Properties

Label 722.6.a.r.1.10
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: \( x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} + \cdots - 34\!\cdots\!72 \) 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 1.10
Root \(-10.7498\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +10.4025 q^{3} +16.0000 q^{4} +95.5276 q^{5} +41.6100 q^{6} +232.219 q^{7} +64.0000 q^{8} -134.788 q^{9} +382.111 q^{10} +345.880 q^{11} +166.440 q^{12} +506.327 q^{13} +928.874 q^{14} +993.725 q^{15} +256.000 q^{16} -797.089 q^{17} -539.153 q^{18} +1528.44 q^{20} +2415.65 q^{21} +1383.52 q^{22} -1190.67 q^{23} +665.760 q^{24} +6000.53 q^{25} +2025.31 q^{26} -3929.94 q^{27} +3715.50 q^{28} -425.364 q^{29} +3974.90 q^{30} +6647.85 q^{31} +1024.00 q^{32} +3598.01 q^{33} -3188.36 q^{34} +22183.3 q^{35} -2156.61 q^{36} -7850.99 q^{37} +5267.06 q^{39} +6113.77 q^{40} -16080.1 q^{41} +9662.61 q^{42} +12922.4 q^{43} +5534.07 q^{44} -12876.0 q^{45} -4762.67 q^{46} -21190.2 q^{47} +2663.04 q^{48} +37118.5 q^{49} +24002.1 q^{50} -8291.71 q^{51} +8101.23 q^{52} -29091.8 q^{53} -15719.8 q^{54} +33041.1 q^{55} +14862.0 q^{56} -1701.45 q^{58} +9346.25 q^{59} +15899.6 q^{60} -8320.98 q^{61} +26591.4 q^{62} -31300.3 q^{63} +4096.00 q^{64} +48368.2 q^{65} +14392.0 q^{66} -14347.1 q^{67} -12753.4 q^{68} -12385.9 q^{69} +88733.1 q^{70} -31200.2 q^{71} -8626.44 q^{72} +1992.28 q^{73} -31404.0 q^{74} +62420.4 q^{75} +80319.7 q^{77} +21068.2 q^{78} -9623.27 q^{79} +24455.1 q^{80} -8127.64 q^{81} -64320.3 q^{82} -34179.5 q^{83} +38650.4 q^{84} -76144.0 q^{85} +51689.4 q^{86} -4424.84 q^{87} +22136.3 q^{88} -111322. q^{89} -51504.0 q^{90} +117578. q^{91} -19050.7 q^{92} +69154.2 q^{93} -84760.6 q^{94} +10652.2 q^{96} +96994.8 q^{97} +148474. q^{98} -46620.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} + 960 q^{8} + 2127 q^{9} + 432 q^{10} + 126 q^{11} - 114 q^{13} + 336 q^{14} + 3840 q^{16} + 4119 q^{17} + 8508 q^{18} + 1728 q^{20} + 3408 q^{21} + 504 q^{22}+ \cdots - 149895 q^{99}+O(q^{100}) \) 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
\(3\) 10.4025 0.667320 0.333660 0.942693i \(-0.391716\pi\)
0.333660 + 0.942693i \(0.391716\pi\)
\(4\) 16.0000 0.500000
\(5\) 95.5276 1.70885 0.854425 0.519575i \(-0.173910\pi\)
0.854425 + 0.519575i \(0.173910\pi\)
\(6\) 41.6100 0.471867
\(7\) 232.219 1.79123 0.895616 0.444828i \(-0.146735\pi\)
0.895616 + 0.444828i \(0.146735\pi\)
\(8\) 64.0000 0.353553
\(9\) −134.788 −0.554684
\(10\) 382.111 1.20834
\(11\) 345.880 0.861873 0.430936 0.902382i \(-0.358183\pi\)
0.430936 + 0.902382i \(0.358183\pi\)
\(12\) 166.440 0.333660
\(13\) 506.327 0.830946 0.415473 0.909606i \(-0.363616\pi\)
0.415473 + 0.909606i \(0.363616\pi\)
\(14\) 928.874 1.26659
\(15\) 993.725 1.14035
\(16\) 256.000 0.250000
\(17\) −797.089 −0.668936 −0.334468 0.942407i \(-0.608557\pi\)
−0.334468 + 0.942407i \(0.608557\pi\)
\(18\) −539.153 −0.392221
\(19\) 0 0
\(20\) 1528.44 0.854425
\(21\) 2415.65 1.19533
\(22\) 1383.52 0.609436
\(23\) −1190.67 −0.469322 −0.234661 0.972077i \(-0.575398\pi\)
−0.234661 + 0.972077i \(0.575398\pi\)
\(24\) 665.760 0.235933
\(25\) 6000.53 1.92017
\(26\) 2025.31 0.587567
\(27\) −3929.94 −1.03747
\(28\) 3715.50 0.895616
\(29\) −425.364 −0.0939216 −0.0469608 0.998897i \(-0.514954\pi\)
−0.0469608 + 0.998897i \(0.514954\pi\)
\(30\) 3974.90 0.806349
\(31\) 6647.85 1.24244 0.621222 0.783635i \(-0.286637\pi\)
0.621222 + 0.783635i \(0.286637\pi\)
\(32\) 1024.00 0.176777
\(33\) 3598.01 0.575145
\(34\) −3188.36 −0.473009
\(35\) 22183.3 3.06095
\(36\) −2156.61 −0.277342
\(37\) −7850.99 −0.942801 −0.471401 0.881919i \(-0.656251\pi\)
−0.471401 + 0.881919i \(0.656251\pi\)
\(38\) 0 0
\(39\) 5267.06 0.554507
\(40\) 6113.77 0.604170
\(41\) −16080.1 −1.49392 −0.746962 0.664867i \(-0.768488\pi\)
−0.746962 + 0.664867i \(0.768488\pi\)
\(42\) 9662.61 0.845223
\(43\) 12922.4 1.06579 0.532894 0.846182i \(-0.321105\pi\)
0.532894 + 0.846182i \(0.321105\pi\)
\(44\) 5534.07 0.430936
\(45\) −12876.0 −0.947871
\(46\) −4762.67 −0.331861
\(47\) −21190.2 −1.39923 −0.699616 0.714519i \(-0.746645\pi\)
−0.699616 + 0.714519i \(0.746645\pi\)
\(48\) 2663.04 0.166830
\(49\) 37118.5 2.20851
\(50\) 24002.1 1.35776
\(51\) −8291.71 −0.446395
\(52\) 8101.23 0.415473
\(53\) −29091.8 −1.42259 −0.711296 0.702893i \(-0.751891\pi\)
−0.711296 + 0.702893i \(0.751891\pi\)
\(54\) −15719.8 −0.733603
\(55\) 33041.1 1.47281
\(56\) 14862.0 0.633296
\(57\) 0 0
\(58\) −1701.45 −0.0664126
\(59\) 9346.25 0.349549 0.174774 0.984609i \(-0.444080\pi\)
0.174774 + 0.984609i \(0.444080\pi\)
\(60\) 15899.6 0.570175
\(61\) −8320.98 −0.286319 −0.143159 0.989700i \(-0.545726\pi\)
−0.143159 + 0.989700i \(0.545726\pi\)
\(62\) 26591.4 0.878541
\(63\) −31300.3 −0.993567
\(64\) 4096.00 0.125000
\(65\) 48368.2 1.41996
\(66\) 14392.0 0.406689
\(67\) −14347.1 −0.390459 −0.195230 0.980758i \(-0.562545\pi\)
−0.195230 + 0.980758i \(0.562545\pi\)
\(68\) −12753.4 −0.334468
\(69\) −12385.9 −0.313188
\(70\) 88733.1 2.16442
\(71\) −31200.2 −0.734532 −0.367266 0.930116i \(-0.619706\pi\)
−0.367266 + 0.930116i \(0.619706\pi\)
\(72\) −8626.44 −0.196110
\(73\) 1992.28 0.0437566 0.0218783 0.999761i \(-0.493035\pi\)
0.0218783 + 0.999761i \(0.493035\pi\)
\(74\) −31404.0 −0.666661
\(75\) 62420.4 1.28137
\(76\) 0 0
\(77\) 80319.7 1.54381
\(78\) 21068.2 0.392096
\(79\) −9623.27 −0.173482 −0.0867411 0.996231i \(-0.527645\pi\)
−0.0867411 + 0.996231i \(0.527645\pi\)
\(80\) 24455.1 0.427213
\(81\) −8127.64 −0.137642
\(82\) −64320.3 −1.05636
\(83\) −34179.5 −0.544590 −0.272295 0.962214i \(-0.587783\pi\)
−0.272295 + 0.962214i \(0.587783\pi\)
\(84\) 38650.4 0.597663
\(85\) −76144.0 −1.14311
\(86\) 51689.4 0.753625
\(87\) −4424.84 −0.0626758
\(88\) 22136.3 0.304718
\(89\) −111322. −1.48973 −0.744863 0.667218i \(-0.767485\pi\)
−0.744863 + 0.667218i \(0.767485\pi\)
\(90\) −51504.0 −0.670246
\(91\) 117578. 1.48842
\(92\) −19050.7 −0.234661
\(93\) 69154.2 0.829108
\(94\) −84760.6 −0.989406
\(95\) 0 0
\(96\) 10652.2 0.117967
\(97\) 96994.8 1.04669 0.523346 0.852120i \(-0.324684\pi\)
0.523346 + 0.852120i \(0.324684\pi\)
\(98\) 148474. 1.56165
\(99\) −46620.5 −0.478067
\(100\) 96008.4 0.960084
\(101\) −170094. −1.65915 −0.829575 0.558396i \(-0.811417\pi\)
−0.829575 + 0.558396i \(0.811417\pi\)
\(102\) −33166.9 −0.315649
\(103\) −72570.9 −0.674015 −0.337008 0.941502i \(-0.609415\pi\)
−0.337008 + 0.941502i \(0.609415\pi\)
\(104\) 32404.9 0.293784
\(105\) 230761. 2.04263
\(106\) −116367. −1.00592
\(107\) −68021.4 −0.574363 −0.287181 0.957876i \(-0.592718\pi\)
−0.287181 + 0.957876i \(0.592718\pi\)
\(108\) −62879.0 −0.518736
\(109\) −19382.6 −0.156259 −0.0781296 0.996943i \(-0.524895\pi\)
−0.0781296 + 0.996943i \(0.524895\pi\)
\(110\) 132164. 1.04144
\(111\) −81669.9 −0.629150
\(112\) 59448.0 0.447808
\(113\) −128260. −0.944917 −0.472458 0.881353i \(-0.656633\pi\)
−0.472458 + 0.881353i \(0.656633\pi\)
\(114\) 0 0
\(115\) −113742. −0.802001
\(116\) −6805.82 −0.0469608
\(117\) −68246.9 −0.460912
\(118\) 37385.0 0.247168
\(119\) −185099. −1.19822
\(120\) 63598.4 0.403175
\(121\) −41418.3 −0.257175
\(122\) −33283.9 −0.202458
\(123\) −167273. −0.996926
\(124\) 106366. 0.621222
\(125\) 274692. 1.57243
\(126\) −125201. −0.702558
\(127\) 94926.0 0.522247 0.261123 0.965305i \(-0.415907\pi\)
0.261123 + 0.965305i \(0.415907\pi\)
\(128\) 16384.0 0.0883883
\(129\) 134425. 0.711221
\(130\) 193473. 1.00406
\(131\) 265435. 1.35139 0.675695 0.737182i \(-0.263844\pi\)
0.675695 + 0.737182i \(0.263844\pi\)
\(132\) 57568.2 0.287573
\(133\) 0 0
\(134\) −57388.2 −0.276096
\(135\) −375418. −1.77288
\(136\) −51013.7 −0.236505
\(137\) −69622.2 −0.316918 −0.158459 0.987366i \(-0.550652\pi\)
−0.158459 + 0.987366i \(0.550652\pi\)
\(138\) −49543.6 −0.221457
\(139\) 369479. 1.62201 0.811004 0.585041i \(-0.198922\pi\)
0.811004 + 0.585041i \(0.198922\pi\)
\(140\) 354933. 1.53047
\(141\) −220430. −0.933735
\(142\) −124801. −0.519393
\(143\) 175128. 0.716170
\(144\) −34505.8 −0.138671
\(145\) −40634.0 −0.160498
\(146\) 7969.13 0.0309406
\(147\) 386124. 1.47378
\(148\) −125616. −0.471401
\(149\) 10845.5 0.0400206 0.0200103 0.999800i \(-0.493630\pi\)
0.0200103 + 0.999800i \(0.493630\pi\)
\(150\) 249682. 0.906064
\(151\) 33593.4 0.119898 0.0599489 0.998201i \(-0.480906\pi\)
0.0599489 + 0.998201i \(0.480906\pi\)
\(152\) 0 0
\(153\) 107438. 0.371048
\(154\) 321279. 1.09164
\(155\) 635053. 2.12315
\(156\) 84273.0 0.277253
\(157\) 571612. 1.85077 0.925385 0.379029i \(-0.123742\pi\)
0.925385 + 0.379029i \(0.123742\pi\)
\(158\) −38493.1 −0.122670
\(159\) −302627. −0.949324
\(160\) 97820.3 0.302085
\(161\) −276495. −0.840665
\(162\) −32510.5 −0.0973277
\(163\) 441057. 1.30025 0.650123 0.759829i \(-0.274717\pi\)
0.650123 + 0.759829i \(0.274717\pi\)
\(164\) −257281. −0.746962
\(165\) 343709. 0.982837
\(166\) −136718. −0.385084
\(167\) −32306.8 −0.0896401 −0.0448200 0.998995i \(-0.514271\pi\)
−0.0448200 + 0.998995i \(0.514271\pi\)
\(168\) 154602. 0.422611
\(169\) −114926. −0.309529
\(170\) −304576. −0.808302
\(171\) 0 0
\(172\) 206758. 0.532894
\(173\) −140107. −0.355915 −0.177957 0.984038i \(-0.556949\pi\)
−0.177957 + 0.984038i \(0.556949\pi\)
\(174\) −17699.4 −0.0443185
\(175\) 1.39343e6 3.43947
\(176\) 88545.2 0.215468
\(177\) 97224.3 0.233261
\(178\) −445288. −1.05339
\(179\) 205087. 0.478416 0.239208 0.970968i \(-0.423112\pi\)
0.239208 + 0.970968i \(0.423112\pi\)
\(180\) −206016. −0.473936
\(181\) −596816. −1.35408 −0.677039 0.735947i \(-0.736737\pi\)
−0.677039 + 0.735947i \(0.736737\pi\)
\(182\) 470314. 1.05247
\(183\) −86558.9 −0.191066
\(184\) −76202.7 −0.165930
\(185\) −749986. −1.61111
\(186\) 276617. 0.586268
\(187\) −275697. −0.576538
\(188\) −339042. −0.699616
\(189\) −912605. −1.85835
\(190\) 0 0
\(191\) 374224. 0.742246 0.371123 0.928584i \(-0.378973\pi\)
0.371123 + 0.928584i \(0.378973\pi\)
\(192\) 42608.6 0.0834150
\(193\) 185090. 0.357676 0.178838 0.983879i \(-0.442766\pi\)
0.178838 + 0.983879i \(0.442766\pi\)
\(194\) 387979. 0.740123
\(195\) 503150. 0.947569
\(196\) 593895. 1.10426
\(197\) 149519. 0.274493 0.137247 0.990537i \(-0.456175\pi\)
0.137247 + 0.990537i \(0.456175\pi\)
\(198\) −186482. −0.338044
\(199\) −305923. −0.547619 −0.273810 0.961784i \(-0.588284\pi\)
−0.273810 + 0.961784i \(0.588284\pi\)
\(200\) 384034. 0.678882
\(201\) −149245. −0.260561
\(202\) −680376. −1.17320
\(203\) −98777.3 −0.168235
\(204\) −132667. −0.223197
\(205\) −1.53609e6 −2.55289
\(206\) −290284. −0.476601
\(207\) 160488. 0.260325
\(208\) 129620. 0.207736
\(209\) 0 0
\(210\) 923046. 1.44436
\(211\) 809738. 1.25210 0.626049 0.779784i \(-0.284671\pi\)
0.626049 + 0.779784i \(0.284671\pi\)
\(212\) −465468. −0.711296
\(213\) −324559. −0.490168
\(214\) −272086. −0.406136
\(215\) 1.23444e6 1.82127
\(216\) −251516. −0.366802
\(217\) 1.54375e6 2.22551
\(218\) −77530.4 −0.110492
\(219\) 20724.7 0.0291997
\(220\) 528657. 0.736406
\(221\) −403588. −0.555849
\(222\) −326679. −0.444876
\(223\) 634589. 0.854536 0.427268 0.904125i \(-0.359476\pi\)
0.427268 + 0.904125i \(0.359476\pi\)
\(224\) 237792. 0.316648
\(225\) −808800. −1.06509
\(226\) −513038. −0.668157
\(227\) −93859.7 −0.120897 −0.0604484 0.998171i \(-0.519253\pi\)
−0.0604484 + 0.998171i \(0.519253\pi\)
\(228\) 0 0
\(229\) 176185. 0.222014 0.111007 0.993820i \(-0.464592\pi\)
0.111007 + 0.993820i \(0.464592\pi\)
\(230\) −454967. −0.567100
\(231\) 835525. 1.03022
\(232\) −27223.3 −0.0332063
\(233\) −226160. −0.272915 −0.136457 0.990646i \(-0.543572\pi\)
−0.136457 + 0.990646i \(0.543572\pi\)
\(234\) −272987. −0.325914
\(235\) −2.02425e6 −2.39108
\(236\) 149540. 0.174774
\(237\) −100106. −0.115768
\(238\) −740395. −0.847269
\(239\) −343996. −0.389546 −0.194773 0.980848i \(-0.562397\pi\)
−0.194773 + 0.980848i \(0.562397\pi\)
\(240\) 254394. 0.285088
\(241\) 1.58686e6 1.75993 0.879966 0.475036i \(-0.157565\pi\)
0.879966 + 0.475036i \(0.157565\pi\)
\(242\) −165673. −0.181850
\(243\) 870427. 0.945620
\(244\) −133136. −0.143159
\(245\) 3.54584e6 3.77402
\(246\) −669092. −0.704933
\(247\) 0 0
\(248\) 425462. 0.439270
\(249\) −355552. −0.363416
\(250\) 1.09877e6 1.11188
\(251\) 745968. 0.747370 0.373685 0.927556i \(-0.378094\pi\)
0.373685 + 0.927556i \(0.378094\pi\)
\(252\) −500805. −0.496784
\(253\) −411828. −0.404496
\(254\) 379704. 0.369284
\(255\) −792088. −0.762821
\(256\) 65536.0 0.0625000
\(257\) 1.92630e6 1.81925 0.909623 0.415436i \(-0.136371\pi\)
0.909623 + 0.415436i \(0.136371\pi\)
\(258\) 537699. 0.502910
\(259\) −1.82315e6 −1.68878
\(260\) 773891. 0.709981
\(261\) 57334.0 0.0520968
\(262\) 1.06174e6 0.955577
\(263\) −964619. −0.859937 −0.429968 0.902844i \(-0.641475\pi\)
−0.429968 + 0.902844i \(0.641475\pi\)
\(264\) 230273. 0.203345
\(265\) −2.77907e6 −2.43100
\(266\) 0 0
\(267\) −1.15803e6 −0.994124
\(268\) −229553. −0.195230
\(269\) −1.61062e6 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(270\) −1.50167e6 −1.25362
\(271\) 1.13650e6 0.940038 0.470019 0.882656i \(-0.344247\pi\)
0.470019 + 0.882656i \(0.344247\pi\)
\(272\) −204055. −0.167234
\(273\) 1.22311e6 0.993250
\(274\) −278489. −0.224095
\(275\) 2.07546e6 1.65494
\(276\) −198175. −0.156594
\(277\) 648448. 0.507780 0.253890 0.967233i \(-0.418290\pi\)
0.253890 + 0.967233i \(0.418290\pi\)
\(278\) 1.47792e6 1.14693
\(279\) −896051. −0.689163
\(280\) 1.41973e6 1.08221
\(281\) −501822. −0.379126 −0.189563 0.981869i \(-0.560707\pi\)
−0.189563 + 0.981869i \(0.560707\pi\)
\(282\) −881722. −0.660251
\(283\) 1.36912e6 1.01619 0.508096 0.861301i \(-0.330350\pi\)
0.508096 + 0.861301i \(0.330350\pi\)
\(284\) −499203. −0.367266
\(285\) 0 0
\(286\) 700513. 0.506408
\(287\) −3.73409e6 −2.67596
\(288\) −138023. −0.0980552
\(289\) −784506. −0.552525
\(290\) −162536. −0.113489
\(291\) 1.00899e6 0.698479
\(292\) 31876.5 0.0218783
\(293\) −684715. −0.465952 −0.232976 0.972483i \(-0.574846\pi\)
−0.232976 + 0.972483i \(0.574846\pi\)
\(294\) 1.54450e6 1.04212
\(295\) 892825. 0.597326
\(296\) −502463. −0.333331
\(297\) −1.35929e6 −0.894169
\(298\) 43382.0 0.0282988
\(299\) −602867. −0.389981
\(300\) 998727. 0.640684
\(301\) 3.00081e6 1.90907
\(302\) 134373. 0.0847805
\(303\) −1.76940e6 −1.10718
\(304\) 0 0
\(305\) −794883. −0.489276
\(306\) 429753. 0.262370
\(307\) −1.43283e6 −0.867656 −0.433828 0.900996i \(-0.642837\pi\)
−0.433828 + 0.900996i \(0.642837\pi\)
\(308\) 1.28511e6 0.771907
\(309\) −754919. −0.449784
\(310\) 2.54021e6 1.50129
\(311\) −286576. −0.168012 −0.0840058 0.996465i \(-0.526771\pi\)
−0.0840058 + 0.996465i \(0.526771\pi\)
\(312\) 337092. 0.196048
\(313\) 326263. 0.188238 0.0941189 0.995561i \(-0.469997\pi\)
0.0941189 + 0.995561i \(0.469997\pi\)
\(314\) 2.28645e6 1.30869
\(315\) −2.99004e6 −1.69786
\(316\) −153972. −0.0867411
\(317\) 3.04382e6 1.70126 0.850631 0.525764i \(-0.176220\pi\)
0.850631 + 0.525764i \(0.176220\pi\)
\(318\) −1.21051e6 −0.671274
\(319\) −147125. −0.0809485
\(320\) 391281. 0.213606
\(321\) −707592. −0.383284
\(322\) −1.10598e6 −0.594440
\(323\) 0 0
\(324\) −130042. −0.0688211
\(325\) 3.03823e6 1.59556
\(326\) 1.76423e6 0.919412
\(327\) −201627. −0.104275
\(328\) −1.02912e6 −0.528182
\(329\) −4.92075e6 −2.50635
\(330\) 1.37484e6 0.694971
\(331\) −2.26302e6 −1.13532 −0.567660 0.823263i \(-0.692151\pi\)
−0.567660 + 0.823263i \(0.692151\pi\)
\(332\) −546871. −0.272295
\(333\) 1.05822e6 0.522956
\(334\) −129227. −0.0633851
\(335\) −1.37054e6 −0.667237
\(336\) 618407. 0.298831
\(337\) 2.34013e6 1.12245 0.561223 0.827665i \(-0.310331\pi\)
0.561223 + 0.827665i \(0.310331\pi\)
\(338\) −459704. −0.218870
\(339\) −1.33422e6 −0.630562
\(340\) −1.21830e6 −0.571556
\(341\) 2.29936e6 1.07083
\(342\) 0 0
\(343\) 4.71670e6 2.16472
\(344\) 827031. 0.376813
\(345\) −1.18320e6 −0.535191
\(346\) −560430. −0.251670
\(347\) −1.61523e6 −0.720128 −0.360064 0.932928i \(-0.617245\pi\)
−0.360064 + 0.932928i \(0.617245\pi\)
\(348\) −70797.5 −0.0313379
\(349\) 49878.2 0.0219203 0.0109602 0.999940i \(-0.496511\pi\)
0.0109602 + 0.999940i \(0.496511\pi\)
\(350\) 5.57374e6 2.43207
\(351\) −1.98983e6 −0.862083
\(352\) 354181. 0.152359
\(353\) 3.46683e6 1.48080 0.740399 0.672167i \(-0.234636\pi\)
0.740399 + 0.672167i \(0.234636\pi\)
\(354\) 388897. 0.164940
\(355\) −2.98048e6 −1.25521
\(356\) −1.78115e6 −0.744863
\(357\) −1.92549e6 −0.799596
\(358\) 820348. 0.338291
\(359\) 2.97986e6 1.22028 0.610140 0.792294i \(-0.291113\pi\)
0.610140 + 0.792294i \(0.291113\pi\)
\(360\) −824063. −0.335123
\(361\) 0 0
\(362\) −2.38726e6 −0.957478
\(363\) −430853. −0.171618
\(364\) 1.88126e6 0.744208
\(365\) 190318. 0.0747735
\(366\) −346236. −0.135104
\(367\) 2.23837e6 0.867495 0.433748 0.901034i \(-0.357191\pi\)
0.433748 + 0.901034i \(0.357191\pi\)
\(368\) −304811. −0.117331
\(369\) 2.16740e6 0.828655
\(370\) −2.99995e6 −1.13922
\(371\) −6.75565e6 −2.54819
\(372\) 1.10647e6 0.414554
\(373\) 1.11334e6 0.414338 0.207169 0.978305i \(-0.433575\pi\)
0.207169 + 0.978305i \(0.433575\pi\)
\(374\) −1.10279e6 −0.407674
\(375\) 2.85748e6 1.04931
\(376\) −1.35617e6 −0.494703
\(377\) −215373. −0.0780437
\(378\) −3.65042e6 −1.31405
\(379\) 3.21599e6 1.15005 0.575026 0.818135i \(-0.304992\pi\)
0.575026 + 0.818135i \(0.304992\pi\)
\(380\) 0 0
\(381\) 987467. 0.348506
\(382\) 1.49690e6 0.524847
\(383\) 491800. 0.171313 0.0856567 0.996325i \(-0.472701\pi\)
0.0856567 + 0.996325i \(0.472701\pi\)
\(384\) 170434. 0.0589833
\(385\) 7.67275e6 2.63815
\(386\) 740360. 0.252915
\(387\) −1.74178e6 −0.591175
\(388\) 1.55192e6 0.523346
\(389\) −2.10114e6 −0.704015 −0.352007 0.935997i \(-0.614501\pi\)
−0.352007 + 0.935997i \(0.614501\pi\)
\(390\) 2.01260e6 0.670032
\(391\) 949068. 0.313946
\(392\) 2.37558e6 0.780827
\(393\) 2.76119e6 0.901809
\(394\) 598077. 0.194096
\(395\) −919288. −0.296455
\(396\) −745928. −0.239033
\(397\) −5.27092e6 −1.67846 −0.839229 0.543778i \(-0.816993\pi\)
−0.839229 + 0.543778i \(0.816993\pi\)
\(398\) −1.22369e6 −0.387225
\(399\) 0 0
\(400\) 1.53614e6 0.480042
\(401\) −4.12441e6 −1.28086 −0.640430 0.768017i \(-0.721244\pi\)
−0.640430 + 0.768017i \(0.721244\pi\)
\(402\) −596981. −0.184245
\(403\) 3.36598e6 1.03240
\(404\) −2.72150e6 −0.829575
\(405\) −776414. −0.235210
\(406\) −395109. −0.118960
\(407\) −2.71550e6 −0.812575
\(408\) −530670. −0.157824
\(409\) −4.71220e6 −1.39289 −0.696443 0.717613i \(-0.745235\pi\)
−0.696443 + 0.717613i \(0.745235\pi\)
\(410\) −6.14437e6 −1.80517
\(411\) −724244. −0.211485
\(412\) −1.16113e6 −0.337008
\(413\) 2.17037e6 0.626122
\(414\) 641951. 0.184078
\(415\) −3.26508e6 −0.930623
\(416\) 518479. 0.146892
\(417\) 3.84350e6 1.08240
\(418\) 0 0
\(419\) −3.47084e6 −0.965827 −0.482914 0.875668i \(-0.660422\pi\)
−0.482914 + 0.875668i \(0.660422\pi\)
\(420\) 3.69218e6 1.02132
\(421\) −1.19223e6 −0.327835 −0.163918 0.986474i \(-0.552413\pi\)
−0.163918 + 0.986474i \(0.552413\pi\)
\(422\) 3.23895e6 0.885367
\(423\) 2.85618e6 0.776131
\(424\) −1.86187e6 −0.502962
\(425\) −4.78295e6 −1.28447
\(426\) −1.29824e6 −0.346601
\(427\) −1.93229e6 −0.512863
\(428\) −1.08834e6 −0.287181
\(429\) 1.82177e6 0.477914
\(430\) 4.93777e6 1.28783
\(431\) −679294. −0.176143 −0.0880714 0.996114i \(-0.528070\pi\)
−0.0880714 + 0.996114i \(0.528070\pi\)
\(432\) −1.00606e6 −0.259368
\(433\) −7.14219e6 −1.83068 −0.915339 0.402685i \(-0.868077\pi\)
−0.915339 + 0.402685i \(0.868077\pi\)
\(434\) 6.17501e6 1.57367
\(435\) −422695. −0.107103
\(436\) −310121. −0.0781296
\(437\) 0 0
\(438\) 82898.8 0.0206473
\(439\) 1.61627e6 0.400271 0.200135 0.979768i \(-0.435862\pi\)
0.200135 + 0.979768i \(0.435862\pi\)
\(440\) 2.11463e6 0.520718
\(441\) −5.00313e6 −1.22503
\(442\) −1.61435e6 −0.393045
\(443\) −3.15786e6 −0.764511 −0.382255 0.924057i \(-0.624853\pi\)
−0.382255 + 0.924057i \(0.624853\pi\)
\(444\) −1.30672e6 −0.314575
\(445\) −1.06343e7 −2.54572
\(446\) 2.53836e6 0.604249
\(447\) 112820. 0.0267066
\(448\) 951167. 0.223904
\(449\) 1.47951e6 0.346338 0.173169 0.984892i \(-0.444599\pi\)
0.173169 + 0.984892i \(0.444599\pi\)
\(450\) −3.23520e6 −0.753130
\(451\) −5.56177e6 −1.28757
\(452\) −2.05215e6 −0.472458
\(453\) 349455. 0.0800102
\(454\) −375439. −0.0854869
\(455\) 1.12320e7 2.54348
\(456\) 0 0
\(457\) −5.30763e6 −1.18880 −0.594402 0.804168i \(-0.702611\pi\)
−0.594402 + 0.804168i \(0.702611\pi\)
\(458\) 704739. 0.156987
\(459\) 3.13251e6 0.694002
\(460\) −1.81987e6 −0.401000
\(461\) −1.54790e6 −0.339227 −0.169613 0.985511i \(-0.554252\pi\)
−0.169613 + 0.985511i \(0.554252\pi\)
\(462\) 3.34210e6 0.728474
\(463\) 7.32198e6 1.58736 0.793682 0.608333i \(-0.208162\pi\)
0.793682 + 0.608333i \(0.208162\pi\)
\(464\) −108893. −0.0234804
\(465\) 6.60614e6 1.41682
\(466\) −904642. −0.192980
\(467\) 6.91574e6 1.46739 0.733696 0.679478i \(-0.237794\pi\)
0.733696 + 0.679478i \(0.237794\pi\)
\(468\) −1.09195e6 −0.230456
\(469\) −3.33165e6 −0.699403
\(470\) −8.09698e6 −1.69075
\(471\) 5.94619e6 1.23506
\(472\) 598160. 0.123584
\(473\) 4.46958e6 0.918573
\(474\) −400424. −0.0818605
\(475\) 0 0
\(476\) −2.96158e6 −0.599110
\(477\) 3.92122e6 0.789089
\(478\) −1.37599e6 −0.275451
\(479\) 3.22667e6 0.642563 0.321281 0.946984i \(-0.395886\pi\)
0.321281 + 0.946984i \(0.395886\pi\)
\(480\) 1.01757e6 0.201587
\(481\) −3.97517e6 −0.783416
\(482\) 6.34744e6 1.24446
\(483\) −2.87624e6 −0.560992
\(484\) −662693. −0.128588
\(485\) 9.26568e6 1.78864
\(486\) 3.48171e6 0.668655
\(487\) −4.64992e6 −0.888430 −0.444215 0.895920i \(-0.646517\pi\)
−0.444215 + 0.895920i \(0.646517\pi\)
\(488\) −532543. −0.101229
\(489\) 4.58809e6 0.867680
\(490\) 1.41834e7 2.66863
\(491\) 8.23773e6 1.54207 0.771034 0.636794i \(-0.219740\pi\)
0.771034 + 0.636794i \(0.219740\pi\)
\(492\) −2.67637e6 −0.498463
\(493\) 339053. 0.0628275
\(494\) 0 0
\(495\) −4.45354e6 −0.816945
\(496\) 1.70185e6 0.310611
\(497\) −7.24526e6 −1.31572
\(498\) −1.42221e6 −0.256974
\(499\) −3.72990e6 −0.670572 −0.335286 0.942116i \(-0.608833\pi\)
−0.335286 + 0.942116i \(0.608833\pi\)
\(500\) 4.39508e6 0.786215
\(501\) −336071. −0.0598186
\(502\) 2.98387e6 0.528470
\(503\) −1.51655e6 −0.267261 −0.133630 0.991031i \(-0.542664\pi\)
−0.133630 + 0.991031i \(0.542664\pi\)
\(504\) −2.00322e6 −0.351279
\(505\) −1.62487e7 −2.83524
\(506\) −1.64731e6 −0.286022
\(507\) −1.19552e6 −0.206555
\(508\) 1.51882e6 0.261123
\(509\) 1.12361e7 1.92231 0.961154 0.276014i \(-0.0890135\pi\)
0.961154 + 0.276014i \(0.0890135\pi\)
\(510\) −3.16835e6 −0.539396
\(511\) 462645. 0.0783782
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) 7.70520e6 1.28640
\(515\) −6.93253e6 −1.15179
\(516\) 2.15080e6 0.355611
\(517\) −7.32924e6 −1.20596
\(518\) −7.29258e6 −1.19414
\(519\) −1.45747e6 −0.237509
\(520\) 3.09557e6 0.502032
\(521\) 4.99957e6 0.806934 0.403467 0.914994i \(-0.367805\pi\)
0.403467 + 0.914994i \(0.367805\pi\)
\(522\) 229336. 0.0368380
\(523\) −4.90563e6 −0.784225 −0.392113 0.919917i \(-0.628256\pi\)
−0.392113 + 0.919917i \(0.628256\pi\)
\(524\) 4.24697e6 0.675695
\(525\) 1.44952e7 2.29523
\(526\) −3.85848e6 −0.608067
\(527\) −5.29893e6 −0.831116
\(528\) 921091. 0.143786
\(529\) −5.01865e6 −0.779737
\(530\) −1.11163e7 −1.71897
\(531\) −1.25976e6 −0.193889
\(532\) 0 0
\(533\) −8.14178e6 −1.24137
\(534\) −4.63211e6 −0.702952
\(535\) −6.49792e6 −0.981500
\(536\) −918212. −0.138048
\(537\) 2.13342e6 0.319256
\(538\) −6.44248e6 −0.959616
\(539\) 1.28385e7 1.90346
\(540\) −6.00668e6 −0.886442
\(541\) 3.83360e6 0.563137 0.281568 0.959541i \(-0.409145\pi\)
0.281568 + 0.959541i \(0.409145\pi\)
\(542\) 4.54599e6 0.664707
\(543\) −6.20837e6 −0.903604
\(544\) −816219. −0.118252
\(545\) −1.85157e6 −0.267024
\(546\) 4.89244e6 0.702334
\(547\) −5.83247e6 −0.833458 −0.416729 0.909031i \(-0.636824\pi\)
−0.416729 + 0.909031i \(0.636824\pi\)
\(548\) −1.11396e6 −0.158459
\(549\) 1.12157e6 0.158816
\(550\) 8.30184e6 1.17022
\(551\) 0 0
\(552\) −792698. −0.110729
\(553\) −2.23470e6 −0.310747
\(554\) 2.59379e6 0.359055
\(555\) −7.80173e6 −1.07512
\(556\) 5.91166e6 0.811004
\(557\) 1.47427e6 0.201344 0.100672 0.994920i \(-0.467901\pi\)
0.100672 + 0.994920i \(0.467901\pi\)
\(558\) −3.58420e6 −0.487312
\(559\) 6.54294e6 0.885611
\(560\) 5.67892e6 0.765237
\(561\) −2.86793e6 −0.384735
\(562\) −2.00729e6 −0.268083
\(563\) −1.71502e6 −0.228033 −0.114016 0.993479i \(-0.536372\pi\)
−0.114016 + 0.993479i \(0.536372\pi\)
\(564\) −3.52689e6 −0.466868
\(565\) −1.22523e7 −1.61472
\(566\) 5.47648e6 0.718556
\(567\) −1.88739e6 −0.246549
\(568\) −1.99681e6 −0.259696
\(569\) 1.09443e7 1.41712 0.708558 0.705652i \(-0.249346\pi\)
0.708558 + 0.705652i \(0.249346\pi\)
\(570\) 0 0
\(571\) −7.80064e6 −1.00124 −0.500622 0.865666i \(-0.666895\pi\)
−0.500622 + 0.865666i \(0.666895\pi\)
\(572\) 2.80205e6 0.358085
\(573\) 3.89286e6 0.495316
\(574\) −1.49364e7 −1.89219
\(575\) −7.14463e6 −0.901177
\(576\) −552092. −0.0693355
\(577\) −8.00141e6 −1.00052 −0.500262 0.865874i \(-0.666763\pi\)
−0.500262 + 0.865874i \(0.666763\pi\)
\(578\) −3.13802e6 −0.390694
\(579\) 1.92540e6 0.238684
\(580\) −650144. −0.0802489
\(581\) −7.93711e6 −0.975488
\(582\) 4.03595e6 0.493899
\(583\) −1.00622e7 −1.22609
\(584\) 127506. 0.0154703
\(585\) −6.51946e6 −0.787630
\(586\) −2.73886e6 −0.329478
\(587\) −7.68911e6 −0.921045 −0.460523 0.887648i \(-0.652338\pi\)
−0.460523 + 0.887648i \(0.652338\pi\)
\(588\) 6.17799e6 0.736892
\(589\) 0 0
\(590\) 3.57130e6 0.422373
\(591\) 1.55537e6 0.183175
\(592\) −2.00985e6 −0.235700
\(593\) 3.87367e6 0.452362 0.226181 0.974085i \(-0.427376\pi\)
0.226181 + 0.974085i \(0.427376\pi\)
\(594\) −5.43714e6 −0.632273
\(595\) −1.76821e7 −2.04758
\(596\) 173528. 0.0200103
\(597\) −3.18236e6 −0.365438
\(598\) −2.41147e6 −0.275758
\(599\) −1.25262e7 −1.42644 −0.713220 0.700940i \(-0.752764\pi\)
−0.713220 + 0.700940i \(0.752764\pi\)
\(600\) 3.99491e6 0.453032
\(601\) 293637. 0.0331608 0.0165804 0.999863i \(-0.494722\pi\)
0.0165804 + 0.999863i \(0.494722\pi\)
\(602\) 1.20032e7 1.34992
\(603\) 1.93381e6 0.216581
\(604\) 537494. 0.0599489
\(605\) −3.95659e6 −0.439474
\(606\) −7.07761e6 −0.782897
\(607\) −4.13932e6 −0.455993 −0.227996 0.973662i \(-0.573217\pi\)
−0.227996 + 0.973662i \(0.573217\pi\)
\(608\) 0 0
\(609\) −1.02753e6 −0.112267
\(610\) −3.17953e6 −0.345970
\(611\) −1.07291e7 −1.16269
\(612\) 1.71901e6 0.185524
\(613\) 5.73897e6 0.616854 0.308427 0.951248i \(-0.400197\pi\)
0.308427 + 0.951248i \(0.400197\pi\)
\(614\) −5.73130e6 −0.613525
\(615\) −1.59792e7 −1.70360
\(616\) 5.14046e6 0.545821
\(617\) −1.15672e7 −1.22325 −0.611627 0.791146i \(-0.709485\pi\)
−0.611627 + 0.791146i \(0.709485\pi\)
\(618\) −3.01967e6 −0.318045
\(619\) −1.84780e7 −1.93833 −0.969167 0.246404i \(-0.920751\pi\)
−0.969167 + 0.246404i \(0.920751\pi\)
\(620\) 1.01608e7 1.06158
\(621\) 4.67925e6 0.486908
\(622\) −1.14630e6 −0.118802
\(623\) −2.58510e7 −2.66844
\(624\) 1.34837e6 0.138627
\(625\) 7.48906e6 0.766879
\(626\) 1.30505e6 0.133104
\(627\) 0 0
\(628\) 9.14580e6 0.925385
\(629\) 6.25794e6 0.630674
\(630\) −1.19602e7 −1.20057
\(631\) 8.95725e6 0.895573 0.447787 0.894140i \(-0.352212\pi\)
0.447787 + 0.894140i \(0.352212\pi\)
\(632\) −615889. −0.0613352
\(633\) 8.42329e6 0.835550
\(634\) 1.21753e7 1.20297
\(635\) 9.06805e6 0.892441
\(636\) −4.84203e6 −0.474662
\(637\) 1.87941e7 1.83515
\(638\) −588498. −0.0572392
\(639\) 4.20541e6 0.407433
\(640\) 1.56512e6 0.151042
\(641\) 1.46979e7 1.41290 0.706450 0.707763i \(-0.250296\pi\)
0.706450 + 0.707763i \(0.250296\pi\)
\(642\) −2.83037e6 −0.271023
\(643\) 1.55533e7 1.48352 0.741762 0.670664i \(-0.233991\pi\)
0.741762 + 0.670664i \(0.233991\pi\)
\(644\) −4.42392e6 −0.420332
\(645\) 1.28413e7 1.21537
\(646\) 0 0
\(647\) −5.91558e6 −0.555567 −0.277784 0.960644i \(-0.589600\pi\)
−0.277784 + 0.960644i \(0.589600\pi\)
\(648\) −520169. −0.0486639
\(649\) 3.23268e6 0.301266
\(650\) 1.21529e7 1.12823
\(651\) 1.60589e7 1.48512
\(652\) 7.05691e6 0.650123
\(653\) 1.27377e6 0.116898 0.0584492 0.998290i \(-0.481384\pi\)
0.0584492 + 0.998290i \(0.481384\pi\)
\(654\) −806509. −0.0737335
\(655\) 2.53564e7 2.30932
\(656\) −4.11650e6 −0.373481
\(657\) −268536. −0.0242711
\(658\) −1.96830e7 −1.77226
\(659\) −6.98055e6 −0.626147 −0.313073 0.949729i \(-0.601359\pi\)
−0.313073 + 0.949729i \(0.601359\pi\)
\(660\) 5.49935e6 0.491419
\(661\) −1.02987e7 −0.916812 −0.458406 0.888743i \(-0.651579\pi\)
−0.458406 + 0.888743i \(0.651579\pi\)
\(662\) −9.05208e6 −0.802792
\(663\) −4.19832e6 −0.370930
\(664\) −2.18749e6 −0.192542
\(665\) 0 0
\(666\) 4.23288e6 0.369786
\(667\) 506467. 0.0440795
\(668\) −516908. −0.0448200
\(669\) 6.60131e6 0.570249
\(670\) −5.48216e6 −0.471808
\(671\) −2.87806e6 −0.246770
\(672\) 2.47363e6 0.211306
\(673\) −5.48362e6 −0.466692 −0.233346 0.972394i \(-0.574967\pi\)
−0.233346 + 0.972394i \(0.574967\pi\)
\(674\) 9.36052e6 0.793689
\(675\) −2.35817e7 −1.99212
\(676\) −1.83882e6 −0.154765
\(677\) 4.99313e6 0.418698 0.209349 0.977841i \(-0.432866\pi\)
0.209349 + 0.977841i \(0.432866\pi\)
\(678\) −5.33687e6 −0.445875
\(679\) 2.25240e7 1.87487
\(680\) −4.87322e6 −0.404151
\(681\) −976375. −0.0806768
\(682\) 9.19742e6 0.757190
\(683\) −6.96239e6 −0.571093 −0.285546 0.958365i \(-0.592175\pi\)
−0.285546 + 0.958365i \(0.592175\pi\)
\(684\) 0 0
\(685\) −6.65084e6 −0.541565
\(686\) 1.88668e7 1.53069
\(687\) 1.83276e6 0.148154
\(688\) 3.30812e6 0.266447
\(689\) −1.47299e7 −1.18210
\(690\) −4.73279e6 −0.378438
\(691\) 9.29521e6 0.740566 0.370283 0.928919i \(-0.379261\pi\)
0.370283 + 0.928919i \(0.379261\pi\)
\(692\) −2.24172e6 −0.177957
\(693\) −1.08261e7 −0.856329
\(694\) −6.46091e6 −0.509208
\(695\) 3.52955e7 2.77177
\(696\) −283190. −0.0221592
\(697\) 1.28173e7 0.999340
\(698\) 199513. 0.0155000
\(699\) −2.35263e6 −0.182121
\(700\) 2.22949e7 1.71973
\(701\) −7.70223e6 −0.591999 −0.296000 0.955188i \(-0.595653\pi\)
−0.296000 + 0.955188i \(0.595653\pi\)
\(702\) −7.95933e6 −0.609585
\(703\) 0 0
\(704\) 1.41672e6 0.107734
\(705\) −2.10572e7 −1.59561
\(706\) 1.38673e7 1.04708
\(707\) −3.94990e7 −2.97192
\(708\) 1.55559e6 0.116630
\(709\) 513218. 0.0383430 0.0191715 0.999816i \(-0.493897\pi\)
0.0191715 + 0.999816i \(0.493897\pi\)
\(710\) −1.19219e7 −0.887565
\(711\) 1.29710e6 0.0962278
\(712\) −7.12461e6 −0.526697
\(713\) −7.91538e6 −0.583106
\(714\) −7.70196e6 −0.565400
\(715\) 1.67296e7 1.22383
\(716\) 3.28139e6 0.239208
\(717\) −3.57842e6 −0.259952
\(718\) 1.19194e7 0.862868
\(719\) 3.80611e6 0.274574 0.137287 0.990531i \(-0.456162\pi\)
0.137287 + 0.990531i \(0.456162\pi\)
\(720\) −3.29625e6 −0.236968
\(721\) −1.68523e7 −1.20732
\(722\) 0 0
\(723\) 1.65073e7 1.17444
\(724\) −9.54905e6 −0.677039
\(725\) −2.55241e6 −0.180345
\(726\) −1.72341e6 −0.121352
\(727\) −2.43735e7 −1.71033 −0.855167 0.518352i \(-0.826546\pi\)
−0.855167 + 0.518352i \(0.826546\pi\)
\(728\) 7.52502e6 0.526235
\(729\) 1.10296e7 0.768674
\(730\) 761272. 0.0528728
\(731\) −1.03003e7 −0.712944
\(732\) −1.38494e6 −0.0955331
\(733\) 4.46559e6 0.306986 0.153493 0.988150i \(-0.450948\pi\)
0.153493 + 0.988150i \(0.450948\pi\)
\(734\) 8.95349e6 0.613412
\(735\) 3.68856e7 2.51848
\(736\) −1.21924e6 −0.0829652
\(737\) −4.96236e6 −0.336526
\(738\) 8.66962e6 0.585948
\(739\) −2.04421e6 −0.137694 −0.0688468 0.997627i \(-0.521932\pi\)
−0.0688468 + 0.997627i \(0.521932\pi\)
\(740\) −1.19998e7 −0.805553
\(741\) 0 0
\(742\) −2.70226e7 −1.80184
\(743\) 1.90490e7 1.26590 0.632950 0.774193i \(-0.281844\pi\)
0.632950 + 0.774193i \(0.281844\pi\)
\(744\) 4.42587e6 0.293134
\(745\) 1.03604e6 0.0683892
\(746\) 4.45335e6 0.292981
\(747\) 4.60699e6 0.302075
\(748\) −4.41115e6 −0.288269
\(749\) −1.57958e7 −1.02882
\(750\) 1.14299e7 0.741978
\(751\) 1.08826e7 0.704097 0.352048 0.935982i \(-0.385485\pi\)
0.352048 + 0.935982i \(0.385485\pi\)
\(752\) −5.42468e6 −0.349808
\(753\) 7.75992e6 0.498735
\(754\) −861492. −0.0551852
\(755\) 3.20909e6 0.204887
\(756\) −1.46017e7 −0.929176
\(757\) 2.24795e6 0.142576 0.0712881 0.997456i \(-0.477289\pi\)
0.0712881 + 0.997456i \(0.477289\pi\)
\(758\) 1.28640e7 0.813209
\(759\) −4.28403e6 −0.269928
\(760\) 0 0
\(761\) 1.37969e7 0.863614 0.431807 0.901966i \(-0.357876\pi\)
0.431807 + 0.901966i \(0.357876\pi\)
\(762\) 3.94987e6 0.246431
\(763\) −4.50100e6 −0.279896
\(764\) 5.98758e6 0.371123
\(765\) 1.02633e7 0.634065
\(766\) 1.96720e6 0.121137
\(767\) 4.73226e6 0.290456
\(768\) 681738. 0.0417075
\(769\) 3.16040e7 1.92720 0.963599 0.267352i \(-0.0861487\pi\)
0.963599 + 0.267352i \(0.0861487\pi\)
\(770\) 3.06910e7 1.86545
\(771\) 2.00383e7 1.21402
\(772\) 2.96144e6 0.178838
\(773\) −2.13708e7 −1.28639 −0.643195 0.765702i \(-0.722392\pi\)
−0.643195 + 0.765702i \(0.722392\pi\)
\(774\) −6.96712e6 −0.418024
\(775\) 3.98906e7 2.38570
\(776\) 6.20767e6 0.370062
\(777\) −1.89653e7 −1.12695
\(778\) −8.40458e6 −0.497814
\(779\) 0 0
\(780\) 8.05040e6 0.473785
\(781\) −1.07915e7 −0.633074
\(782\) 3.79627e6 0.221994
\(783\) 1.67165e6 0.0974410
\(784\) 9.50233e6 0.552128
\(785\) 5.46048e7 3.16269
\(786\) 1.10448e7 0.637676
\(787\) −1.91269e7 −1.10080 −0.550399 0.834901i \(-0.685525\pi\)
−0.550399 + 0.834901i \(0.685525\pi\)
\(788\) 2.39231e6 0.137247
\(789\) −1.00344e7 −0.573853
\(790\) −3.67715e6 −0.209625
\(791\) −2.97842e7 −1.69256
\(792\) −2.98371e6 −0.169022
\(793\) −4.21313e6 −0.237915
\(794\) −2.10837e7 −1.18685
\(795\) −2.89092e7 −1.62225
\(796\) −4.89476e6 −0.273810
\(797\) −8.28296e6 −0.461891 −0.230946 0.972967i \(-0.574182\pi\)
−0.230946 + 0.972967i \(0.574182\pi\)
\(798\) 0 0
\(799\) 1.68904e7 0.935996
\(800\) 6.14454e6 0.339441
\(801\) 1.50049e7 0.826326
\(802\) −1.64977e7 −0.905704
\(803\) 689090. 0.0377126
\(804\) −2.38792e6 −0.130281
\(805\) −2.64129e7 −1.43657
\(806\) 1.34639e7 0.730019
\(807\) −1.67545e7 −0.905622
\(808\) −1.08860e7 −0.586598
\(809\) −8.91658e6 −0.478991 −0.239495 0.970898i \(-0.576982\pi\)
−0.239495 + 0.970898i \(0.576982\pi\)
\(810\) −3.10565e6 −0.166319
\(811\) −1.52149e7 −0.812302 −0.406151 0.913806i \(-0.633129\pi\)
−0.406151 + 0.913806i \(0.633129\pi\)
\(812\) −1.58044e6 −0.0841177
\(813\) 1.18224e7 0.627306
\(814\) −1.08620e7 −0.574577
\(815\) 4.21331e7 2.22192
\(816\) −2.12268e6 −0.111599
\(817\) 0 0
\(818\) −1.88488e7 −0.984918
\(819\) −1.58482e7 −0.825600
\(820\) −2.45775e7 −1.27645
\(821\) 2.23907e7 1.15934 0.579669 0.814852i \(-0.303182\pi\)
0.579669 + 0.814852i \(0.303182\pi\)
\(822\) −2.89698e6 −0.149543
\(823\) −2.10545e7 −1.08354 −0.541772 0.840526i \(-0.682246\pi\)
−0.541772 + 0.840526i \(0.682246\pi\)
\(824\) −4.64454e6 −0.238300
\(825\) 2.15900e7 1.10438
\(826\) 8.68149e6 0.442735
\(827\) 2.23530e7 1.13651 0.568253 0.822854i \(-0.307619\pi\)
0.568253 + 0.822854i \(0.307619\pi\)
\(828\) 2.56781e6 0.130163
\(829\) −3.04175e7 −1.53723 −0.768613 0.639714i \(-0.779053\pi\)
−0.768613 + 0.639714i \(0.779053\pi\)
\(830\) −1.30603e7 −0.658050
\(831\) 6.74548e6 0.338852
\(832\) 2.07391e6 0.103868
\(833\) −2.95867e7 −1.47735
\(834\) 1.53740e7 0.765371
\(835\) −3.08619e6 −0.153181
\(836\) 0 0
\(837\) −2.61256e7 −1.28900
\(838\) −1.38834e7 −0.682943
\(839\) −2.41182e7 −1.18288 −0.591439 0.806350i \(-0.701440\pi\)
−0.591439 + 0.806350i \(0.701440\pi\)
\(840\) 1.47687e7 0.722179
\(841\) −2.03302e7 −0.991179
\(842\) −4.76893e6 −0.231814
\(843\) −5.22020e6 −0.252999
\(844\) 1.29558e7 0.626049
\(845\) −1.09786e7 −0.528939
\(846\) 1.14247e7 0.548807
\(847\) −9.61810e6 −0.460660
\(848\) −7.44749e6 −0.355648
\(849\) 1.42423e7 0.678125
\(850\) −1.91318e7 −0.908257
\(851\) 9.34792e6 0.442477
\(852\) −5.19295e6 −0.245084
\(853\) 1.65084e7 0.776842 0.388421 0.921482i \(-0.373021\pi\)
0.388421 + 0.921482i \(0.373021\pi\)
\(854\) −7.72914e6 −0.362649
\(855\) 0 0
\(856\) −4.35337e6 −0.203068
\(857\) −1.13507e7 −0.527923 −0.263962 0.964533i \(-0.585029\pi\)
−0.263962 + 0.964533i \(0.585029\pi\)
\(858\) 7.28708e6 0.337937
\(859\) 2.00932e7 0.929108 0.464554 0.885545i \(-0.346215\pi\)
0.464554 + 0.885545i \(0.346215\pi\)
\(860\) 1.97511e7 0.910635
\(861\) −3.88439e7 −1.78573
\(862\) −2.71718e6 −0.124552
\(863\) 1.97337e7 0.901950 0.450975 0.892537i \(-0.351076\pi\)
0.450975 + 0.892537i \(0.351076\pi\)
\(864\) −4.02426e6 −0.183401
\(865\) −1.33841e7 −0.608205
\(866\) −2.85688e7 −1.29448
\(867\) −8.16082e6 −0.368711
\(868\) 2.47001e7 1.11275
\(869\) −3.32849e6 −0.149520
\(870\) −1.69078e6 −0.0757336
\(871\) −7.26430e6 −0.324451
\(872\) −1.24049e6 −0.0552460
\(873\) −1.30737e7 −0.580583
\(874\) 0 0
\(875\) 6.37887e7 2.81659
\(876\) 331595. 0.0145998
\(877\) 1.84073e7 0.808149 0.404074 0.914726i \(-0.367594\pi\)
0.404074 + 0.914726i \(0.367594\pi\)
\(878\) 6.46510e6 0.283034
\(879\) −7.12275e6 −0.310939
\(880\) 8.45851e6 0.368203
\(881\) 1.42310e7 0.617727 0.308864 0.951106i \(-0.400051\pi\)
0.308864 + 0.951106i \(0.400051\pi\)
\(882\) −2.00125e7 −0.866224
\(883\) 1.06212e7 0.458430 0.229215 0.973376i \(-0.426384\pi\)
0.229215 + 0.973376i \(0.426384\pi\)
\(884\) −6.45740e6 −0.277925
\(885\) 9.28761e6 0.398608
\(886\) −1.26314e7 −0.540591
\(887\) 4.79046e6 0.204441 0.102220 0.994762i \(-0.467405\pi\)
0.102220 + 0.994762i \(0.467405\pi\)
\(888\) −5.22687e6 −0.222438
\(889\) 2.20436e7 0.935465
\(890\) −4.25373e7 −1.80009
\(891\) −2.81118e6 −0.118630
\(892\) 1.01534e7 0.427268
\(893\) 0 0
\(894\) 451281. 0.0188844
\(895\) 1.95915e7 0.817541
\(896\) 3.80467e6 0.158324
\(897\) −6.27132e6 −0.260242
\(898\) 5.91802e6 0.244898
\(899\) −2.82775e6 −0.116692
\(900\) −1.29408e7 −0.532543
\(901\) 2.31887e7 0.951623
\(902\) −2.22471e7 −0.910451
\(903\) 3.12159e7 1.27396
\(904\) −8.20861e6 −0.334078
\(905\) −5.70124e7 −2.31392
\(906\) 1.39782e6 0.0565758
\(907\) −3.87079e7 −1.56236 −0.781180 0.624306i \(-0.785382\pi\)
−0.781180 + 0.624306i \(0.785382\pi\)
\(908\) −1.50176e6 −0.0604484
\(909\) 2.29267e7 0.920303
\(910\) 4.49280e7 1.79851
\(911\) 2.75114e7 1.09829 0.549146 0.835727i \(-0.314953\pi\)
0.549146 + 0.835727i \(0.314953\pi\)
\(912\) 0 0
\(913\) −1.18220e7 −0.469368
\(914\) −2.12305e7 −0.840612
\(915\) −8.26877e6 −0.326504
\(916\) 2.81896e6 0.111007
\(917\) 6.16390e7 2.42065
\(918\) 1.25300e7 0.490734
\(919\) −1.32340e7 −0.516896 −0.258448 0.966025i \(-0.583211\pi\)
−0.258448 + 0.966025i \(0.583211\pi\)
\(920\) −7.27947e6 −0.283550
\(921\) −1.49050e7 −0.579004
\(922\) −6.19159e6 −0.239870
\(923\) −1.57975e7 −0.610357
\(924\) 1.33684e7 0.515109
\(925\) −4.71101e7 −1.81034
\(926\) 2.92879e7 1.12244
\(927\) 9.78170e6 0.373865
\(928\) −435572. −0.0166031
\(929\) −1.94057e7 −0.737717 −0.368858 0.929486i \(-0.620251\pi\)
−0.368858 + 0.929486i \(0.620251\pi\)
\(930\) 2.64245e7 1.00184
\(931\) 0 0
\(932\) −3.61857e6 −0.136457
\(933\) −2.98111e6 −0.112118
\(934\) 2.76629e7 1.03760
\(935\) −2.63367e7 −0.985217
\(936\) −4.36780e6 −0.162957
\(937\) 4.70727e7 1.75154 0.875770 0.482729i \(-0.160354\pi\)
0.875770 + 0.482729i \(0.160354\pi\)
\(938\) −1.33266e7 −0.494553
\(939\) 3.39395e6 0.125615
\(940\) −3.23879e7 −1.19554
\(941\) 5.01568e7 1.84653 0.923264 0.384167i \(-0.125511\pi\)
0.923264 + 0.384167i \(0.125511\pi\)
\(942\) 2.37848e7 0.873317
\(943\) 1.91460e7 0.701131
\(944\) 2.39264e6 0.0873871
\(945\) −8.71790e7 −3.17565
\(946\) 1.78783e7 0.649529
\(947\) 1.85612e7 0.672561 0.336281 0.941762i \(-0.390831\pi\)
0.336281 + 0.941762i \(0.390831\pi\)
\(948\) −1.60170e6 −0.0578841
\(949\) 1.00875e6 0.0363594
\(950\) 0 0
\(951\) 3.16633e7 1.13529
\(952\) −1.18463e7 −0.423635
\(953\) 3.60296e7 1.28507 0.642536 0.766255i \(-0.277882\pi\)
0.642536 + 0.766255i \(0.277882\pi\)
\(954\) 1.56849e7 0.557970
\(955\) 3.57487e7 1.26839
\(956\) −5.50394e6 −0.194773
\(957\) −1.53046e6 −0.0540185
\(958\) 1.29067e7 0.454361
\(959\) −1.61676e7 −0.567673
\(960\) 4.07030e6 0.142544
\(961\) 1.55647e7 0.543667
\(962\) −1.59007e7 −0.553959
\(963\) 9.16848e6 0.318590
\(964\) 2.53898e7 0.879966
\(965\) 1.76812e7 0.611215
\(966\) −1.15050e7 −0.396682
\(967\) 1.22925e7 0.422740 0.211370 0.977406i \(-0.432207\pi\)
0.211370 + 0.977406i \(0.432207\pi\)
\(968\) −2.65077e6 −0.0909251
\(969\) 0 0
\(970\) 3.70627e7 1.26476
\(971\) 1.73779e7 0.591493 0.295746 0.955267i \(-0.404432\pi\)
0.295746 + 0.955267i \(0.404432\pi\)
\(972\) 1.39268e7 0.472810
\(973\) 8.57999e7 2.90539
\(974\) −1.85997e7 −0.628215
\(975\) 3.16051e7 1.06475
\(976\) −2.13017e6 −0.0715797
\(977\) −1.54100e7 −0.516496 −0.258248 0.966079i \(-0.583145\pi\)
−0.258248 + 0.966079i \(0.583145\pi\)
\(978\) 1.83524e7 0.613542
\(979\) −3.85040e7 −1.28395
\(980\) 5.67334e7 1.88701
\(981\) 2.61254e6 0.0866744
\(982\) 3.29509e7 1.09041
\(983\) 5.45907e7 1.80192 0.900959 0.433903i \(-0.142864\pi\)
0.900959 + 0.433903i \(0.142864\pi\)
\(984\) −1.07055e7 −0.352466
\(985\) 1.42832e7 0.469068
\(986\) 1.35621e6 0.0444258
\(987\) −5.11880e7 −1.67254
\(988\) 0 0
\(989\) −1.53862e7 −0.500197
\(990\) −1.78142e7 −0.577667
\(991\) 4.29839e7 1.39034 0.695171 0.718845i \(-0.255329\pi\)
0.695171 + 0.718845i \(0.255329\pi\)
\(992\) 6.80740e6 0.219635
\(993\) −2.35410e7 −0.757622
\(994\) −2.89810e7 −0.930353
\(995\) −2.92241e7 −0.935800
\(996\) −5.68883e6 −0.181708
\(997\) −5.32456e7 −1.69647 −0.848234 0.529622i \(-0.822334\pi\)
−0.848234 + 0.529622i \(0.822334\pi\)
\(998\) −1.49196e7 −0.474166
\(999\) 3.08539e7 0.978130
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.r.1.10 15
19.6 even 9 38.6.e.b.17.4 yes 30
19.16 even 9 38.6.e.b.9.4 30
19.18 odd 2 722.6.a.q.1.6 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.6.e.b.9.4 30 19.16 even 9
38.6.e.b.17.4 yes 30 19.6 even 9
722.6.a.q.1.6 15 19.18 odd 2
722.6.a.r.1.10 15 1.1 even 1 trivial