382
\(43\) −13412.4 −1.10620 −0.553100 0.833115i \(-0.686555\pi\)
−0.553100 + 0.833115i \(0.686555\pi\)
\(44\) −393.499 −0.0306416
\(45\) −2025.00 −0.149071
\(46\) 15120.3 1.05358
\(47\) −24103.2 −1.59159 −0.795793 0.605569i \(-0.792945\pi\)
−0.795793 + 0.605569i \(0.792945\pi\)
\(48\) 10057.4 0.630062
\(49\) −5665.52 −0.337093
\(50\) −3710.84 −0.209917
\(51\) −15898.1 −0.855892
\(52\) 404.060 0.0207223
\(53\) −28493.8 −1.39335 −0.696675 0.717387i \(-0.745338\pi\)
−0.696675 + 0.717387i \(0.745338\pi\)
\(54\) 4328.32 0.201993
\(55\) 3025.00 0.134840
\(56\) −18016.5 −0.767716
\(57\) −17988.3 −0.733335
\(58\) 1902.88 0.0742747
\(59\) −18203.6 −0.680813 −0.340407 0.940278i \(-0.610565\pi\)
−0.340407 + 0.940278i \(0.610565\pi\)
\(60\) 731.713 0.0262399
\(61\) 4696.58 0.161606 0.0808030 0.996730i \(-0.474252\pi\)
0.0808030 + 0.996730i \(0.474252\pi\)
\(62\) −50790.3 −1.67804
\(63\) −8549.81 −0.271397
\(64\) 28795.4 0.878767
\(65\) −3106.19 −0.0911893
\(66\) −6465.77 −0.182709
\(67\) 5374.30 0.146263 0.0731315 0.997322i \(-0.476701\pi\)
0.0731315 + 0.997322i \(0.476701\pi\)
\(68\) 5744.60 0.150656
\(69\) 22919.9 0.579548
\(70\) −15667.6 −0.382172
\(71\) −16054.9 −0.377974 −0.188987 0.981980i \(-0.560520\pi\)
−0.188987 + 0.981980i \(0.560520\pi\)
\(72\) 13825.6 0.314306
\(73\) 26176.0 0.574904 0.287452 0.957795i \(-0.407192\pi\)
0.287452 + 0.957795i \(0.407192\pi\)
\(74\) −24499.2 −0.520082
\(75\) −5625.00 −0.115470
\(76\) 6499.88 0.129084
\(77\) 12771.9 0.245488
\(78\) 6639.30 0.123562
\(79\) −21818.8 −0.393336 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(80\) 27937.2 0.488044
\(81\) 6561.00 0.111111
\(82\) −26906.8 −0.441904
\(83\) −65176.7 −1.03848 −0.519239 0.854629i \(-0.673784\pi\)
−0.519239 + 0.854629i \(0.673784\pi\)
\(84\) 3089.39 0.0477720
\(85\) −44161.3 −0.662971
\(86\) 79633.8 1.16105
\(87\) 2884.44 0.0408567
\(88\) −20653.1 −0.284301
\(89\) 28370.0 0.379651 0.189826 0.981818i \(-0.439208\pi\)
0.189826 + 0.981818i \(0.439208\pi\)
\(90\) 12023.1 0.156463
\(91\) −13114.7 −0.166018
\(92\) −8281.86 −0.102014
\(93\) −76989.5 −0.923047
\(94\) 143109. 1.67050
\(95\) −49967.5 −0.568039
\(96\) −10556.6 −0.116909
\(97\) −51795.2 −0.558934 −0.279467 0.960155i \(-0.590158\pi\)
−0.279467 + 0.960155i \(0.590158\pi\)
\(98\) 33638.1 0.353807
\(99\) −9801.00 −0.100504
\(100\) 2032.54 0.0203254
\(101\) 152512. 1.48765 0.743823 0.668376i \(-0.233010\pi\)
0.743823 + 0.668376i \(0.233010\pi\)
\(102\) 94392.3 0.898331
\(103\) 190367. 1.76807 0.884035 0.467420i \(-0.154816\pi\)
0.884035 + 0.467420i \(0.154816\pi\)
\(104\) 21207.4 0.192266
\(105\) −23749.5 −0.210223
\(106\) 169177. 1.46244
\(107\) −68674.3 −0.579875 −0.289938 0.957046i \(-0.593635\pi\)
−0.289938 + 0.957046i \(0.593635\pi\)
\(108\) −2370.75 −0.0195581
\(109\) 29463.1 0.237526 0.118763 0.992923i \(-0.462107\pi\)
0.118763 + 0.992923i \(0.462107\pi\)
\(110\) −17960.5 −0.141526
\(111\) −37136.6 −0.286084
\(112\) 117955. 0.888526
\(113\) −99307.3 −0.731619 −0.365810 0.930690i \(-0.619208\pi\)
−0.365810 + 0.930690i \(0.619208\pi\)
\(114\) 106803. 0.769697
\(115\) 63666.3 0.448916
\(116\) −1042.26 −0.00719170
\(117\) 10064.0 0.0679685
\(118\) 108081. 0.714571
\(119\) −186455. −1.20700
\(120\) 38404.4 0.243460
\(121\) 14641.0 0.0909091
\(122\) −27885.2 −0.169619
\(123\) −40786.2 −0.243080
\(124\) 27819.3 0.162477
\(125\) −15625.0 −0.0894427
\(126\) 50763.2 0.284854
\(127\) 192783. 1.06062 0.530308 0.847805i \(-0.322076\pi\)
0.530308 + 0.847805i \(0.322076\pi\)
\(128\) −208503. −1.12483
\(129\) 120711. 0.638665
\(130\) 18442.5 0.0957109
\(131\) 194663. 0.991074 0.495537 0.868587i \(-0.334971\pi\)
0.495537 + 0.868587i \(0.334971\pi\)
\(132\) 3541.49 0.0176909
\(133\) −210969. −1.03416
\(134\) −31909.1 −0.153515
\(135\) 18225.0 0.0860663
\(136\) 301509. 1.39783
\(137\) −219837. −1.00069 −0.500345 0.865826i \(-0.666793\pi\)
−0.500345 + 0.865826i \(0.666793\pi\)
\(138\) −136083. −0.608284
\(139\) −60583.0 −0.265959 −0.132979 0.991119i \(-0.542454\pi\)
−0.132979 + 0.991119i \(0.542454\pi\)
\(140\) 8581.63 0.0370041
\(141\) 216929. 0.918902
\(142\) 95323.4 0.396715
\(143\) −15033.9 −0.0614798
\(144\) −90516.7 −0.363766
\(145\) 8012.33 0.0316475
\(146\) −155416. −0.603410
\(147\) 50989.7 0.194621
\(148\) 13418.9 0.0503573
\(149\) 61572.6 0.227207 0.113604 0.993526i \(-0.463761\pi\)
0.113604 + 0.993526i \(0.463761\pi\)
\(150\) 33397.6 0.121196
\(151\) −194637. −0.694676 −0.347338 0.937740i \(-0.612914\pi\)
−0.347338 + 0.937740i \(0.612914\pi\)
\(152\) 341151. 1.19767
\(153\) 143083. 0.494150
\(154\) −75831.4 −0.257660
\(155\) −213860. −0.714989
\(156\) −3636.54 −0.0119640
\(157\) 76011.6 0.246111 0.123055 0.992400i \(-0.460731\pi\)
0.123055 + 0.992400i \(0.460731\pi\)
\(158\) 129546. 0.412839
\(159\) 256444. 0.804451
\(160\) −29323.9 −0.0905570
\(161\) 268807. 0.817290
\(162\) −38954.9 −0.116620
\(163\) −498250. −1.46885 −0.734426 0.678689i \(-0.762548\pi\)
−0.734426 + 0.678689i \(0.762548\pi\)
\(164\) 14737.7 0.0427877
\(165\) −27225.0 −0.0778499
\(166\) 386976. 1.08997
\(167\) −650928. −1.80610 −0.903050 0.429536i \(-0.858677\pi\)
−0.903050 + 0.429536i \(0.858677\pi\)
\(168\) 162149. 0.443241
\(169\) −355856. −0.958423
\(170\) 262201. 0.695844
\(171\) 161895. 0.423391
\(172\) −43617.7 −0.112420
\(173\) −400605. −1.01766 −0.508828 0.860868i \(-0.669921\pi\)
−0.508828 + 0.860868i \(0.669921\pi\)
\(174\) −17125.9 −0.0428825
\(175\) −65970.8 −0.162838
\(176\) 135216. 0.329039
\(177\) 163833. 0.393068
\(178\) −168443. −0.398476
\(179\) −820215. −1.91335 −0.956677 0.291151i \(-0.905962\pi\)
−0.956677 + 0.291151i \(0.905962\pi\)
\(180\) −6585.41 −0.0151496
\(181\) 639863. 1.45175 0.725873 0.687828i \(-0.241436\pi\)
0.725873 + 0.687828i \(0.241436\pi\)
\(182\) 77866.6 0.174250
\(183\) −42269.2 −0.0933033
\(184\) −434679. −0.946507
\(185\) −103157. −0.221600
\(186\) 457113. 0.968816
\(187\) −213741. −0.446975
\(188\) −78384.9 −0.161748
\(189\) 76948.3 0.156691
\(190\) 296674. 0.596205
\(191\) 707635. 1.40354 0.701772 0.712402i \(-0.252393\pi\)
0.701772 + 0.712402i \(0.252393\pi\)
\(192\) −259159. −0.507356
\(193\) −993614. −1.92010 −0.960051 0.279825i \(-0.909724\pi\)
−0.960051 + 0.279825i \(0.909724\pi\)
\(194\) 307526. 0.586648
\(195\) 27955.7 0.0526482
\(196\) −18424.6 −0.0342576
\(197\) −661275. −1.21399 −0.606997 0.794704i \(-0.707626\pi\)
−0.606997 + 0.794704i \(0.707626\pi\)
\(198\) 58191.9 0.105487
\(199\) −414819. −0.742551 −0.371276 0.928523i \(-0.621080\pi\)
−0.371276 + 0.928523i \(0.621080\pi\)
\(200\) 106679. 0.188584
\(201\) −48368.7 −0.0844450
\(202\) −905514. −1.56141
\(203\) 33829.1 0.0576169
\(204\) −51701.4 −0.0869815
\(205\) −113295. −0.188289
\(206\) −1.13028e6 −1.85574
\(207\) −206279. −0.334602
\(208\) −138845. −0.222522
\(209\) −241842. −0.382972
\(210\) 141009. 0.220647
\(211\) −519315. −0.803017 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(212\) −92663.3 −0.141602
\(213\) 144494. 0.218223
\(214\) 407743. 0.608628
\(215\) 335309. 0.494708
\(216\) −124430. −0.181465
\(217\) −902943. −1.30170
\(218\) −174933. −0.249304
\(219\) −235584. −0.331921
\(220\) 9837.47 0.0137033
\(221\) 219477. 0.302279
\(222\) 220492. 0.300270
\(223\) 1.10323e6 1.48561 0.742804 0.669509i \(-0.233495\pi\)
0.742804 + 0.669509i \(0.233495\pi\)
\(224\) −123809. −0.164867
\(225\) 50625.0 0.0666667
\(226\) 589622. 0.767896
\(227\) 1.07714e6 1.38742 0.693712 0.720253i \(-0.255974\pi\)
0.693712 + 0.720253i \(0.255974\pi\)
\(228\) −58498.9 −0.0745265
\(229\) 669513. 0.843666 0.421833 0.906674i \(-0.361387\pi\)
0.421833 + 0.906674i \(0.361387\pi\)
\(230\) −378009. −0.471175
\(231\) −114947. −0.141732
\(232\) −54703.8 −0.0667264
\(233\) 100605. 0.121403 0.0607016 0.998156i \(-0.480666\pi\)
0.0607016 + 0.998156i \(0.480666\pi\)
\(234\) −59753.7 −0.0713387
\(235\) 602580. 0.711779
\(236\) −59199.2 −0.0691889
\(237\) 196370. 0.227093
\(238\) 1.10705e6 1.26684
\(239\) −867105. −0.981921 −0.490961 0.871182i \(-0.663354\pi\)
−0.490961 + 0.871182i \(0.663354\pi\)
\(240\) −251435. −0.281772
\(241\) −618174. −0.685595 −0.342798 0.939409i \(-0.611375\pi\)
−0.342798 + 0.939409i \(0.611375\pi\)
\(242\) −86928.7 −0.0954167
\(243\) −59049.0 −0.0641500
\(244\) 15273.6 0.0164235
\(245\) 141638. 0.150752
\(246\) 242161. 0.255133
\(247\) 248333. 0.258995
\(248\) 1.46012e6 1.50750
\(249\) 586590. 0.599565
\(250\) 92771.0 0.0938777
\(251\) 630505. 0.631691 0.315845 0.948811i \(-0.397712\pi\)
0.315845 + 0.948811i \(0.397712\pi\)
\(252\) −27804.5 −0.0275812
\(253\) 308145. 0.302659
\(254\) −1.14462e6 −1.11321
\(255\) 397452. 0.382767
\(256\) 316500. 0.301838
\(257\) 475889. 0.449441 0.224720 0.974423i \(-0.427853\pi\)
0.224720 + 0.974423i \(0.427853\pi\)
\(258\) −716704. −0.670333
\(259\) −435543. −0.403442
\(260\) −10101.5 −0.00926728
\(261\) −25959.9 −0.0235886
\(262\) −1.15578e6 −1.04022
\(263\) 415568. 0.370470 0.185235 0.982694i \(-0.440695\pi\)
0.185235 + 0.982694i \(0.440695\pi\)
\(264\) 185878. 0.164141
\(265\) 712344. 0.623125
\(266\) 1.25260e6 1.08544
\(267\) −255330. −0.219192
\(268\) 17477.5 0.0148642
\(269\) −1.86987e6 −1.57554 −0.787772 0.615967i \(-0.788765\pi\)
−0.787772 + 0.615967i \(0.788765\pi\)
\(270\) −108208. −0.0903338
\(271\) −1.23898e6 −1.02481 −0.512404 0.858744i \(-0.671245\pi\)
−0.512404 + 0.858744i \(0.671245\pi\)
\(272\) −1.97399e6 −1.61779
\(273\) 118032. 0.0958506
\(274\) 1.30525e6 1.05031
\(275\) −75625.0 −0.0603023
\(276\) 74536.7 0.0588976
\(277\) 846334. 0.662739 0.331370 0.943501i \(-0.392489\pi\)
0.331370 + 0.943501i \(0.392489\pi\)
\(278\) 359702. 0.279146
\(279\) 692905. 0.532922
\(280\) 450413. 0.343333
\(281\) −1.07789e6 −0.814347 −0.407174 0.913351i \(-0.633486\pi\)
−0.407174 + 0.913351i \(0.633486\pi\)
\(282\) −1.28798e6 −0.964465
\(283\) −2.59466e6 −1.92581 −0.962906 0.269836i \(-0.913031\pi\)
−0.962906 + 0.269836i \(0.913031\pi\)
\(284\) −52211.4 −0.0384122
\(285\) 449707. 0.327957
\(286\) 89261.7 0.0645283
\(287\) −478346. −0.342797
\(288\) 95009.5 0.0674972
\(289\) 1.70050e6 1.19765
\(290\) −47572.0 −0.0332167
\(291\) 466157. 0.322700
\(292\) 85125.7 0.0584256
\(293\) −155161. −0.105588 −0.0527939 0.998605i \(-0.516813\pi\)
−0.0527939 + 0.998605i \(0.516813\pi\)
\(294\) −302743. −0.204271
\(295\) 455091. 0.304469
\(296\) 704301. 0.467228
\(297\) 88209.0 0.0580259
\(298\) −365578. −0.238473
\(299\) −316415. −0.204682
\(300\) −18292.8 −0.0117348
\(301\) 1.41572e6 0.900659
\(302\) 1.15562e6 0.729121
\(303\) −1.37261e6 −0.858893
\(304\) −2.23352e6 −1.38614
\(305\) −117415. −0.0722724
\(306\) −849531. −0.518652
\(307\) −1.94896e6 −1.18020 −0.590102 0.807329i \(-0.700912\pi\)
−0.590102 + 0.807329i \(0.700912\pi\)
\(308\) 41535.1 0.0249481
\(309\) −1.71331e6 −1.02080
\(310\) 1.26976e6 0.750441
\(311\) 853359. 0.500300 0.250150 0.968207i \(-0.419520\pi\)
0.250150 + 0.968207i \(0.419520\pi\)
\(312\) −190866. −0.111005
\(313\) 594619. 0.343066 0.171533 0.985178i \(-0.445128\pi\)
0.171533 + 0.985178i \(0.445128\pi\)
\(314\) −451307. −0.258314
\(315\) 213745. 0.121372
\(316\) −70956.1 −0.0399735
\(317\) 3.25318e6 1.81827 0.909137 0.416497i \(-0.136743\pi\)
0.909137 + 0.416497i \(0.136743\pi\)
\(318\) −1.52260e6 −0.844339
\(319\) 38779.7 0.0213367
\(320\) −719886. −0.392996
\(321\) 618069. 0.334791
\(322\) −1.59600e6 −0.857815
\(323\) 3.53061e6 1.88297
\(324\) 21336.7 0.0112919
\(325\) 77654.6 0.0407811
\(326\) 2.95828e6 1.54168
\(327\) −265168. −0.137136
\(328\) 773516. 0.396995
\(329\) 2.54417e6 1.29585
\(330\) 161644. 0.0817100
\(331\) −2.05862e6 −1.03278 −0.516389 0.856354i \(-0.672724\pi\)
−0.516389 + 0.856354i \(0.672724\pi\)
\(332\) −211958. −0.105537
\(333\) 334229. 0.165171
\(334\) 3.86478e6 1.89565
\(335\) −134357. −0.0654108
\(336\) −1.06159e6 −0.512991
\(337\) 87610.8 0.0420226 0.0210113 0.999779i \(-0.493311\pi\)
0.0210113 + 0.999779i \(0.493311\pi\)
\(338\) 2.11284e6 1.00595
\(339\) 893766. 0.422400
\(340\) −143615. −0.0673756
\(341\) −1.03508e6 −0.482046
\(342\) −961224. −0.444385
\(343\) 2.37205e6 1.08865
\(344\) −2.28931e6 −1.04306
\(345\) −572997. −0.259182
\(346\) 2.37853e6 1.06812
\(347\) −3.19161e6 −1.42294 −0.711470 0.702716i \(-0.751970\pi\)
−0.711470 + 0.702716i \(0.751970\pi\)
\(348\) 9380.36 0.00415213
\(349\) −1.36137e6 −0.598293 −0.299146 0.954207i \(-0.596702\pi\)
−0.299146 + 0.954207i \(0.596702\pi\)
\(350\) 391691. 0.170912
\(351\) −90576.4 −0.0392416
\(352\) −141928. −0.0610535
\(353\) −1.84373e6 −0.787517 −0.393758 0.919214i \(-0.628825\pi\)
−0.393758 + 0.919214i \(0.628825\pi\)
\(354\) −972731. −0.412558
\(355\) 401372. 0.169035
\(356\) 92261.0 0.0385827
\(357\) 1.67809e6 0.696860
\(358\) 4.86990e6 2.00823
\(359\) −345148. −0.141341 −0.0706707 0.997500i \(-0.522514\pi\)
−0.0706707 + 0.997500i \(0.522514\pi\)
\(360\) −345640. −0.140562
\(361\) 1.51869e6 0.613342
\(362\) −3.79909e6 −1.52373
\(363\) −131769. −0.0524864
\(364\) −42649.8 −0.0168719
\(365\) −654399. −0.257105
\(366\) 250967. 0.0979296
\(367\) 810586. 0.314148 0.157074 0.987587i \(-0.449794\pi\)
0.157074 + 0.987587i \(0.449794\pi\)
\(368\) 2.84586e6 1.09545
\(369\) 367075. 0.140343
\(370\) 612479. 0.232588
\(371\) 3.00761e6 1.13445
\(372\) −250374. −0.0938063
\(373\) −637008. −0.237068 −0.118534 0.992950i \(-0.537819\pi\)
−0.118534 + 0.992950i \(0.537819\pi\)
\(374\) 1.26905e6 0.469138
\(375\) 140625. 0.0516398
\(376\) −4.11409e6 −1.50073
\(377\) −39820.4 −0.0144295
\(378\) −456869. −0.164461
\(379\) −3.62357e6 −1.29580 −0.647901 0.761725i \(-0.724353\pi\)
−0.647901 + 0.761725i \(0.724353\pi\)
\(380\) −162497. −0.0577280
\(381\) −1.73504e6 −0.612347
\(382\) −4.20147e6 −1.47314
\(383\) 419232. 0.146035 0.0730175 0.997331i \(-0.476737\pi\)
0.0730175 + 0.997331i \(0.476737\pi\)
\(384\) 1.87653e6 0.649422
\(385\) −319298. −0.109786
\(386\) 5.89943e6 2.01531
\(387\) −1.08640e6 −0.368734
\(388\) −168441. −0.0568026
\(389\) −662976. −0.222139 −0.111069 0.993813i \(-0.535428\pi\)
−0.111069 + 0.993813i \(0.535428\pi\)
\(390\) −165982. −0.0552587
\(391\) −4.49854e6 −1.48809
\(392\) −967027. −0.317851
\(393\) −1.75197e6 −0.572197
\(394\) 3.92622e6 1.27419
\(395\) 545471. 0.175905
\(396\) −31873.4 −0.0102139
\(397\) 3.68928e6 1.17480 0.587402 0.809295i \(-0.300151\pi\)
0.587402 + 0.809295i \(0.300151\pi\)
\(398\) 2.46293e6 0.779370
\(399\) 1.89872e6 0.597075
\(400\) −698431. −0.218260
\(401\) 4.03113e6 1.25189 0.625945 0.779867i \(-0.284713\pi\)
0.625945 + 0.779867i \(0.284713\pi\)
\(402\) 287181. 0.0886322
\(403\) 1.06286e6 0.325997
\(404\) 495977. 0.151185
\(405\) −164025. −0.0496904
\(406\) −200855. −0.0604738
\(407\) −499280. −0.149403
\(408\) −2.71359e6 −0.807036
\(409\) 4.07904e6 1.20573 0.602864 0.797844i \(-0.294026\pi\)
0.602864 + 0.797844i \(0.294026\pi\)
\(410\) 672671. 0.197625
\(411\) 1.97853e6 0.577748
\(412\) 619086. 0.179683
\(413\) 1.92145e6 0.554312
\(414\) 1.22475e6 0.351193
\(415\) 1.62942e6 0.464421
\(416\) 145737. 0.0412891
\(417\) 545247. 0.153551
\(418\) 1.43590e6 0.401961
\(419\) −1.22736e6 −0.341537 −0.170769 0.985311i \(-0.554625\pi\)
−0.170769 + 0.985311i \(0.554625\pi\)
\(420\) −77234.6 −0.0213643
\(421\) −16159.8 −0.00444357 −0.00222178 0.999998i \(-0.500707\pi\)
−0.00222178 + 0.999998i \(0.500707\pi\)
\(422\) 3.08335e6 0.842834
\(423\) −1.95236e6 −0.530528
\(424\) −4.86350e6 −1.31381
\(425\) 1.10403e6 0.296490
\(426\) −857911. −0.229044
\(427\) −495739. −0.131578
\(428\) −223333. −0.0589309
\(429\) 135305. 0.0354954
\(430\) −1.99084e6 −0.519238
\(431\) 185090. 0.0479944 0.0239972 0.999712i \(-0.492361\pi\)
0.0239972 + 0.999712i \(0.492361\pi\)
\(432\) 814650. 0.210021
\(433\) −2.83808e6 −0.727453 −0.363727 0.931506i \(-0.618496\pi\)
−0.363727 + 0.931506i \(0.618496\pi\)
\(434\) 5.36108e6 1.36624
\(435\) −72111.0 −0.0182717
\(436\) 95815.6 0.0241390
\(437\) −5.08999e6 −1.27501
\(438\) 1.39874e6 0.348379
\(439\) −1.17189e6 −0.290218 −0.145109 0.989416i \(-0.546353\pi\)
−0.145109 + 0.989416i \(0.546353\pi\)
\(440\) 516326. 0.127143
\(441\) −458907. −0.112364
\(442\) −1.30311e6 −0.317268
\(443\) −6.32946e6 −1.53235 −0.766174 0.642634i \(-0.777842\pi\)
−0.766174 + 0.642634i \(0.777842\pi\)
\(444\) −120770. −0.0290738
\(445\) −709251. −0.169785
\(446\) −6.55026e6 −1.55927
\(447\) −554153. −0.131178
\(448\) −3.03945e6 −0.715484
\(449\) −2.81992e6 −0.660116 −0.330058 0.943961i \(-0.607068\pi\)
−0.330058 + 0.943961i \(0.607068\pi\)
\(450\) −300578. −0.0699723
\(451\) −548347. −0.126945
\(452\) −322953. −0.0743521
\(453\) 1.75173e6 0.401071
\(454\) −6.39537e6 −1.45622
\(455\) 327868. 0.0742455
\(456\) −3.07036e6 −0.691475
\(457\) 624549. 0.139887 0.0699433 0.997551i \(-0.477718\pi\)
0.0699433 + 0.997551i \(0.477718\pi\)
\(458\) −3.97513e6 −0.885498
\(459\) −1.28774e6 −0.285297
\(460\) 207046. 0.0456219
\(461\) 3.84656e6 0.842985 0.421492 0.906832i \(-0.361506\pi\)
0.421492 + 0.906832i \(0.361506\pi\)
\(462\) 682483. 0.148760
\(463\) −7.90622e6 −1.71402 −0.857011 0.515298i \(-0.827681\pi\)
−0.857011 + 0.515298i \(0.827681\pi\)
\(464\) 358148. 0.0772267
\(465\) 1.92474e6 0.412799
\(466\) −597327. −0.127423
\(467\) 7.13863e6 1.51469 0.757343 0.653017i \(-0.226497\pi\)
0.757343 + 0.653017i \(0.226497\pi\)
\(468\) 32728.8 0.00690742
\(469\) −567274. −0.119086
\(470\) −3.57772e6 −0.747071
\(471\) −684105. −0.142092
\(472\) −3.10711e6 −0.641951
\(473\) 1.62290e6 0.333532
\(474\) −1.16591e6 −0.238353
\(475\) 1.24919e6 0.254035
\(476\) −606361. −0.122663
\(477\) −2.30799e6 −0.464450
\(478\) 5.14830e6 1.03061
\(479\) −6.55862e6 −1.30609 −0.653046 0.757319i \(-0.726509\pi\)
−0.653046 + 0.757319i \(0.726509\pi\)
\(480\) 263915. 0.0522831
\(481\) 512680. 0.101038
\(482\) 3.67031e6 0.719590
\(483\) −2.41927e6 −0.471863
\(484\) 47613.4 0.00923880
\(485\) 1.29488e6 0.249963
\(486\) 350594. 0.0673309
\(487\) −6.92845e6 −1.32377 −0.661887 0.749604i \(-0.730244\pi\)
−0.661887 + 0.749604i \(0.730244\pi\)
\(488\) 801643. 0.152381
\(489\) 4.48425e6 0.848042
\(490\) −840953. −0.158227
\(491\) −9.60498e6 −1.79801 −0.899006 0.437936i \(-0.855710\pi\)
−0.899006 + 0.437936i \(0.855710\pi\)
\(492\) −132639. −0.0247035
\(493\) −566136. −0.104907
\(494\) −1.47444e6 −0.271838
\(495\) 245025. 0.0449467
\(496\) −9.55944e6 −1.74473
\(497\) 1.69465e6 0.307743
\(498\) −3.48279e6 −0.629294
\(499\) 6.72389e6 1.20884 0.604421 0.796665i \(-0.293405\pi\)
0.604421 + 0.796665i \(0.293405\pi\)
\(500\) −50813.4 −0.00908977
\(501\) 5.85835e6 1.04275
\(502\) −3.74353e6 −0.663012
\(503\) −142175. −0.0250555 −0.0125277 0.999922i \(-0.503988\pi\)
−0.0125277 + 0.999922i \(0.503988\pi\)
\(504\) −1.45934e6 −0.255905
\(505\) −3.81279e6 −0.665296
\(506\) −1.82956e6 −0.317666
\(507\) 3.20270e6 0.553346
\(508\) 626940. 0.107787
\(509\) −1.05255e7 −1.80072 −0.900362 0.435143i \(-0.856698\pi\)
−0.900362 + 0.435143i \(0.856698\pi\)
\(510\) −2.35981e6 −0.401746
\(511\) −2.76296e6 −0.468082
\(512\) 4.79293e6 0.808027
\(513\) −1.45705e6 −0.244445
\(514\) −2.82552e6 −0.471726
\(515\) −4.75919e6 −0.790705
\(516\) 392560. 0.0649055
\(517\) 2.91649e6 0.479881
\(518\) 2.58597e6 0.423446
\(519\) 3.60544e6 0.587544
\(520\) −530184. −0.0859841
\(521\) 2.99141e6 0.482816 0.241408 0.970424i \(-0.422391\pi\)
0.241408 + 0.970424i \(0.422391\pi\)
\(522\) 154133. 0.0247582
\(523\) −5.44567e6 −0.870557 −0.435279 0.900296i \(-0.643350\pi\)
−0.435279 + 0.900296i \(0.643350\pi\)
\(524\) 633057. 0.100720
\(525\) 593737. 0.0940147
\(526\) −2.46737e6 −0.388839
\(527\) 1.51109e7 2.37009
\(528\) −1.21695e6 −0.189971
\(529\) 49092.3 0.00762736
\(530\) −4.22943e6 −0.654022
\(531\) −1.47449e6 −0.226938
\(532\) −686083. −0.105099
\(533\) 563064. 0.0858499
\(534\) 1.51598e6 0.230060
\(535\) 1.71686e6 0.259328
\(536\) 917320. 0.137914
\(537\) 7.38194e6 1.10468
\(538\) 1.11021e7 1.65367
\(539\) 685528. 0.101637
\(540\) 59268.7 0.00874664
\(541\) −340079. −0.0499559 −0.0249780 0.999688i \(-0.507952\pi\)
−0.0249780 + 0.999688i \(0.507952\pi\)
\(542\) 7.35628e6 1.07562
\(543\) −5.75877e6 −0.838166
\(544\) 2.07197e6 0.300183
\(545\) −736577. −0.106225
\(546\) −700799. −0.100603
\(547\) 8.20219e6 1.17209 0.586046 0.810278i \(-0.300684\pi\)
0.586046 + 0.810278i \(0.300684\pi\)
\(548\) −714922. −0.101697
\(549\) 380423. 0.0538687
\(550\) 449012. 0.0632923
\(551\) −640569. −0.0898849
\(552\) 3.91211e6 0.546466
\(553\) 2.30305e6 0.320251
\(554\) −5.02498e6 −0.695600
\(555\) 928414. 0.127941
\(556\) −197019. −0.0270285
\(557\) 6.21617e6 0.848956 0.424478 0.905438i \(-0.360458\pi\)
0.424478 + 0.905438i \(0.360458\pi\)
\(558\) −4.11402e6 −0.559346
\(559\) −1.66645e6 −0.225560
\(560\) −2.94887e6 −0.397361
\(561\) 1.92367e6 0.258061
\(562\) 6.39982e6 0.854726
\(563\) −7.83265e6 −1.04145 −0.520724 0.853725i \(-0.674338\pi\)
−0.520724 + 0.853725i \(0.674338\pi\)
\(564\) 705464. 0.0933850
\(565\) 2.48268e6 0.327190
\(566\) 1.54054e7 2.02130
\(567\) −692535. −0.0904657
\(568\) −2.74035e6 −0.356398
\(569\) −479265. −0.0620576 −0.0310288 0.999518i \(-0.509878\pi\)
−0.0310288 + 0.999518i \(0.509878\pi\)
\(570\) −2.67007e6 −0.344219
\(571\) −5.13344e6 −0.658898 −0.329449 0.944173i \(-0.606863\pi\)
−0.329449 + 0.944173i \(0.606863\pi\)
\(572\) −48891.2 −0.00624800
\(573\) −6.36871e6 −0.810336
\(574\) 2.84010e6 0.359794
\(575\) −1.59166e6 −0.200761
\(576\) 2.33243e6 0.292922
\(577\) 7.94103e6 0.992973 0.496486 0.868045i \(-0.334623\pi\)
0.496486 + 0.868045i \(0.334623\pi\)
\(578\) −1.00964e7 −1.25704
\(579\) 8.94252e6 1.10857
\(580\) 26056.5 0.00321623
\(581\) 6.87961e6 0.845519
\(582\) −2.76773e6 −0.338701
\(583\) 3.44774e6 0.420111
\(584\) 4.46788e6 0.542088
\(585\) −251601. −0.0303964
\(586\) 921244. 0.110823
\(587\) −1.01573e7 −1.21669 −0.608347 0.793671i \(-0.708167\pi\)
−0.608347 + 0.793671i \(0.708167\pi\)
\(588\) 165821. 0.0197787
\(589\) 1.70976e7 2.03071
\(590\) −2.70203e6 −0.319566
\(591\) 5.95147e6 0.700900
\(592\) −4.61108e6 −0.540752
\(593\) −9.94917e6 −1.16185 −0.580925 0.813957i \(-0.697309\pi\)
−0.580925 + 0.813957i \(0.697309\pi\)
\(594\) −523727. −0.0609030
\(595\) 4.66137e6 0.539785
\(596\) 200238. 0.0230903
\(597\) 3.73338e6 0.428712
\(598\) 1.87866e6 0.214831
\(599\) 1.18037e7 1.34416 0.672080 0.740478i \(-0.265401\pi\)
0.672080 + 0.740478i \(0.265401\pi\)
\(600\) −960111. −0.108879
\(601\) −8.52084e6 −0.962268 −0.481134 0.876647i \(-0.659775\pi\)
−0.481134 + 0.876647i \(0.659775\pi\)
\(602\) −8.40560e6 −0.945317
\(603\) 435318. 0.0487544
\(604\) −632969. −0.0705977
\(605\) −366025. −0.0406558
\(606\) 8.14963e6 0.901480
\(607\) 1.48363e7 1.63438 0.817191 0.576368i \(-0.195530\pi\)
0.817191 + 0.576368i \(0.195530\pi\)
\(608\) 2.34439e6 0.257199
\(609\) −304462. −0.0332652
\(610\) 697131. 0.0758559
\(611\) −2.99476e6 −0.324533
\(612\) 465313. 0.0502188
\(613\) 1.31231e7 1.41054 0.705272 0.708936i \(-0.250825\pi\)
0.705272 + 0.708936i \(0.250825\pi\)
\(614\) 1.15716e7 1.23872
\(615\) 1.01965e6 0.108709
\(616\) 2.18000e6 0.231475
\(617\) 4.62938e6 0.489564 0.244782 0.969578i \(-0.421284\pi\)
0.244782 + 0.969578i \(0.421284\pi\)
\(618\) 1.01725e7 1.07141
\(619\) 9.49235e6 0.995743 0.497872 0.867251i \(-0.334115\pi\)
0.497872 + 0.867251i \(0.334115\pi\)
\(620\) −695484. −0.0726621
\(621\) 1.85651e6 0.193183
\(622\) −5.06669e6 −0.525107
\(623\) −2.99455e6 −0.309109
\(624\) 1.24961e6 0.128473
\(625\) 390625. 0.0400000
\(626\) −3.53045e6 −0.360077
\(627\) 2.17658e6 0.221109
\(628\) 247194. 0.0250115
\(629\) 7.28888e6 0.734572
\(630\) −1.26908e6 −0.127391
\(631\) 1.14798e7 1.14779 0.573893 0.818930i \(-0.305432\pi\)
0.573893 + 0.818930i \(0.305432\pi\)
\(632\) −3.72418e6 −0.370884
\(633\) 4.67384e6 0.463622
\(634\) −1.93152e7 −1.90843
\(635\) −4.81957e6 −0.474322
\(636\) 833970. 0.0817537
\(637\) −703926. −0.0687351
\(638\) −230248. −0.0223947
\(639\) −1.30045e6 −0.125991
\(640\) 5.21257e6 0.503040
\(641\) −1.85269e7 −1.78097 −0.890486 0.455010i \(-0.849636\pi\)
−0.890486 + 0.455010i \(0.849636\pi\)
\(642\) −3.66969e6 −0.351392
\(643\) 1.34056e7 1.27867 0.639337 0.768927i \(-0.279209\pi\)
0.639337 + 0.768927i \(0.279209\pi\)
\(644\) 874176. 0.0830586
\(645\) −3.01778e6 −0.285620
\(646\) −2.09624e7 −1.97633
\(647\) 8.21634e6 0.771645 0.385823 0.922573i \(-0.373918\pi\)
0.385823 + 0.922573i \(0.373918\pi\)
\(648\) 1.11987e6 0.104769
\(649\) 2.20264e6 0.205273
\(650\) −461062. −0.0428032
\(651\) 8.12649e6 0.751537
\(652\) −1.62034e6 −0.149275
\(653\) 4.81154e6 0.441571 0.220786 0.975322i \(-0.429138\pi\)
0.220786 + 0.975322i \(0.429138\pi\)
\(654\) 1.57439e6 0.143936
\(655\) −4.86659e6 −0.443222
\(656\) −5.06424e6 −0.459467
\(657\) 2.12025e6 0.191635
\(658\) −1.51056e7 −1.36011
\(659\) −6.57414e6 −0.589693 −0.294846 0.955545i \(-0.595268\pi\)
−0.294846 + 0.955545i \(0.595268\pi\)
\(660\) −88537.2 −0.00791163
\(661\) 1.07312e7 0.955315 0.477657 0.878546i \(-0.341486\pi\)
0.477657 + 0.878546i \(0.341486\pi\)
\(662\) 1.22228e7 1.08399
\(663\) −1.97529e6 −0.174521
\(664\) −1.11248e7 −0.979199
\(665\) 5.27423e6 0.462492
\(666\) −1.98443e6 −0.173361
\(667\) 816184. 0.0710352
\(668\) −2.11685e6 −0.183548
\(669\) −9.92908e6 −0.857716
\(670\) 797726. 0.0686542
\(671\) −568287. −0.0487260
\(672\) 1.11428e6 0.0951859
\(673\) 7.52813e6 0.640693 0.320346 0.947301i \(-0.396201\pi\)
0.320346 + 0.947301i \(0.396201\pi\)
\(674\) −520176. −0.0441063
\(675\) −455625. −0.0384900
\(676\) −1.15726e6 −0.0974014
\(677\) −1.63689e7 −1.37262 −0.686308 0.727311i \(-0.740770\pi\)
−0.686308 + 0.727311i \(0.740770\pi\)
\(678\) −5.30659e6 −0.443345
\(679\) 5.46715e6 0.455079
\(680\) −7.53774e6 −0.625128
\(681\) −9.69429e6 −0.801029
\(682\) 6.14563e6 0.505948
\(683\) 2.56590e6 0.210469 0.105234 0.994447i \(-0.466441\pi\)
0.105234 + 0.994447i \(0.466441\pi\)
\(684\) 526490. 0.0430279
\(685\) 5.49592e6 0.447522
\(686\) −1.40837e7 −1.14263
\(687\) −6.02562e6 −0.487091
\(688\) 1.49882e7 1.20720
\(689\) −3.54028e6 −0.284112
\(690\) 3.40208e6 0.272033
\(691\) −8.83595e6 −0.703977 −0.351988 0.936004i \(-0.614494\pi\)
−0.351988 + 0.936004i \(0.614494\pi\)
\(692\) −1.30279e6 −0.103421
\(693\) 1.03453e6 0.0818293
\(694\) 1.89497e7 1.49350
\(695\) 1.51458e6 0.118940
\(696\) 492334. 0.0385245
\(697\) 8.00520e6 0.624152
\(698\) 8.08294e6 0.627959
\(699\) −905446. −0.0700922
\(700\) −214541. −0.0165487
\(701\) 43232.3 0.00332287 0.00166143 0.999999i \(-0.499471\pi\)
0.00166143 + 0.999999i \(0.499471\pi\)
\(702\) 537783. 0.0411874
\(703\) 8.24719e6 0.629387
\(704\) −3.48425e6 −0.264958
\(705\) −5.42322e6 −0.410946
\(706\) 1.09468e7 0.826565
\(707\) −1.60981e7 −1.21123
\(708\) 532793. 0.0399462
\(709\) −5.33741e6 −0.398763 −0.199382 0.979922i \(-0.563893\pi\)
−0.199382 + 0.979922i \(0.563893\pi\)
\(710\) −2.38309e6 −0.177416
\(711\) −1.76733e6 −0.131112
\(712\) 4.84238e6 0.357980
\(713\) −2.17850e7 −1.60485
\(714\) −9.96341e6 −0.731413
\(715\) 375848. 0.0274946
\(716\) −2.66739e6 −0.194448
\(717\) 7.80394e6 0.566913
\(718\) 2.04926e6 0.148350
\(719\) −9.14720e6 −0.659881 −0.329941 0.944002i \(-0.607029\pi\)
−0.329941 + 0.944002i \(0.607029\pi\)
\(720\) 2.26292e6 0.162681
\(721\) −2.00939e7 −1.43955
\(722\) −9.01701e6 −0.643754
\(723\) 5.56356e6 0.395829
\(724\) 2.08087e6 0.147536
\(725\) −200308. −0.0141532
\(726\) 782358. 0.0550889
\(727\) −1.61992e6 −0.113673 −0.0568367 0.998383i \(-0.518101\pi\)
−0.0568367 + 0.998383i \(0.518101\pi\)
\(728\) −2.23850e6 −0.156541
\(729\) 531441. 0.0370370
\(730\) 3.88539e6 0.269853
\(731\) −2.36923e7 −1.63989
\(732\) −137462. −0.00948211
\(733\) 2.29112e7 1.57503 0.787515 0.616296i \(-0.211367\pi\)
0.787515 + 0.616296i \(0.211367\pi\)
\(734\) −4.81273e6 −0.329724
\(735\) −1.27474e6 −0.0870370
\(736\) −2.98711e6 −0.203262
\(737\) −650290. −0.0441000
\(738\) −2.17945e6 −0.147301
\(739\) 5.61873e6 0.378466 0.189233 0.981932i \(-0.439400\pi\)
0.189233 + 0.981932i \(0.439400\pi\)
\(740\) −335473. −0.0225205
\(741\) −2.23500e6 −0.149531
\(742\) −1.78572e7 −1.19070
\(743\) −2.58649e7 −1.71885 −0.859426 0.511261i \(-0.829179\pi\)
−0.859426 + 0.511261i \(0.829179\pi\)
\(744\) −1.31411e7 −0.870358
\(745\) −1.53931e6 −0.101610
\(746\) 3.78214e6 0.248823
\(747\) −5.27931e6 −0.346159
\(748\) −695097. −0.0454246
\(749\) 7.24879e6 0.472129
\(750\) −834939. −0.0542003
\(751\) −2.70229e7 −1.74837 −0.874183 0.485596i \(-0.838603\pi\)
−0.874183 + 0.485596i \(0.838603\pi\)
\(752\) 2.69351e7 1.73689
\(753\) −5.67455e6 −0.364707
\(754\) 236428. 0.0151450
\(755\) 4.86592e6 0.310669
\(756\) 250240. 0.0159240
\(757\) −1.63080e6 −0.103433 −0.0517166 0.998662i \(-0.516469\pi\)
−0.0517166 + 0.998662i \(0.516469\pi\)
\(758\) 2.15144e7 1.36005
\(759\) −2.77330e6 −0.174740
\(760\) −8.52877e6 −0.535614
\(761\) 1.46228e7 0.915311 0.457656 0.889130i \(-0.348689\pi\)
0.457656 + 0.889130i \(0.348689\pi\)
\(762\) 1.03016e7 0.642710
\(763\) −3.10992e6 −0.193392
\(764\) 2.30127e6 0.142638
\(765\) −3.57707e6 −0.220990
\(766\) −2.48912e6 −0.153276
\(767\) −2.26175e6 −0.138822
\(768\) −2.84850e6 −0.174266
\(769\) 2.12198e7 1.29398 0.646988 0.762500i \(-0.276028\pi\)
0.646988 + 0.762500i \(0.276028\pi\)
\(770\) 1.89579e6 0.115229
\(771\) −4.28300e6 −0.259485
\(772\) −3.23129e6 −0.195134
\(773\) −7.33809e6 −0.441707 −0.220853 0.975307i \(-0.570884\pi\)
−0.220853 + 0.975307i \(0.570884\pi\)
\(774\) 6.45034e6 0.387017
\(775\) 5.34649e6 0.319753
\(776\) −8.84074e6 −0.527029
\(777\) 3.91988e6 0.232927
\(778\) 3.93632e6 0.233153
\(779\) 9.05769e6 0.534778
\(780\) 90913.4 0.00535046
\(781\) 1.94264e6 0.113963
\(782\) 2.67094e7 1.56188
\(783\) 233640. 0.0136189
\(784\) 6.33116e6 0.367869
\(785\) −1.90029e6 −0.110064
\(786\) 1.04021e7 0.600569
\(787\) 6.32394e6 0.363958 0.181979 0.983302i \(-0.441750\pi\)
0.181979 + 0.983302i \(0.441750\pi\)
\(788\) −2.15050e6 −0.123374
\(789\) −3.74011e6 −0.213891
\(790\) −3.23865e6 −0.184627
\(791\) 1.04822e7 0.595678
\(792\) −1.67290e6 −0.0947669
\(793\) 583538. 0.0329524
\(794\) −2.19045e7 −1.23306
\(795\) −6.41109e6 −0.359761
\(796\) −1.34902e6 −0.0754631
\(797\) −2.37699e7 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(798\) −1.12734e7 −0.626680
\(799\) −4.25771e7 −2.35944
\(800\) 733098. 0.0404983
\(801\) 2.29797e6 0.126550
\(802\) −2.39342e7 −1.31396
\(803\) −3.16729e6 −0.173340
\(804\) −157298. −0.00858187
\(805\) −6.72018e6 −0.365503
\(806\) −6.31057e6 −0.342161
\(807\) 1.68288e7 0.909641
\(808\) 2.60317e7 1.40273
\(809\) 3.06904e7 1.64866 0.824331 0.566108i \(-0.191551\pi\)
0.824331 + 0.566108i \(0.191551\pi\)
\(810\) 973873. 0.0521543
\(811\) −1.80033e7 −0.961172 −0.480586 0.876948i \(-0.659576\pi\)
−0.480586 + 0.876948i \(0.659576\pi\)
\(812\) 110014. 0.00585542
\(813\) 1.11509e7 0.591673
\(814\) 2.96440e6 0.156811
\(815\) 1.24562e7 0.656890
\(816\) 1.77659e7 0.934034
\(817\) −2.68073e7 −1.40507
\(818\) −2.42187e7 −1.26551
\(819\) −1.06229e6 −0.0553394
\(820\) −368441. −0.0191352
\(821\) −1.63239e7 −0.845213 −0.422607 0.906313i \(-0.638885\pi\)
−0.422607 + 0.906313i \(0.638885\pi\)
\(822\) −1.17472e7 −0.606396
\(823\) 1.82069e7 0.936992 0.468496 0.883466i \(-0.344796\pi\)
0.468496 + 0.883466i \(0.344796\pi\)
\(824\) 3.24931e7 1.66715
\(825\) 680625. 0.0348155
\(826\) −1.14083e7 −0.581797
\(827\) −3.78345e7 −1.92364 −0.961820 0.273683i \(-0.911758\pi\)
−0.961820 + 0.273683i \(0.911758\pi\)
\(828\) −670830. −0.0340045
\(829\) 5.02252e6 0.253826 0.126913 0.991914i \(-0.459493\pi\)
0.126913 + 0.991914i \(0.459493\pi\)
\(830\) −9.67441e6 −0.487449
\(831\) −7.61701e6 −0.382633
\(832\) 3.57776e6 0.179185
\(833\) −1.00079e7 −0.499723
\(834\) −3.23732e6 −0.161165
\(835\) 1.62732e7 0.807712
\(836\) −786485. −0.0389202
\(837\) −6.23615e6 −0.307682
\(838\) 7.28728e6 0.358472
\(839\) −2.82732e7 −1.38666 −0.693331 0.720619i \(-0.743858\pi\)
−0.693331 + 0.720619i \(0.743858\pi\)
\(840\) −4.05371e6 −0.198223
\(841\) −2.04084e7 −0.994992
\(842\) 95946.5 0.00466390
\(843\) 9.70103e6 0.470164
\(844\) −1.68884e6 −0.0816080
\(845\) 8.89639e6 0.428620
\(846\) 1.15918e7 0.556834
\(847\) −1.54540e6 −0.0740174
\(848\) 3.18415e7 1.52056
\(849\) 2.33519e7 1.11187
\(850\) −6.55502e6 −0.311191
\(851\) −1.05082e7 −0.497399
\(852\) 469903. 0.0221773
\(853\) 7.27728e6 0.342449 0.171225 0.985232i \(-0.445228\pi\)
0.171225 + 0.985232i \(0.445228\pi\)
\(854\) 2.94338e6 0.138102
\(855\) −4.04736e6 −0.189346
\(856\) −1.17218e7 −0.546775
\(857\) 2.49180e7 1.15894 0.579469 0.814994i \(-0.303260\pi\)
0.579469 + 0.814994i \(0.303260\pi\)
\(858\) −803355. −0.0372554
\(859\) −3.01693e7 −1.39502 −0.697512 0.716573i \(-0.745710\pi\)
−0.697512 + 0.716573i \(0.745710\pi\)
\(860\) 1.09044e6 0.0502756
\(861\) 4.30511e6 0.197914
\(862\) −1.09894e6 −0.0503742
\(863\) 3.86426e7 1.76620 0.883099 0.469187i \(-0.155453\pi\)
0.883099 + 0.469187i \(0.155453\pi\)
\(864\) −855085. −0.0389695
\(865\) 1.00151e7 0.455110
\(866\) 1.68507e7 0.763524
\(867\) −1.53045e7 −0.691466
\(868\) −2.93642e6 −0.132288
\(869\) 2.64008e6 0.118595
\(870\) 428148. 0.0191776
\(871\) 667743. 0.0298239
\(872\) 5.02895e6 0.223968
\(873\) −4.19541e6 −0.186311
\(874\) 3.02210e7 1.33823
\(875\) 1.64927e6 0.0728235
\(876\) −766131. −0.0337321
\(877\) 1.35160e7 0.593404 0.296702 0.954970i \(-0.404113\pi\)
0.296702 + 0.954970i \(0.404113\pi\)
\(878\) 6.95790e6 0.304608
\(879\) 1.39645e6 0.0609611
\(880\) −3.38041e6 −0.147151
\(881\) 1.56537e7 0.679479 0.339739 0.940520i \(-0.389661\pi\)
0.339739 + 0.940520i \(0.389661\pi\)
\(882\) 2.72469e6 0.117936
\(883\) −7.31854e6 −0.315880 −0.157940 0.987449i \(-0.550485\pi\)
−0.157940 + 0.987449i \(0.550485\pi\)
\(884\) 713752. 0.0307197
\(885\) −4.09582e6 −0.175785
\(886\) 3.75802e7 1.60833
\(887\) 3.94393e6 0.168314 0.0841570 0.996453i \(-0.473180\pi\)
0.0841570 + 0.996453i \(0.473180\pi\)
\(888\) −6.33871e6 −0.269754
\(889\) −2.03488e7 −0.863545
\(890\) 4.21107e6 0.178204
\(891\) −793881. −0.0335013
\(892\) 3.58777e6 0.150978
\(893\) −4.81750e7 −2.02159
\(894\) 3.29020e6 0.137682
\(895\) 2.05054e7 0.855678
\(896\) 2.20082e7 0.915828
\(897\) 2.84773e6 0.118173
\(898\) 1.67428e7 0.692847
\(899\) −2.74162e6 −0.113138
\(900\) 164635. 0.00677512
\(901\) −5.03329e7 −2.06557
\(902\) 3.25573e6 0.133239
\(903\) −1.27415e7 −0.519996
\(904\) −1.69504e7 −0.689857
\(905\) −1.59966e7 −0.649241
\(906\) −1.04006e7 −0.420958
\(907\) −2.09119e7 −0.844064 −0.422032 0.906581i \(-0.638683\pi\)
−0.422032 + 0.906581i \(0.638683\pi\)
\(908\) 3.50293e6 0.140999
\(909\) 1.23534e7 0.495882
\(910\) −1.94666e6 −0.0779269
\(911\) 2.63760e7 1.05296 0.526481 0.850187i \(-0.323511\pi\)
0.526481 + 0.850187i \(0.323511\pi\)
\(912\) 2.01017e7 0.800288
\(913\) 7.88638e6 0.313113
\(914\) −3.70816e6 −0.146823
\(915\) 1.05673e6 0.0417265
\(916\) 2.17729e6 0.0857390
\(917\) −2.05474e7 −0.806924
\(918\) 7.64578e6 0.299444
\(919\) −1.74127e7 −0.680109 −0.340054 0.940406i \(-0.610445\pi\)
−0.340054 + 0.940406i \(0.610445\pi\)
\(920\) 1.08670e7 0.423291
\(921\) 1.75406e7 0.681391
\(922\) −2.28383e7 −0.884783
\(923\) −1.99478e6 −0.0770709
\(924\) −373816. −0.0144038
\(925\) 2.57893e6 0.0991025
\(926\) 4.69419e7 1.79901
\(927\) 1.54198e7 0.589357
\(928\) −375924. −0.0143295
\(929\) 1.23992e7 0.471360 0.235680 0.971831i \(-0.424268\pi\)
0.235680 + 0.971831i \(0.424268\pi\)
\(930\) −1.14278e7 −0.433268
\(931\) −1.13237e7 −0.428166
\(932\) 327174. 0.0123378
\(933\) −7.68023e6 −0.288848
\(934\) −4.23845e7 −1.58979
\(935\) 5.34352e6 0.199893
\(936\) 1.71780e6 0.0640888
\(937\) 6.45952e6 0.240354 0.120177 0.992752i \(-0.461654\pi\)
0.120177 + 0.992752i \(0.461654\pi\)
\(938\) 3.36810e6 0.124991
\(939\) −5.35157e6 −0.198069
\(940\) 1.95962e6 0.0723357
\(941\) −4.98442e6 −0.183502 −0.0917510 0.995782i \(-0.529246\pi\)
−0.0917510 + 0.995782i \(0.529246\pi\)
\(942\) 4.06176e6 0.149138
\(943\) −1.15409e7 −0.422630
\(944\) 2.03424e7 0.742971
\(945\) −1.92371e6 −0.0700744
\(946\) −9.63569e6 −0.350070
\(947\) 8.49575e6 0.307841 0.153921 0.988083i \(-0.450810\pi\)
0.153921 + 0.988083i \(0.450810\pi\)
\(948\) 638605. 0.0230787
\(949\) 3.25230e6 0.117226
\(950\) −7.41685e6 −0.266631
\(951\) −2.92786e7 −1.04978
\(952\) −3.18253e7 −1.13810
\(953\) 3.18137e7 1.13470 0.567351 0.823476i \(-0.307968\pi\)
0.567351 + 0.823476i \(0.307968\pi\)
\(954\) 1.37034e7 0.487479
\(955\) −1.76909e7 −0.627684
\(956\) −2.81987e6 −0.0997895
\(957\) −349017. −0.0123188
\(958\) 3.89408e7 1.37085
\(959\) 2.32045e7 0.814753
\(960\) 6.47897e6 0.226897
\(961\) 4.45484e7 1.55605
\(962\) −3.04396e6 −0.106048
\(963\) −5.56262e6 −0.193292
\(964\) −2.01034e6 −0.0696748
\(965\) 2.48403e7 0.858696
\(966\) 1.43640e7 0.495260
\(967\) −4.98824e6 −0.171546 −0.0857732 0.996315i \(-0.527336\pi\)
−0.0857732 + 0.996315i \(0.527336\pi\)
\(968\) 2.49902e6 0.0857198
\(969\) −3.17754e7 −1.08713
\(970\) −7.68815e6 −0.262357
\(971\) 1.81815e7 0.618845 0.309422 0.950925i \(-0.399864\pi\)
0.309422 + 0.950925i \(0.399864\pi\)
\(972\) −192031. −0.00651936
\(973\) 6.39473e6 0.216541
\(974\) 4.11366e7 1.38941
\(975\) −698892. −0.0235450
\(976\) −5.24838e6 −0.176360
\(977\) 1.72113e7 0.576870 0.288435 0.957499i \(-0.406865\pi\)
0.288435 + 0.957499i \(0.406865\pi\)
\(978\) −2.66245e7 −0.890091
\(979\) −3.43277e6 −0.114469
\(980\) 460615. 0.0153205
\(981\) 2.38651e6 0.0791755
\(982\) 5.70281e7 1.88717
\(983\) −4.14848e7 −1.36932 −0.684661 0.728861i \(-0.740050\pi\)
−0.684661 + 0.728861i \(0.740050\pi\)
\(984\) −6.96164e6 −0.229205
\(985\) 1.65319e7 0.542915
\(986\) 3.36134e6 0.110108
\(987\) −2.28975e7 −0.748162
\(988\) 807593. 0.0263209
\(989\) 3.41566e7 1.11041
\(990\) −1.45480e6 −0.0471753
\(991\) −1.58584e7 −0.512951 −0.256475 0.966551i \(-0.582561\pi\)
−0.256475 + 0.966551i \(0.582561\pi\)
\(992\) 1.00339e7 0.323736
\(993\) 1.85276e7 0.596274
\(994\) −1.00617e7 −0.323002
\(995\) 1.03705e7 0.332079
\(996\) 1.90762e6 0.0609319
\(997\) −4.08647e7 −1.30200 −0.651000 0.759078i \(-0.725650\pi\)
−0.651000 + 0.759078i \(0.725650\pi\)
\(998\) −3.99221e7 −1.26878
\(999\) −3.00806e6 −0.0953615
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 165.6.a.g.1.2 5
3.2 odd 2 495.6.a.i.1.4 5
5.4 even 2 825.6.a.k.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.g.1.2 5 1.1 even 1 trivial
495.6.a.i.1.4 5 3.2 odd 2
825.6.a.k.1.4 5 5.4 even 2