Properties

Label 927.2.a.b.1.2
Level $927$
Weight $2$
Character 927.1
Self dual yes
Analytic conductor $7.402$
Analytic rank $0$
Dimension $2$
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] = [927,2,Mod(1,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 927.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61803 q^{2} +4.85410 q^{4} +0.381966 q^{5} -1.00000 q^{7} +7.47214 q^{8} +O(q^{10})\) \(q+2.61803 q^{2} +4.85410 q^{4} +0.381966 q^{5} -1.00000 q^{7} +7.47214 q^{8} +1.00000 q^{10} +2.61803 q^{11} -4.85410 q^{13} -2.61803 q^{14} +9.85410 q^{16} +5.61803 q^{17} +5.85410 q^{19} +1.85410 q^{20} +6.85410 q^{22} -4.47214 q^{23} -4.85410 q^{25} -12.7082 q^{26} -4.85410 q^{28} +5.23607 q^{29} -6.70820 q^{31} +10.8541 q^{32} +14.7082 q^{34} -0.381966 q^{35} +6.70820 q^{37} +15.3262 q^{38} +2.85410 q^{40} -8.94427 q^{41} -8.70820 q^{43} +12.7082 q^{44} -11.7082 q^{46} -4.09017 q^{47} -6.00000 q^{49} -12.7082 q^{50} -23.5623 q^{52} -1.09017 q^{53} +1.00000 q^{55} -7.47214 q^{56} +13.7082 q^{58} -6.38197 q^{59} +4.14590 q^{61} -17.5623 q^{62} +8.70820 q^{64} -1.85410 q^{65} +14.4164 q^{67} +27.2705 q^{68} -1.00000 q^{70} +4.09017 q^{71} -10.8541 q^{73} +17.5623 q^{74} +28.4164 q^{76} -2.61803 q^{77} -6.56231 q^{79} +3.76393 q^{80} -23.4164 q^{82} +6.32624 q^{83} +2.14590 q^{85} -22.7984 q^{86} +19.5623 q^{88} +2.29180 q^{89} +4.85410 q^{91} -21.7082 q^{92} -10.7082 q^{94} +2.23607 q^{95} -1.70820 q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 13 q^{16} + 9 q^{17} + 5 q^{19} - 3 q^{20} + 7 q^{22} - 3 q^{25} - 12 q^{26} - 3 q^{28} + 6 q^{29} + 15 q^{32} + 16 q^{34} - 3 q^{35} + 15 q^{38} - q^{40} - 4 q^{43} + 12 q^{44} - 10 q^{46} + 3 q^{47} - 12 q^{49} - 12 q^{50} - 27 q^{52} + 9 q^{53} + 2 q^{55} - 6 q^{56} + 14 q^{58} - 15 q^{59} + 15 q^{61} - 15 q^{62} + 4 q^{64} + 3 q^{65} + 2 q^{67} + 21 q^{68} - 2 q^{70} - 3 q^{71} - 15 q^{73} + 15 q^{74} + 30 q^{76} - 3 q^{77} + 7 q^{79} + 12 q^{80} - 20 q^{82} - 3 q^{83} + 11 q^{85} - 21 q^{86} + 19 q^{88} + 18 q^{89} + 3 q^{91} - 30 q^{92} - 8 q^{94} + 10 q^{97} - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61803 1.85123 0.925615 0.378467i \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) 0 0
\(4\) 4.85410 2.42705
\(5\) 0.381966 0.170820 0.0854102 0.996346i \(-0.472780\pi\)
0.0854102 + 0.996346i \(0.472780\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 7.47214 2.64180
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.61803 0.789367 0.394683 0.918817i \(-0.370854\pi\)
0.394683 + 0.918817i \(0.370854\pi\)
\(12\) 0 0
\(13\) −4.85410 −1.34629 −0.673143 0.739512i \(-0.735056\pi\)
−0.673143 + 0.739512i \(0.735056\pi\)
\(14\) −2.61803 −0.699699
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) 5.61803 1.36257 0.681287 0.732017i \(-0.261421\pi\)
0.681287 + 0.732017i \(0.261421\pi\)
\(18\) 0 0
\(19\) 5.85410 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) 1.85410 0.414590
\(21\) 0 0
\(22\) 6.85410 1.46130
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) −4.85410 −0.970820
\(26\) −12.7082 −2.49228
\(27\) 0 0
\(28\) −4.85410 −0.917339
\(29\) 5.23607 0.972313 0.486157 0.873872i \(-0.338398\pi\)
0.486157 + 0.873872i \(0.338398\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) 14.7082 2.52244
\(35\) −0.381966 −0.0645640
\(36\) 0 0
\(37\) 6.70820 1.10282 0.551411 0.834234i \(-0.314090\pi\)
0.551411 + 0.834234i \(0.314090\pi\)
\(38\) 15.3262 2.48624
\(39\) 0 0
\(40\) 2.85410 0.451273
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) 0 0
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) 12.7082 1.91583
\(45\) 0 0
\(46\) −11.7082 −1.72628
\(47\) −4.09017 −0.596613 −0.298306 0.954470i \(-0.596422\pi\)
−0.298306 + 0.954470i \(0.596422\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −12.7082 −1.79721
\(51\) 0 0
\(52\) −23.5623 −3.26750
\(53\) −1.09017 −0.149746 −0.0748732 0.997193i \(-0.523855\pi\)
−0.0748732 + 0.997193i \(0.523855\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −7.47214 −0.998506
\(57\) 0 0
\(58\) 13.7082 1.79998
\(59\) −6.38197 −0.830861 −0.415431 0.909625i \(-0.636369\pi\)
−0.415431 + 0.909625i \(0.636369\pi\)
\(60\) 0 0
\(61\) 4.14590 0.530828 0.265414 0.964135i \(-0.414491\pi\)
0.265414 + 0.964135i \(0.414491\pi\)
\(62\) −17.5623 −2.23042
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) −1.85410 −0.229973
\(66\) 0 0
\(67\) 14.4164 1.76124 0.880622 0.473819i \(-0.157125\pi\)
0.880622 + 0.473819i \(0.157125\pi\)
\(68\) 27.2705 3.30704
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 4.09017 0.485414 0.242707 0.970100i \(-0.421965\pi\)
0.242707 + 0.970100i \(0.421965\pi\)
\(72\) 0 0
\(73\) −10.8541 −1.27038 −0.635188 0.772357i \(-0.719077\pi\)
−0.635188 + 0.772357i \(0.719077\pi\)
\(74\) 17.5623 2.04158
\(75\) 0 0
\(76\) 28.4164 3.25959
\(77\) −2.61803 −0.298353
\(78\) 0 0
\(79\) −6.56231 −0.738317 −0.369159 0.929366i \(-0.620354\pi\)
−0.369159 + 0.929366i \(0.620354\pi\)
\(80\) 3.76393 0.420820
\(81\) 0 0
\(82\) −23.4164 −2.58591
\(83\) 6.32624 0.694395 0.347197 0.937792i \(-0.387133\pi\)
0.347197 + 0.937792i \(0.387133\pi\)
\(84\) 0 0
\(85\) 2.14590 0.232755
\(86\) −22.7984 −2.45841
\(87\) 0 0
\(88\) 19.5623 2.08535
\(89\) 2.29180 0.242930 0.121465 0.992596i \(-0.461241\pi\)
0.121465 + 0.992596i \(0.461241\pi\)
\(90\) 0 0
\(91\) 4.85410 0.508848
\(92\) −21.7082 −2.26324
\(93\) 0 0
\(94\) −10.7082 −1.10447
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) −1.70820 −0.173442 −0.0867209 0.996233i \(-0.527639\pi\)
−0.0867209 + 0.996233i \(0.527639\pi\)
\(98\) −15.7082 −1.58677
\(99\) 0 0
\(100\) −23.5623 −2.35623
\(101\) −1.90983 −0.190035 −0.0950176 0.995476i \(-0.530291\pi\)
−0.0950176 + 0.995476i \(0.530291\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329
\(104\) −36.2705 −3.55662
\(105\) 0 0
\(106\) −2.85410 −0.277215
\(107\) 7.09017 0.685433 0.342716 0.939439i \(-0.388653\pi\)
0.342716 + 0.939439i \(0.388653\pi\)
\(108\) 0 0
\(109\) −9.56231 −0.915903 −0.457951 0.888977i \(-0.651417\pi\)
−0.457951 + 0.888977i \(0.651417\pi\)
\(110\) 2.61803 0.249620
\(111\) 0 0
\(112\) −9.85410 −0.931125
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) −1.70820 −0.159291
\(116\) 25.4164 2.35985
\(117\) 0 0
\(118\) −16.7082 −1.53811
\(119\) −5.61803 −0.515004
\(120\) 0 0
\(121\) −4.14590 −0.376900
\(122\) 10.8541 0.982684
\(123\) 0 0
\(124\) −32.5623 −2.92418
\(125\) −3.76393 −0.336656
\(126\) 0 0
\(127\) 15.2705 1.35504 0.677519 0.735505i \(-0.263055\pi\)
0.677519 + 0.735505i \(0.263055\pi\)
\(128\) 1.09017 0.0963583
\(129\) 0 0
\(130\) −4.85410 −0.425733
\(131\) 2.23607 0.195366 0.0976831 0.995218i \(-0.468857\pi\)
0.0976831 + 0.995218i \(0.468857\pi\)
\(132\) 0 0
\(133\) −5.85410 −0.507615
\(134\) 37.7426 3.26047
\(135\) 0 0
\(136\) 41.9787 3.59965
\(137\) 12.7082 1.08574 0.542868 0.839818i \(-0.317339\pi\)
0.542868 + 0.839818i \(0.317339\pi\)
\(138\) 0 0
\(139\) −11.1459 −0.945383 −0.472691 0.881228i \(-0.656717\pi\)
−0.472691 + 0.881228i \(0.656717\pi\)
\(140\) −1.85410 −0.156700
\(141\) 0 0
\(142\) 10.7082 0.898613
\(143\) −12.7082 −1.06271
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) −28.4164 −2.35176
\(147\) 0 0
\(148\) 32.5623 2.67661
\(149\) 7.47214 0.612141 0.306071 0.952009i \(-0.400986\pi\)
0.306071 + 0.952009i \(0.400986\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 43.7426 3.54800
\(153\) 0 0
\(154\) −6.85410 −0.552319
\(155\) −2.56231 −0.205809
\(156\) 0 0
\(157\) 16.7082 1.33346 0.666730 0.745299i \(-0.267693\pi\)
0.666730 + 0.745299i \(0.267693\pi\)
\(158\) −17.1803 −1.36679
\(159\) 0 0
\(160\) 4.14590 0.327762
\(161\) 4.47214 0.352454
\(162\) 0 0
\(163\) 2.70820 0.212123 0.106061 0.994360i \(-0.466176\pi\)
0.106061 + 0.994360i \(0.466176\pi\)
\(164\) −43.4164 −3.39025
\(165\) 0 0
\(166\) 16.5623 1.28548
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 10.5623 0.812485
\(170\) 5.61803 0.430884
\(171\) 0 0
\(172\) −42.2705 −3.22310
\(173\) −16.0344 −1.21908 −0.609538 0.792757i \(-0.708645\pi\)
−0.609538 + 0.792757i \(0.708645\pi\)
\(174\) 0 0
\(175\) 4.85410 0.366936
\(176\) 25.7984 1.94463
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 7.85410 0.587043 0.293522 0.955952i \(-0.405173\pi\)
0.293522 + 0.955952i \(0.405173\pi\)
\(180\) 0 0
\(181\) 3.85410 0.286473 0.143237 0.989688i \(-0.454249\pi\)
0.143237 + 0.989688i \(0.454249\pi\)
\(182\) 12.7082 0.941995
\(183\) 0 0
\(184\) −33.4164 −2.46349
\(185\) 2.56231 0.188384
\(186\) 0 0
\(187\) 14.7082 1.07557
\(188\) −19.8541 −1.44801
\(189\) 0 0
\(190\) 5.85410 0.424701
\(191\) −5.61803 −0.406507 −0.203253 0.979126i \(-0.565152\pi\)
−0.203253 + 0.979126i \(0.565152\pi\)
\(192\) 0 0
\(193\) 20.1246 1.44860 0.724301 0.689484i \(-0.242163\pi\)
0.724301 + 0.689484i \(0.242163\pi\)
\(194\) −4.47214 −0.321081
\(195\) 0 0
\(196\) −29.1246 −2.08033
\(197\) 16.4164 1.16962 0.584810 0.811170i \(-0.301169\pi\)
0.584810 + 0.811170i \(0.301169\pi\)
\(198\) 0 0
\(199\) −23.4164 −1.65995 −0.829973 0.557804i \(-0.811644\pi\)
−0.829973 + 0.557804i \(0.811644\pi\)
\(200\) −36.2705 −2.56471
\(201\) 0 0
\(202\) −5.00000 −0.351799
\(203\) −5.23607 −0.367500
\(204\) 0 0
\(205\) −3.41641 −0.238612
\(206\) −2.61803 −0.182407
\(207\) 0 0
\(208\) −47.8328 −3.31661
\(209\) 15.3262 1.06014
\(210\) 0 0
\(211\) 14.8541 1.02260 0.511299 0.859403i \(-0.329164\pi\)
0.511299 + 0.859403i \(0.329164\pi\)
\(212\) −5.29180 −0.363442
\(213\) 0 0
\(214\) 18.5623 1.26889
\(215\) −3.32624 −0.226848
\(216\) 0 0
\(217\) 6.70820 0.455383
\(218\) −25.0344 −1.69555
\(219\) 0 0
\(220\) 4.85410 0.327263
\(221\) −27.2705 −1.83441
\(222\) 0 0
\(223\) 7.70820 0.516180 0.258090 0.966121i \(-0.416907\pi\)
0.258090 + 0.966121i \(0.416907\pi\)
\(224\) −10.8541 −0.725220
\(225\) 0 0
\(226\) 39.2705 2.61224
\(227\) −2.94427 −0.195418 −0.0977091 0.995215i \(-0.531151\pi\)
−0.0977091 + 0.995215i \(0.531151\pi\)
\(228\) 0 0
\(229\) 6.70820 0.443291 0.221645 0.975127i \(-0.428857\pi\)
0.221645 + 0.975127i \(0.428857\pi\)
\(230\) −4.47214 −0.294884
\(231\) 0 0
\(232\) 39.1246 2.56866
\(233\) 26.8885 1.76153 0.880764 0.473556i \(-0.157030\pi\)
0.880764 + 0.473556i \(0.157030\pi\)
\(234\) 0 0
\(235\) −1.56231 −0.101914
\(236\) −30.9787 −2.01654
\(237\) 0 0
\(238\) −14.7082 −0.953391
\(239\) −8.67376 −0.561059 −0.280530 0.959845i \(-0.590510\pi\)
−0.280530 + 0.959845i \(0.590510\pi\)
\(240\) 0 0
\(241\) −8.27051 −0.532750 −0.266375 0.963869i \(-0.585826\pi\)
−0.266375 + 0.963869i \(0.585826\pi\)
\(242\) −10.8541 −0.697728
\(243\) 0 0
\(244\) 20.1246 1.28835
\(245\) −2.29180 −0.146417
\(246\) 0 0
\(247\) −28.4164 −1.80809
\(248\) −50.1246 −3.18292
\(249\) 0 0
\(250\) −9.85410 −0.623228
\(251\) −11.2361 −0.709214 −0.354607 0.935015i \(-0.615385\pi\)
−0.354607 + 0.935015i \(0.615385\pi\)
\(252\) 0 0
\(253\) −11.7082 −0.736088
\(254\) 39.9787 2.50849
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) 13.4721 0.840369 0.420184 0.907439i \(-0.361965\pi\)
0.420184 + 0.907439i \(0.361965\pi\)
\(258\) 0 0
\(259\) −6.70820 −0.416828
\(260\) −9.00000 −0.558156
\(261\) 0 0
\(262\) 5.85410 0.361668
\(263\) 20.6180 1.27136 0.635681 0.771952i \(-0.280719\pi\)
0.635681 + 0.771952i \(0.280719\pi\)
\(264\) 0 0
\(265\) −0.416408 −0.0255797
\(266\) −15.3262 −0.939712
\(267\) 0 0
\(268\) 69.9787 4.27463
\(269\) −12.3262 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 55.3607 3.35673
\(273\) 0 0
\(274\) 33.2705 2.00995
\(275\) −12.7082 −0.766334
\(276\) 0 0
\(277\) 4.70820 0.282889 0.141444 0.989946i \(-0.454825\pi\)
0.141444 + 0.989946i \(0.454825\pi\)
\(278\) −29.1803 −1.75012
\(279\) 0 0
\(280\) −2.85410 −0.170565
\(281\) 31.4721 1.87747 0.938735 0.344640i \(-0.111999\pi\)
0.938735 + 0.344640i \(0.111999\pi\)
\(282\) 0 0
\(283\) −19.7082 −1.17153 −0.585766 0.810481i \(-0.699206\pi\)
−0.585766 + 0.810481i \(0.699206\pi\)
\(284\) 19.8541 1.17812
\(285\) 0 0
\(286\) −33.2705 −1.96733
\(287\) 8.94427 0.527964
\(288\) 0 0
\(289\) 14.5623 0.856606
\(290\) 5.23607 0.307472
\(291\) 0 0
\(292\) −52.6869 −3.08327
\(293\) 21.6525 1.26495 0.632476 0.774580i \(-0.282039\pi\)
0.632476 + 0.774580i \(0.282039\pi\)
\(294\) 0 0
\(295\) −2.43769 −0.141928
\(296\) 50.1246 2.91343
\(297\) 0 0
\(298\) 19.5623 1.13321
\(299\) 21.7082 1.25542
\(300\) 0 0
\(301\) 8.70820 0.501933
\(302\) −49.7426 −2.86237
\(303\) 0 0
\(304\) 57.6869 3.30857
\(305\) 1.58359 0.0906762
\(306\) 0 0
\(307\) −2.85410 −0.162892 −0.0814461 0.996678i \(-0.525954\pi\)
−0.0814461 + 0.996678i \(0.525954\pi\)
\(308\) −12.7082 −0.724117
\(309\) 0 0
\(310\) −6.70820 −0.381000
\(311\) −2.88854 −0.163794 −0.0818971 0.996641i \(-0.526098\pi\)
−0.0818971 + 0.996641i \(0.526098\pi\)
\(312\) 0 0
\(313\) 16.2918 0.920867 0.460433 0.887694i \(-0.347694\pi\)
0.460433 + 0.887694i \(0.347694\pi\)
\(314\) 43.7426 2.46854
\(315\) 0 0
\(316\) −31.8541 −1.79193
\(317\) −28.4164 −1.59602 −0.798012 0.602641i \(-0.794115\pi\)
−0.798012 + 0.602641i \(0.794115\pi\)
\(318\) 0 0
\(319\) 13.7082 0.767512
\(320\) 3.32624 0.185942
\(321\) 0 0
\(322\) 11.7082 0.652473
\(323\) 32.8885 1.82997
\(324\) 0 0
\(325\) 23.5623 1.30700
\(326\) 7.09017 0.392688
\(327\) 0 0
\(328\) −66.8328 −3.69022
\(329\) 4.09017 0.225498
\(330\) 0 0
\(331\) −16.1459 −0.887459 −0.443729 0.896161i \(-0.646345\pi\)
−0.443729 + 0.896161i \(0.646345\pi\)
\(332\) 30.7082 1.68533
\(333\) 0 0
\(334\) −23.5623 −1.28927
\(335\) 5.50658 0.300856
\(336\) 0 0
\(337\) −2.43769 −0.132790 −0.0663948 0.997793i \(-0.521150\pi\)
−0.0663948 + 0.997793i \(0.521150\pi\)
\(338\) 27.6525 1.50410
\(339\) 0 0
\(340\) 10.4164 0.564909
\(341\) −17.5623 −0.951052
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −65.0689 −3.50828
\(345\) 0 0
\(346\) −41.9787 −2.25679
\(347\) 1.47214 0.0790284 0.0395142 0.999219i \(-0.487419\pi\)
0.0395142 + 0.999219i \(0.487419\pi\)
\(348\) 0 0
\(349\) 15.4164 0.825221 0.412611 0.910907i \(-0.364617\pi\)
0.412611 + 0.910907i \(0.364617\pi\)
\(350\) 12.7082 0.679282
\(351\) 0 0
\(352\) 28.4164 1.51460
\(353\) 25.0344 1.33245 0.666224 0.745751i \(-0.267909\pi\)
0.666224 + 0.745751i \(0.267909\pi\)
\(354\) 0 0
\(355\) 1.56231 0.0829186
\(356\) 11.1246 0.589603
\(357\) 0 0
\(358\) 20.5623 1.08675
\(359\) −14.6738 −0.774452 −0.387226 0.921985i \(-0.626567\pi\)
−0.387226 + 0.921985i \(0.626567\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) 10.0902 0.530328
\(363\) 0 0
\(364\) 23.5623 1.23500
\(365\) −4.14590 −0.217006
\(366\) 0 0
\(367\) 16.4377 0.858041 0.429020 0.903295i \(-0.358859\pi\)
0.429020 + 0.903295i \(0.358859\pi\)
\(368\) −44.0689 −2.29725
\(369\) 0 0
\(370\) 6.70820 0.348743
\(371\) 1.09017 0.0565988
\(372\) 0 0
\(373\) −22.6869 −1.17468 −0.587342 0.809339i \(-0.699826\pi\)
−0.587342 + 0.809339i \(0.699826\pi\)
\(374\) 38.5066 1.99113
\(375\) 0 0
\(376\) −30.5623 −1.57613
\(377\) −25.4164 −1.30901
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 10.8541 0.556804
\(381\) 0 0
\(382\) −14.7082 −0.752537
\(383\) −0.819660 −0.0418827 −0.0209413 0.999781i \(-0.506666\pi\)
−0.0209413 + 0.999781i \(0.506666\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 52.6869 2.68169
\(387\) 0 0
\(388\) −8.29180 −0.420952
\(389\) −7.41641 −0.376027 −0.188013 0.982166i \(-0.560205\pi\)
−0.188013 + 0.982166i \(0.560205\pi\)
\(390\) 0 0
\(391\) −25.1246 −1.27061
\(392\) −44.8328 −2.26440
\(393\) 0 0
\(394\) 42.9787 2.16524
\(395\) −2.50658 −0.126120
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −61.3050 −3.07294
\(399\) 0 0
\(400\) −47.8328 −2.39164
\(401\) 23.8885 1.19294 0.596468 0.802637i \(-0.296570\pi\)
0.596468 + 0.802637i \(0.296570\pi\)
\(402\) 0 0
\(403\) 32.5623 1.62204
\(404\) −9.27051 −0.461225
\(405\) 0 0
\(406\) −13.7082 −0.680327
\(407\) 17.5623 0.870531
\(408\) 0 0
\(409\) 36.7082 1.81510 0.907552 0.419940i \(-0.137949\pi\)
0.907552 + 0.419940i \(0.137949\pi\)
\(410\) −8.94427 −0.441726
\(411\) 0 0
\(412\) −4.85410 −0.239144
\(413\) 6.38197 0.314036
\(414\) 0 0
\(415\) 2.41641 0.118617
\(416\) −52.6869 −2.58319
\(417\) 0 0
\(418\) 40.1246 1.96256
\(419\) 4.09017 0.199818 0.0999089 0.994997i \(-0.468145\pi\)
0.0999089 + 0.994997i \(0.468145\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) 38.8885 1.89306
\(423\) 0 0
\(424\) −8.14590 −0.395600
\(425\) −27.2705 −1.32281
\(426\) 0 0
\(427\) −4.14590 −0.200634
\(428\) 34.4164 1.66358
\(429\) 0 0
\(430\) −8.70820 −0.419947
\(431\) −34.3607 −1.65510 −0.827548 0.561395i \(-0.810265\pi\)
−0.827548 + 0.561395i \(0.810265\pi\)
\(432\) 0 0
\(433\) 14.4164 0.692808 0.346404 0.938085i \(-0.387403\pi\)
0.346404 + 0.938085i \(0.387403\pi\)
\(434\) 17.5623 0.843018
\(435\) 0 0
\(436\) −46.4164 −2.22294
\(437\) −26.1803 −1.25238
\(438\) 0 0
\(439\) 29.5623 1.41093 0.705466 0.708744i \(-0.250738\pi\)
0.705466 + 0.708744i \(0.250738\pi\)
\(440\) 7.47214 0.356220
\(441\) 0 0
\(442\) −71.3951 −3.39592
\(443\) 15.4377 0.733467 0.366733 0.930326i \(-0.380476\pi\)
0.366733 + 0.930326i \(0.380476\pi\)
\(444\) 0 0
\(445\) 0.875388 0.0414974
\(446\) 20.1803 0.955567
\(447\) 0 0
\(448\) −8.70820 −0.411424
\(449\) 13.3607 0.630529 0.315265 0.949004i \(-0.397907\pi\)
0.315265 + 0.949004i \(0.397907\pi\)
\(450\) 0 0
\(451\) −23.4164 −1.10264
\(452\) 72.8115 3.42477
\(453\) 0 0
\(454\) −7.70820 −0.361764
\(455\) 1.85410 0.0869216
\(456\) 0 0
\(457\) −9.85410 −0.460955 −0.230478 0.973078i \(-0.574029\pi\)
−0.230478 + 0.973078i \(0.574029\pi\)
\(458\) 17.5623 0.820633
\(459\) 0 0
\(460\) −8.29180 −0.386607
\(461\) −12.2148 −0.568899 −0.284450 0.958691i \(-0.591811\pi\)
−0.284450 + 0.958691i \(0.591811\pi\)
\(462\) 0 0
\(463\) −21.4164 −0.995305 −0.497652 0.867377i \(-0.665804\pi\)
−0.497652 + 0.867377i \(0.665804\pi\)
\(464\) 51.5967 2.39532
\(465\) 0 0
\(466\) 70.3951 3.26099
\(467\) 3.65248 0.169016 0.0845082 0.996423i \(-0.473068\pi\)
0.0845082 + 0.996423i \(0.473068\pi\)
\(468\) 0 0
\(469\) −14.4164 −0.665688
\(470\) −4.09017 −0.188665
\(471\) 0 0
\(472\) −47.6869 −2.19497
\(473\) −22.7984 −1.04827
\(474\) 0 0
\(475\) −28.4164 −1.30383
\(476\) −27.2705 −1.24994
\(477\) 0 0
\(478\) −22.7082 −1.03865
\(479\) −8.18034 −0.373769 −0.186885 0.982382i \(-0.559839\pi\)
−0.186885 + 0.982382i \(0.559839\pi\)
\(480\) 0 0
\(481\) −32.5623 −1.48471
\(482\) −21.6525 −0.986243
\(483\) 0 0
\(484\) −20.1246 −0.914755
\(485\) −0.652476 −0.0296274
\(486\) 0 0
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 30.9787 1.40234
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −35.7771 −1.61460 −0.807299 0.590143i \(-0.799071\pi\)
−0.807299 + 0.590143i \(0.799071\pi\)
\(492\) 0 0
\(493\) 29.4164 1.32485
\(494\) −74.3951 −3.34719
\(495\) 0 0
\(496\) −66.1033 −2.96813
\(497\) −4.09017 −0.183469
\(498\) 0 0
\(499\) 13.2705 0.594070 0.297035 0.954867i \(-0.404002\pi\)
0.297035 + 0.954867i \(0.404002\pi\)
\(500\) −18.2705 −0.817082
\(501\) 0 0
\(502\) −29.4164 −1.31292
\(503\) −13.3607 −0.595723 −0.297862 0.954609i \(-0.596273\pi\)
−0.297862 + 0.954609i \(0.596273\pi\)
\(504\) 0 0
\(505\) −0.729490 −0.0324619
\(506\) −30.6525 −1.36267
\(507\) 0 0
\(508\) 74.1246 3.28875
\(509\) −26.6180 −1.17982 −0.589912 0.807468i \(-0.700837\pi\)
−0.589912 + 0.807468i \(0.700837\pi\)
\(510\) 0 0
\(511\) 10.8541 0.480157
\(512\) −40.3050 −1.78124
\(513\) 0 0
\(514\) 35.2705 1.55572
\(515\) −0.381966 −0.0168314
\(516\) 0 0
\(517\) −10.7082 −0.470946
\(518\) −17.5623 −0.771643
\(519\) 0 0
\(520\) −13.8541 −0.607543
\(521\) −17.1803 −0.752684 −0.376342 0.926481i \(-0.622818\pi\)
−0.376342 + 0.926481i \(0.622818\pi\)
\(522\) 0 0
\(523\) −10.5836 −0.462788 −0.231394 0.972860i \(-0.574329\pi\)
−0.231394 + 0.972860i \(0.574329\pi\)
\(524\) 10.8541 0.474164
\(525\) 0 0
\(526\) 53.9787 2.35358
\(527\) −37.6869 −1.64167
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) −1.09017 −0.0473540
\(531\) 0 0
\(532\) −28.4164 −1.23201
\(533\) 43.4164 1.88057
\(534\) 0 0
\(535\) 2.70820 0.117086
\(536\) 107.721 4.65285
\(537\) 0 0
\(538\) −32.2705 −1.39128
\(539\) −15.7082 −0.676600
\(540\) 0 0
\(541\) 9.14590 0.393213 0.196606 0.980482i \(-0.437008\pi\)
0.196606 + 0.980482i \(0.437008\pi\)
\(542\) 2.61803 0.112454
\(543\) 0 0
\(544\) 60.9787 2.61444
\(545\) −3.65248 −0.156455
\(546\) 0 0
\(547\) −2.27051 −0.0970800 −0.0485400 0.998821i \(-0.515457\pi\)
−0.0485400 + 0.998821i \(0.515457\pi\)
\(548\) 61.6869 2.63513
\(549\) 0 0
\(550\) −33.2705 −1.41866
\(551\) 30.6525 1.30584
\(552\) 0 0
\(553\) 6.56231 0.279058
\(554\) 12.3262 0.523692
\(555\) 0 0
\(556\) −54.1033 −2.29449
\(557\) 31.6869 1.34262 0.671309 0.741178i \(-0.265732\pi\)
0.671309 + 0.741178i \(0.265732\pi\)
\(558\) 0 0
\(559\) 42.2705 1.78785
\(560\) −3.76393 −0.159055
\(561\) 0 0
\(562\) 82.3951 3.47563
\(563\) 1.20163 0.0506425 0.0253213 0.999679i \(-0.491939\pi\)
0.0253213 + 0.999679i \(0.491939\pi\)
\(564\) 0 0
\(565\) 5.72949 0.241041
\(566\) −51.5967 −2.16877
\(567\) 0 0
\(568\) 30.5623 1.28237
\(569\) 27.1591 1.13857 0.569283 0.822141i \(-0.307221\pi\)
0.569283 + 0.822141i \(0.307221\pi\)
\(570\) 0 0
\(571\) −22.5623 −0.944203 −0.472102 0.881544i \(-0.656504\pi\)
−0.472102 + 0.881544i \(0.656504\pi\)
\(572\) −61.6869 −2.57926
\(573\) 0 0
\(574\) 23.4164 0.977382
\(575\) 21.7082 0.905295
\(576\) 0 0
\(577\) −36.8328 −1.53337 −0.766685 0.642023i \(-0.778095\pi\)
−0.766685 + 0.642023i \(0.778095\pi\)
\(578\) 38.1246 1.58577
\(579\) 0 0
\(580\) 9.70820 0.403111
\(581\) −6.32624 −0.262457
\(582\) 0 0
\(583\) −2.85410 −0.118205
\(584\) −81.1033 −3.35608
\(585\) 0 0
\(586\) 56.6869 2.34171
\(587\) 47.0689 1.94274 0.971370 0.237570i \(-0.0763510\pi\)
0.971370 + 0.237570i \(0.0763510\pi\)
\(588\) 0 0
\(589\) −39.2705 −1.61811
\(590\) −6.38197 −0.262741
\(591\) 0 0
\(592\) 66.1033 2.71683
\(593\) −2.18034 −0.0895358 −0.0447679 0.998997i \(-0.514255\pi\)
−0.0447679 + 0.998997i \(0.514255\pi\)
\(594\) 0 0
\(595\) −2.14590 −0.0879732
\(596\) 36.2705 1.48570
\(597\) 0 0
\(598\) 56.8328 2.32407
\(599\) 35.4508 1.44848 0.724241 0.689547i \(-0.242190\pi\)
0.724241 + 0.689547i \(0.242190\pi\)
\(600\) 0 0
\(601\) 3.56231 0.145309 0.0726547 0.997357i \(-0.476853\pi\)
0.0726547 + 0.997357i \(0.476853\pi\)
\(602\) 22.7984 0.929192
\(603\) 0 0
\(604\) −92.2279 −3.75270
\(605\) −1.58359 −0.0643822
\(606\) 0 0
\(607\) 5.70820 0.231689 0.115844 0.993267i \(-0.463043\pi\)
0.115844 + 0.993267i \(0.463043\pi\)
\(608\) 63.5410 2.57693
\(609\) 0 0
\(610\) 4.14590 0.167863
\(611\) 19.8541 0.803211
\(612\) 0 0
\(613\) 24.4164 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(614\) −7.47214 −0.301551
\(615\) 0 0
\(616\) −19.5623 −0.788188
\(617\) −3.27051 −0.131666 −0.0658329 0.997831i \(-0.520970\pi\)
−0.0658329 + 0.997831i \(0.520970\pi\)
\(618\) 0 0
\(619\) −28.6869 −1.15302 −0.576512 0.817088i \(-0.695587\pi\)
−0.576512 + 0.817088i \(0.695587\pi\)
\(620\) −12.4377 −0.499510
\(621\) 0 0
\(622\) −7.56231 −0.303221
\(623\) −2.29180 −0.0918189
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 42.6525 1.70474
\(627\) 0 0
\(628\) 81.1033 3.23638
\(629\) 37.6869 1.50268
\(630\) 0 0
\(631\) 42.2705 1.68276 0.841381 0.540442i \(-0.181743\pi\)
0.841381 + 0.540442i \(0.181743\pi\)
\(632\) −49.0344 −1.95049
\(633\) 0 0
\(634\) −74.3951 −2.95461
\(635\) 5.83282 0.231468
\(636\) 0 0
\(637\) 29.1246 1.15396
\(638\) 35.8885 1.42084
\(639\) 0 0
\(640\) 0.416408 0.0164600
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 21.7082 0.855423
\(645\) 0 0
\(646\) 86.1033 3.38769
\(647\) −1.25735 −0.0494317 −0.0247158 0.999695i \(-0.507868\pi\)
−0.0247158 + 0.999695i \(0.507868\pi\)
\(648\) 0 0
\(649\) −16.7082 −0.655854
\(650\) 61.6869 2.41956
\(651\) 0 0
\(652\) 13.1459 0.514833
\(653\) 5.23607 0.204903 0.102452 0.994738i \(-0.467331\pi\)
0.102452 + 0.994738i \(0.467331\pi\)
\(654\) 0 0
\(655\) 0.854102 0.0333725
\(656\) −88.1378 −3.44120
\(657\) 0 0
\(658\) 10.7082 0.417449
\(659\) −36.5967 −1.42561 −0.712803 0.701364i \(-0.752575\pi\)
−0.712803 + 0.701364i \(0.752575\pi\)
\(660\) 0 0
\(661\) −31.5623 −1.22763 −0.613816 0.789449i \(-0.710366\pi\)
−0.613816 + 0.789449i \(0.710366\pi\)
\(662\) −42.2705 −1.64289
\(663\) 0 0
\(664\) 47.2705 1.83445
\(665\) −2.23607 −0.0867110
\(666\) 0 0
\(667\) −23.4164 −0.906687
\(668\) −43.6869 −1.69030
\(669\) 0 0
\(670\) 14.4164 0.556954
\(671\) 10.8541 0.419018
\(672\) 0 0
\(673\) −24.2918 −0.936380 −0.468190 0.883628i \(-0.655094\pi\)
−0.468190 + 0.883628i \(0.655094\pi\)
\(674\) −6.38197 −0.245824
\(675\) 0 0
\(676\) 51.2705 1.97194
\(677\) −40.0344 −1.53865 −0.769324 0.638858i \(-0.779407\pi\)
−0.769324 + 0.638858i \(0.779407\pi\)
\(678\) 0 0
\(679\) 1.70820 0.0655549
\(680\) 16.0344 0.614893
\(681\) 0 0
\(682\) −45.9787 −1.76062
\(683\) −47.2361 −1.80744 −0.903719 0.428126i \(-0.859174\pi\)
−0.903719 + 0.428126i \(0.859174\pi\)
\(684\) 0 0
\(685\) 4.85410 0.185466
\(686\) 34.0344 1.29944
\(687\) 0 0
\(688\) −85.8115 −3.27153
\(689\) 5.29180 0.201601
\(690\) 0 0
\(691\) 7.85410 0.298784 0.149392 0.988778i \(-0.452268\pi\)
0.149392 + 0.988778i \(0.452268\pi\)
\(692\) −77.8328 −2.95876
\(693\) 0 0
\(694\) 3.85410 0.146300
\(695\) −4.25735 −0.161491
\(696\) 0 0
\(697\) −50.2492 −1.90333
\(698\) 40.3607 1.52767
\(699\) 0 0
\(700\) 23.5623 0.890571
\(701\) 25.7984 0.974391 0.487196 0.873293i \(-0.338020\pi\)
0.487196 + 0.873293i \(0.338020\pi\)
\(702\) 0 0
\(703\) 39.2705 1.48112
\(704\) 22.7984 0.859246
\(705\) 0 0
\(706\) 65.5410 2.46667
\(707\) 1.90983 0.0718266
\(708\) 0 0
\(709\) −1.02129 −0.0383552 −0.0191776 0.999816i \(-0.506105\pi\)
−0.0191776 + 0.999816i \(0.506105\pi\)
\(710\) 4.09017 0.153501
\(711\) 0 0
\(712\) 17.1246 0.641772
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) −4.85410 −0.181533
\(716\) 38.1246 1.42478
\(717\) 0 0
\(718\) −38.4164 −1.43369
\(719\) −39.3262 −1.46662 −0.733311 0.679894i \(-0.762026\pi\)
−0.733311 + 0.679894i \(0.762026\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 39.9787 1.48785
\(723\) 0 0
\(724\) 18.7082 0.695285
\(725\) −25.4164 −0.943942
\(726\) 0 0
\(727\) −4.72949 −0.175407 −0.0877035 0.996147i \(-0.527953\pi\)
−0.0877035 + 0.996147i \(0.527953\pi\)
\(728\) 36.2705 1.34427
\(729\) 0 0
\(730\) −10.8541 −0.401728
\(731\) −48.9230 −1.80948
\(732\) 0 0
\(733\) 14.7082 0.543260 0.271630 0.962402i \(-0.412437\pi\)
0.271630 + 0.962402i \(0.412437\pi\)
\(734\) 43.0344 1.58843
\(735\) 0 0
\(736\) −48.5410 −1.78925
\(737\) 37.7426 1.39027
\(738\) 0 0
\(739\) −16.8328 −0.619205 −0.309603 0.950866i \(-0.600196\pi\)
−0.309603 + 0.950866i \(0.600196\pi\)
\(740\) 12.4377 0.457219
\(741\) 0 0
\(742\) 2.85410 0.104777
\(743\) −30.7082 −1.12657 −0.563287 0.826261i \(-0.690464\pi\)
−0.563287 + 0.826261i \(0.690464\pi\)
\(744\) 0 0
\(745\) 2.85410 0.104566
\(746\) −59.3951 −2.17461
\(747\) 0 0
\(748\) 71.3951 2.61046
\(749\) −7.09017 −0.259069
\(750\) 0 0
\(751\) −44.1246 −1.61013 −0.805065 0.593187i \(-0.797870\pi\)
−0.805065 + 0.593187i \(0.797870\pi\)
\(752\) −40.3050 −1.46977
\(753\) 0 0
\(754\) −66.5410 −2.42328
\(755\) −7.25735 −0.264122
\(756\) 0 0
\(757\) −21.2918 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(758\) 13.0902 0.475456
\(759\) 0 0
\(760\) 16.7082 0.606070
\(761\) 7.47214 0.270865 0.135432 0.990787i \(-0.456758\pi\)
0.135432 + 0.990787i \(0.456758\pi\)
\(762\) 0 0
\(763\) 9.56231 0.346179
\(764\) −27.2705 −0.986612
\(765\) 0 0
\(766\) −2.14590 −0.0775344
\(767\) 30.9787 1.11858
\(768\) 0 0
\(769\) 31.3951 1.13214 0.566069 0.824358i \(-0.308464\pi\)
0.566069 + 0.824358i \(0.308464\pi\)
\(770\) −2.61803 −0.0943474
\(771\) 0 0
\(772\) 97.6869 3.51583
\(773\) 16.5279 0.594466 0.297233 0.954805i \(-0.403936\pi\)
0.297233 + 0.954805i \(0.403936\pi\)
\(774\) 0 0
\(775\) 32.5623 1.16967
\(776\) −12.7639 −0.458198
\(777\) 0 0
\(778\) −19.4164 −0.696112
\(779\) −52.3607 −1.87602
\(780\) 0 0
\(781\) 10.7082 0.383170
\(782\) −65.7771 −2.35218
\(783\) 0 0
\(784\) −59.1246 −2.11159
\(785\) 6.38197 0.227782
\(786\) 0 0
\(787\) 43.4164 1.54763 0.773814 0.633413i \(-0.218347\pi\)
0.773814 + 0.633413i \(0.218347\pi\)
\(788\) 79.6869 2.83873
\(789\) 0 0
\(790\) −6.56231 −0.233476
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) −20.1246 −0.714646
\(794\) 52.3607 1.85821
\(795\) 0 0
\(796\) −113.666 −4.02877
\(797\) 50.1246 1.77550 0.887752 0.460321i \(-0.152266\pi\)
0.887752 + 0.460321i \(0.152266\pi\)
\(798\) 0 0
\(799\) −22.9787 −0.812928
\(800\) −52.6869 −1.86276
\(801\) 0 0
\(802\) 62.5410 2.20840
\(803\) −28.4164 −1.00279
\(804\) 0 0
\(805\) 1.70820 0.0602063
\(806\) 85.2492 3.00278
\(807\) 0 0
\(808\) −14.2705 −0.502035
\(809\) −2.94427 −0.103515 −0.0517575 0.998660i \(-0.516482\pi\)
−0.0517575 + 0.998660i \(0.516482\pi\)
\(810\) 0 0
\(811\) 21.5410 0.756408 0.378204 0.925722i \(-0.376542\pi\)
0.378204 + 0.925722i \(0.376542\pi\)
\(812\) −25.4164 −0.891941
\(813\) 0 0
\(814\) 45.9787 1.61155
\(815\) 1.03444 0.0362349
\(816\) 0 0
\(817\) −50.9787 −1.78352
\(818\) 96.1033 3.36017
\(819\) 0 0
\(820\) −16.5836 −0.579124
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 0 0
\(823\) 3.43769 0.119830 0.0599152 0.998203i \(-0.480917\pi\)
0.0599152 + 0.998203i \(0.480917\pi\)
\(824\) −7.47214 −0.260304
\(825\) 0 0
\(826\) 16.7082 0.581353
\(827\) −6.70820 −0.233267 −0.116634 0.993175i \(-0.537210\pi\)
−0.116634 + 0.993175i \(0.537210\pi\)
\(828\) 0 0
\(829\) 14.2705 0.495635 0.247818 0.968807i \(-0.420287\pi\)
0.247818 + 0.968807i \(0.420287\pi\)
\(830\) 6.32624 0.219587
\(831\) 0 0
\(832\) −42.2705 −1.46547
\(833\) −33.7082 −1.16792
\(834\) 0 0
\(835\) −3.43769 −0.118966
\(836\) 74.3951 2.57301
\(837\) 0 0
\(838\) 10.7082 0.369909
\(839\) −23.6180 −0.815385 −0.407693 0.913119i \(-0.633666\pi\)
−0.407693 + 0.913119i \(0.633666\pi\)
\(840\) 0 0
\(841\) −1.58359 −0.0546066
\(842\) 7.85410 0.270670
\(843\) 0 0
\(844\) 72.1033 2.48190
\(845\) 4.03444 0.138789
\(846\) 0 0
\(847\) 4.14590 0.142455
\(848\) −10.7426 −0.368904
\(849\) 0 0
\(850\) −71.3951 −2.44883
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) −44.2705 −1.51579 −0.757897 0.652375i \(-0.773773\pi\)
−0.757897 + 0.652375i \(0.773773\pi\)
\(854\) −10.8541 −0.371420
\(855\) 0 0
\(856\) 52.9787 1.81078
\(857\) −8.23607 −0.281339 −0.140669 0.990057i \(-0.544925\pi\)
−0.140669 + 0.990057i \(0.544925\pi\)
\(858\) 0 0
\(859\) 10.5623 0.360381 0.180191 0.983632i \(-0.442329\pi\)
0.180191 + 0.983632i \(0.442329\pi\)
\(860\) −16.1459 −0.550571
\(861\) 0 0
\(862\) −89.9574 −3.06396
\(863\) 21.7082 0.738956 0.369478 0.929240i \(-0.379537\pi\)
0.369478 + 0.929240i \(0.379537\pi\)
\(864\) 0 0
\(865\) −6.12461 −0.208243
\(866\) 37.7426 1.28255
\(867\) 0 0
\(868\) 32.5623 1.10524
\(869\) −17.1803 −0.582803
\(870\) 0 0
\(871\) −69.9787 −2.37114
\(872\) −71.4508 −2.41963
\(873\) 0 0
\(874\) −68.5410 −2.31843
\(875\) 3.76393 0.127244
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 77.3951 2.61196
\(879\) 0 0
\(880\) 9.85410 0.332182
\(881\) −29.8885 −1.00697 −0.503485 0.864004i \(-0.667949\pi\)
−0.503485 + 0.864004i \(0.667949\pi\)
\(882\) 0 0
\(883\) 5.87539 0.197723 0.0988613 0.995101i \(-0.468480\pi\)
0.0988613 + 0.995101i \(0.468480\pi\)
\(884\) −132.374 −4.45221
\(885\) 0 0
\(886\) 40.4164 1.35782
\(887\) −40.1935 −1.34957 −0.674783 0.738016i \(-0.735763\pi\)
−0.674783 + 0.738016i \(0.735763\pi\)
\(888\) 0 0
\(889\) −15.2705 −0.512156
\(890\) 2.29180 0.0768212
\(891\) 0 0
\(892\) 37.4164 1.25279
\(893\) −23.9443 −0.801265
\(894\) 0 0
\(895\) 3.00000 0.100279
\(896\) −1.09017 −0.0364200
\(897\) 0 0
\(898\) 34.9787 1.16725
\(899\) −35.1246 −1.17147
\(900\) 0 0
\(901\) −6.12461 −0.204040
\(902\) −61.3050 −2.04123
\(903\) 0 0
\(904\) 112.082 3.72779
\(905\) 1.47214 0.0489355
\(906\) 0 0
\(907\) −7.12461 −0.236569 −0.118284 0.992980i \(-0.537739\pi\)
−0.118284 + 0.992980i \(0.537739\pi\)
\(908\) −14.2918 −0.474290
\(909\) 0 0
\(910\) 4.85410 0.160912
\(911\) 10.0344 0.332456 0.166228 0.986087i \(-0.446841\pi\)
0.166228 + 0.986087i \(0.446841\pi\)
\(912\) 0 0
\(913\) 16.5623 0.548132
\(914\) −25.7984 −0.853334
\(915\) 0 0
\(916\) 32.5623 1.07589
\(917\) −2.23607 −0.0738415
\(918\) 0 0
\(919\) −27.9787 −0.922933 −0.461466 0.887158i \(-0.652676\pi\)
−0.461466 + 0.887158i \(0.652676\pi\)
\(920\) −12.7639 −0.420814
\(921\) 0 0
\(922\) −31.9787 −1.05316
\(923\) −19.8541 −0.653506
\(924\) 0 0
\(925\) −32.5623 −1.07064
\(926\) −56.0689 −1.84254
\(927\) 0 0
\(928\) 56.8328 1.86563
\(929\) −14.9443 −0.490306 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(930\) 0 0
\(931\) −35.1246 −1.15116
\(932\) 130.520 4.27532
\(933\) 0 0
\(934\) 9.56231 0.312888
\(935\) 5.61803 0.183729
\(936\) 0 0
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) −37.7426 −1.23234
\(939\) 0 0
\(940\) −7.58359 −0.247350
\(941\) 23.3951 0.762659 0.381330 0.924439i \(-0.375466\pi\)
0.381330 + 0.924439i \(0.375466\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) −62.8885 −2.04685
\(945\) 0 0
\(946\) −59.6869 −1.94059
\(947\) 41.0132 1.33275 0.666374 0.745617i \(-0.267845\pi\)
0.666374 + 0.745617i \(0.267845\pi\)
\(948\) 0 0
\(949\) 52.6869 1.71029
\(950\) −74.3951 −2.41370
\(951\) 0 0
\(952\) −41.9787 −1.36054
\(953\) −13.3607 −0.432795 −0.216397 0.976305i \(-0.569431\pi\)
−0.216397 + 0.976305i \(0.569431\pi\)
\(954\) 0 0
\(955\) −2.14590 −0.0694396
\(956\) −42.1033 −1.36172
\(957\) 0 0
\(958\) −21.4164 −0.691933
\(959\) −12.7082 −0.410369
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) −85.2492 −2.74855
\(963\) 0 0
\(964\) −40.1459 −1.29301
\(965\) 7.68692 0.247451
\(966\) 0 0
\(967\) −14.4164 −0.463600 −0.231800 0.972763i \(-0.574462\pi\)
−0.231800 + 0.972763i \(0.574462\pi\)
\(968\) −30.9787 −0.995694
\(969\) 0 0
\(970\) −1.70820 −0.0548471
\(971\) 53.0132 1.70127 0.850637 0.525754i \(-0.176217\pi\)
0.850637 + 0.525754i \(0.176217\pi\)
\(972\) 0 0
\(973\) 11.1459 0.357321
\(974\) −60.2148 −1.92941
\(975\) 0 0
\(976\) 40.8541 1.30771
\(977\) −46.7426 −1.49543 −0.747715 0.664020i \(-0.768849\pi\)
−0.747715 + 0.664020i \(0.768849\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −11.1246 −0.355363
\(981\) 0 0
\(982\) −93.6656 −2.98899
\(983\) −18.6525 −0.594922 −0.297461 0.954734i \(-0.596140\pi\)
−0.297461 + 0.954734i \(0.596140\pi\)
\(984\) 0 0
\(985\) 6.27051 0.199795
\(986\) 77.0132 2.45260
\(987\) 0 0
\(988\) −137.936 −4.38833
\(989\) 38.9443 1.23836
\(990\) 0 0
\(991\) 24.2705 0.770978 0.385489 0.922712i \(-0.374033\pi\)
0.385489 + 0.922712i \(0.374033\pi\)
\(992\) −72.8115 −2.31177
\(993\) 0 0
\(994\) −10.7082 −0.339644
\(995\) −8.94427 −0.283552
\(996\) 0 0
\(997\) −39.2918 −1.24438 −0.622192 0.782865i \(-0.713758\pi\)
−0.622192 + 0.782865i \(0.713758\pi\)
\(998\) 34.7426 1.09976
\(999\) 0 0
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 927.2.a.b.1.2 2
3.2 odd 2 103.2.a.a.1.1 2
12.11 even 2 1648.2.a.f.1.2 2
15.14 odd 2 2575.2.a.g.1.2 2
21.20 even 2 5047.2.a.a.1.1 2
24.5 odd 2 6592.2.a.t.1.1 2
24.11 even 2 6592.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 3.2 odd 2
927.2.a.b.1.2 2 1.1 even 1 trivial
1648.2.a.f.1.2 2 12.11 even 2
2575.2.a.g.1.2 2 15.14 odd 2
5047.2.a.a.1.1 2 21.20 even 2
6592.2.a.h.1.1 2 24.11 even 2
6592.2.a.t.1.1 2 24.5 odd 2