Properties

Label 1150.2.b.i.599.3
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,2,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.i.599.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} -0.618034i q^{7} -1.00000i q^{8} +0.381966 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} -0.618034i q^{7} -1.00000i q^{8} +0.381966 q^{9} -2.85410 q^{11} +1.61803i q^{12} +7.09017i q^{13} +0.618034 q^{14} +1.00000 q^{16} +6.09017i q^{17} +0.381966i q^{18} -1.85410 q^{19} -1.00000 q^{21} -2.85410i q^{22} -1.00000i q^{23} -1.61803 q^{24} -7.09017 q^{26} -5.47214i q^{27} +0.618034i q^{28} +9.23607 q^{29} +9.09017 q^{31} +1.00000i q^{32} +4.61803i q^{33} -6.09017 q^{34} -0.381966 q^{36} +6.47214i q^{37} -1.85410i q^{38} +11.4721 q^{39} +3.32624 q^{41} -1.00000i q^{42} +2.85410 q^{44} +1.00000 q^{46} -3.70820i q^{47} -1.61803i q^{48} +6.61803 q^{49} +9.85410 q^{51} -7.09017i q^{52} -0.472136i q^{53} +5.47214 q^{54} -0.618034 q^{56} +3.00000i q^{57} +9.23607i q^{58} -1.70820 q^{59} -9.32624 q^{61} +9.09017i q^{62} -0.236068i q^{63} -1.00000 q^{64} -4.61803 q^{66} +14.4721i q^{67} -6.09017i q^{68} -1.61803 q^{69} -4.09017 q^{71} -0.381966i q^{72} -3.23607i q^{73} -6.47214 q^{74} +1.85410 q^{76} +1.76393i q^{77} +11.4721i q^{78} -1.52786 q^{79} -7.70820 q^{81} +3.32624i q^{82} +6.94427i q^{83} +1.00000 q^{84} -14.9443i q^{87} +2.85410i q^{88} +10.4721 q^{89} +4.38197 q^{91} +1.00000i q^{92} -14.7082i q^{93} +3.70820 q^{94} +1.61803 q^{96} +12.3820i q^{97} +6.61803i q^{98} -1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46} + 22 q^{49} + 26 q^{51} + 4 q^{54} + 2 q^{56} + 20 q^{59} - 6 q^{61} - 4 q^{64} - 14 q^{66} - 2 q^{69} + 6 q^{71} - 8 q^{74} - 6 q^{76} - 24 q^{79} - 4 q^{81} + 4 q^{84} + 24 q^{89} + 22 q^{91} - 12 q^{94} + 2 q^{96} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.61803i − 0.934172i −0.884212 0.467086i \(-0.845304\pi\)
0.884212 0.467086i \(-0.154696\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) − 0.618034i − 0.233595i −0.993156 0.116797i \(-0.962737\pi\)
0.993156 0.116797i \(-0.0372628\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) −2.85410 −0.860544 −0.430272 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) 1.61803i 0.467086i
\(13\) 7.09017i 1.96646i 0.182372 + 0.983230i \(0.441623\pi\)
−0.182372 + 0.983230i \(0.558377\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.09017i 1.47708i 0.674208 + 0.738542i \(0.264485\pi\)
−0.674208 + 0.738542i \(0.735515\pi\)
\(18\) 0.381966i 0.0900303i
\(19\) −1.85410 −0.425360 −0.212680 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) − 2.85410i − 0.608497i
\(23\) − 1.00000i − 0.208514i
\(24\) −1.61803 −0.330280
\(25\) 0 0
\(26\) −7.09017 −1.39050
\(27\) − 5.47214i − 1.05311i
\(28\) 0.618034i 0.116797i
\(29\) 9.23607 1.71509 0.857547 0.514405i \(-0.171987\pi\)
0.857547 + 0.514405i \(0.171987\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.61803i 0.803897i
\(34\) −6.09017 −1.04446
\(35\) 0 0
\(36\) −0.381966 −0.0636610
\(37\) 6.47214i 1.06401i 0.846740 + 0.532006i \(0.178562\pi\)
−0.846740 + 0.532006i \(0.821438\pi\)
\(38\) − 1.85410i − 0.300775i
\(39\) 11.4721 1.83701
\(40\) 0 0
\(41\) 3.32624 0.519471 0.259736 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 2.85410 0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 3.70820i − 0.540897i −0.962734 0.270449i \(-0.912828\pi\)
0.962734 0.270449i \(-0.0871720\pi\)
\(48\) − 1.61803i − 0.233543i
\(49\) 6.61803 0.945433
\(50\) 0 0
\(51\) 9.85410 1.37985
\(52\) − 7.09017i − 0.983230i
\(53\) − 0.472136i − 0.0648529i −0.999474 0.0324264i \(-0.989677\pi\)
0.999474 0.0324264i \(-0.0103235\pi\)
\(54\) 5.47214 0.744663
\(55\) 0 0
\(56\) −0.618034 −0.0825883
\(57\) 3.00000i 0.397360i
\(58\) 9.23607i 1.21276i
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) 9.09017i 1.15445i
\(63\) − 0.236068i − 0.0297418i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.61803 −0.568441
\(67\) 14.4721i 1.76805i 0.467437 + 0.884026i \(0.345177\pi\)
−0.467437 + 0.884026i \(0.654823\pi\)
\(68\) − 6.09017i − 0.738542i
\(69\) −1.61803 −0.194788
\(70\) 0 0
\(71\) −4.09017 −0.485414 −0.242707 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) − 0.381966i − 0.0450151i
\(73\) − 3.23607i − 0.378753i −0.981905 0.189377i \(-0.939353\pi\)
0.981905 0.189377i \(-0.0606467\pi\)
\(74\) −6.47214 −0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) 1.76393i 0.201019i
\(78\) 11.4721i 1.29896i
\(79\) −1.52786 −0.171898 −0.0859491 0.996300i \(-0.527392\pi\)
−0.0859491 + 0.996300i \(0.527392\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 3.32624i 0.367322i
\(83\) 6.94427i 0.762233i 0.924527 + 0.381116i \(0.124460\pi\)
−0.924527 + 0.381116i \(0.875540\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) − 14.9443i − 1.60219i
\(88\) 2.85410i 0.304248i
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) 1.00000i 0.104257i
\(93\) − 14.7082i − 1.52517i
\(94\) 3.70820 0.382472
\(95\) 0 0
\(96\) 1.61803 0.165140
\(97\) 12.3820i 1.25720i 0.777730 + 0.628599i \(0.216371\pi\)
−0.777730 + 0.628599i \(0.783629\pi\)
\(98\) 6.61803i 0.668522i
\(99\) −1.09017 −0.109566
\(100\) 0 0
\(101\) −0.291796 −0.0290348 −0.0145174 0.999895i \(-0.504621\pi\)
−0.0145174 + 0.999895i \(0.504621\pi\)
\(102\) 9.85410i 0.975701i
\(103\) − 16.5623i − 1.63193i −0.578100 0.815966i \(-0.696205\pi\)
0.578100 0.815966i \(-0.303795\pi\)
\(104\) 7.09017 0.695248
\(105\) 0 0
\(106\) 0.472136 0.0458579
\(107\) 18.1803i 1.75756i 0.477227 + 0.878780i \(0.341642\pi\)
−0.477227 + 0.878780i \(0.658358\pi\)
\(108\) 5.47214i 0.526557i
\(109\) 11.5623 1.10747 0.553734 0.832694i \(-0.313202\pi\)
0.553734 + 0.832694i \(0.313202\pi\)
\(110\) 0 0
\(111\) 10.4721 0.993971
\(112\) − 0.618034i − 0.0583987i
\(113\) − 1.05573i − 0.0993145i −0.998766 0.0496573i \(-0.984187\pi\)
0.998766 0.0496573i \(-0.0158129\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −9.23607 −0.857547
\(117\) 2.70820i 0.250374i
\(118\) − 1.70820i − 0.157253i
\(119\) 3.76393 0.345039
\(120\) 0 0
\(121\) −2.85410 −0.259464
\(122\) − 9.32624i − 0.844358i
\(123\) − 5.38197i − 0.485276i
\(124\) −9.09017 −0.816321
\(125\) 0 0
\(126\) 0.236068 0.0210306
\(127\) 16.1803i 1.43577i 0.696160 + 0.717886i \(0.254890\pi\)
−0.696160 + 0.717886i \(0.745110\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.94427 0.257242 0.128621 0.991694i \(-0.458945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(132\) − 4.61803i − 0.401948i
\(133\) 1.14590i 0.0993620i
\(134\) −14.4721 −1.25020
\(135\) 0 0
\(136\) 6.09017 0.522228
\(137\) − 10.3262i − 0.882230i −0.897451 0.441115i \(-0.854583\pi\)
0.897451 0.441115i \(-0.145417\pi\)
\(138\) − 1.61803i − 0.137736i
\(139\) −12.7639 −1.08262 −0.541311 0.840822i \(-0.682072\pi\)
−0.541311 + 0.840822i \(0.682072\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 4.09017i − 0.343239i
\(143\) − 20.2361i − 1.69223i
\(144\) 0.381966 0.0318305
\(145\) 0 0
\(146\) 3.23607 0.267819
\(147\) − 10.7082i − 0.883198i
\(148\) − 6.47214i − 0.532006i
\(149\) 7.85410 0.643433 0.321717 0.946836i \(-0.395740\pi\)
0.321717 + 0.946836i \(0.395740\pi\)
\(150\) 0 0
\(151\) −2.56231 −0.208517 −0.104259 0.994550i \(-0.533247\pi\)
−0.104259 + 0.994550i \(0.533247\pi\)
\(152\) 1.85410i 0.150388i
\(153\) 2.32624i 0.188065i
\(154\) −1.76393 −0.142142
\(155\) 0 0
\(156\) −11.4721 −0.918506
\(157\) − 3.70820i − 0.295947i −0.988991 0.147973i \(-0.952725\pi\)
0.988991 0.147973i \(-0.0472750\pi\)
\(158\) − 1.52786i − 0.121550i
\(159\) −0.763932 −0.0605838
\(160\) 0 0
\(161\) −0.618034 −0.0487079
\(162\) − 7.70820i − 0.605614i
\(163\) − 1.38197i − 0.108244i −0.998534 0.0541220i \(-0.982764\pi\)
0.998534 0.0541220i \(-0.0172360\pi\)
\(164\) −3.32624 −0.259736
\(165\) 0 0
\(166\) −6.94427 −0.538980
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) −37.2705 −2.86696
\(170\) 0 0
\(171\) −0.708204 −0.0541577
\(172\) 0 0
\(173\) 1.43769i 0.109306i 0.998505 + 0.0546529i \(0.0174052\pi\)
−0.998505 + 0.0546529i \(0.982595\pi\)
\(174\) 14.9443 1.13292
\(175\) 0 0
\(176\) −2.85410 −0.215136
\(177\) 2.76393i 0.207750i
\(178\) 10.4721i 0.784920i
\(179\) −2.18034 −0.162966 −0.0814831 0.996675i \(-0.525966\pi\)
−0.0814831 + 0.996675i \(0.525966\pi\)
\(180\) 0 0
\(181\) −12.1459 −0.902797 −0.451399 0.892322i \(-0.649075\pi\)
−0.451399 + 0.892322i \(0.649075\pi\)
\(182\) 4.38197i 0.324813i
\(183\) 15.0902i 1.11550i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 14.7082 1.07846
\(187\) − 17.3820i − 1.27110i
\(188\) 3.70820i 0.270449i
\(189\) −3.38197 −0.246002
\(190\) 0 0
\(191\) −13.7082 −0.991891 −0.495945 0.868354i \(-0.665178\pi\)
−0.495945 + 0.868354i \(0.665178\pi\)
\(192\) 1.61803i 0.116772i
\(193\) − 0.763932i − 0.0549890i −0.999622 0.0274945i \(-0.991247\pi\)
0.999622 0.0274945i \(-0.00875288\pi\)
\(194\) −12.3820 −0.888973
\(195\) 0 0
\(196\) −6.61803 −0.472717
\(197\) − 22.5623i − 1.60750i −0.594969 0.803749i \(-0.702836\pi\)
0.594969 0.803749i \(-0.297164\pi\)
\(198\) − 1.09017i − 0.0774750i
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 23.4164 1.65167
\(202\) − 0.291796i − 0.0205307i
\(203\) − 5.70820i − 0.400637i
\(204\) −9.85410 −0.689925
\(205\) 0 0
\(206\) 16.5623 1.15395
\(207\) − 0.381966i − 0.0265485i
\(208\) 7.09017i 0.491615i
\(209\) 5.29180 0.366041
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0.472136i 0.0324264i
\(213\) 6.61803i 0.453460i
\(214\) −18.1803 −1.24278
\(215\) 0 0
\(216\) −5.47214 −0.372332
\(217\) − 5.61803i − 0.381377i
\(218\) 11.5623i 0.783098i
\(219\) −5.23607 −0.353821
\(220\) 0 0
\(221\) −43.1803 −2.90462
\(222\) 10.4721i 0.702844i
\(223\) 20.9443i 1.40253i 0.712900 + 0.701266i \(0.247382\pi\)
−0.712900 + 0.701266i \(0.752618\pi\)
\(224\) 0.618034 0.0412941
\(225\) 0 0
\(226\) 1.05573 0.0702260
\(227\) − 18.7639i − 1.24541i −0.782458 0.622703i \(-0.786035\pi\)
0.782458 0.622703i \(-0.213965\pi\)
\(228\) − 3.00000i − 0.198680i
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 2.85410 0.187786
\(232\) − 9.23607i − 0.606378i
\(233\) − 6.29180i − 0.412189i −0.978532 0.206095i \(-0.933925\pi\)
0.978532 0.206095i \(-0.0660755\pi\)
\(234\) −2.70820 −0.177041
\(235\) 0 0
\(236\) 1.70820 0.111195
\(237\) 2.47214i 0.160582i
\(238\) 3.76393i 0.243979i
\(239\) 20.3607 1.31702 0.658511 0.752571i \(-0.271186\pi\)
0.658511 + 0.752571i \(0.271186\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) − 2.85410i − 0.183469i
\(243\) − 3.94427i − 0.253025i
\(244\) 9.32624 0.597051
\(245\) 0 0
\(246\) 5.38197 0.343142
\(247\) − 13.1459i − 0.836453i
\(248\) − 9.09017i − 0.577226i
\(249\) 11.2361 0.712057
\(250\) 0 0
\(251\) 6.14590 0.387926 0.193963 0.981009i \(-0.437866\pi\)
0.193963 + 0.981009i \(0.437866\pi\)
\(252\) 0.236068i 0.0148709i
\(253\) 2.85410i 0.179436i
\(254\) −16.1803 −1.01524
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 7.81966i − 0.487777i −0.969803 0.243888i \(-0.921577\pi\)
0.969803 0.243888i \(-0.0784231\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 3.52786 0.218369
\(262\) 2.94427i 0.181898i
\(263\) − 20.7426i − 1.27905i −0.768772 0.639523i \(-0.779132\pi\)
0.768772 0.639523i \(-0.220868\pi\)
\(264\) 4.61803 0.284220
\(265\) 0 0
\(266\) −1.14590 −0.0702595
\(267\) − 16.9443i − 1.03697i
\(268\) − 14.4721i − 0.884026i
\(269\) 14.1803 0.864591 0.432295 0.901732i \(-0.357704\pi\)
0.432295 + 0.901732i \(0.357704\pi\)
\(270\) 0 0
\(271\) −30.3262 −1.84219 −0.921094 0.389341i \(-0.872703\pi\)
−0.921094 + 0.389341i \(0.872703\pi\)
\(272\) 6.09017i 0.369271i
\(273\) − 7.09017i − 0.429117i
\(274\) 10.3262 0.623831
\(275\) 0 0
\(276\) 1.61803 0.0973942
\(277\) 29.4164i 1.76746i 0.467996 + 0.883730i \(0.344976\pi\)
−0.467996 + 0.883730i \(0.655024\pi\)
\(278\) − 12.7639i − 0.765530i
\(279\) 3.47214 0.207871
\(280\) 0 0
\(281\) 22.7639 1.35798 0.678991 0.734146i \(-0.262417\pi\)
0.678991 + 0.734146i \(0.262417\pi\)
\(282\) − 6.00000i − 0.357295i
\(283\) 26.9443i 1.60167i 0.598885 + 0.800835i \(0.295611\pi\)
−0.598885 + 0.800835i \(0.704389\pi\)
\(284\) 4.09017 0.242707
\(285\) 0 0
\(286\) 20.2361 1.19658
\(287\) − 2.05573i − 0.121346i
\(288\) 0.381966i 0.0225076i
\(289\) −20.0902 −1.18177
\(290\) 0 0
\(291\) 20.0344 1.17444
\(292\) 3.23607i 0.189377i
\(293\) 19.8885i 1.16190i 0.813939 + 0.580951i \(0.197319\pi\)
−0.813939 + 0.580951i \(0.802681\pi\)
\(294\) 10.7082 0.624515
\(295\) 0 0
\(296\) 6.47214 0.376185
\(297\) 15.6180i 0.906250i
\(298\) 7.85410i 0.454976i
\(299\) 7.09017 0.410035
\(300\) 0 0
\(301\) 0 0
\(302\) − 2.56231i − 0.147444i
\(303\) 0.472136i 0.0271235i
\(304\) −1.85410 −0.106340
\(305\) 0 0
\(306\) −2.32624 −0.132982
\(307\) − 28.4508i − 1.62378i −0.583813 0.811888i \(-0.698440\pi\)
0.583813 0.811888i \(-0.301560\pi\)
\(308\) − 1.76393i − 0.100509i
\(309\) −26.7984 −1.52451
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) − 11.4721i − 0.649482i
\(313\) − 12.7984i − 0.723407i −0.932293 0.361703i \(-0.882195\pi\)
0.932293 0.361703i \(-0.117805\pi\)
\(314\) 3.70820 0.209266
\(315\) 0 0
\(316\) 1.52786 0.0859491
\(317\) − 11.0902i − 0.622886i −0.950265 0.311443i \(-0.899188\pi\)
0.950265 0.311443i \(-0.100812\pi\)
\(318\) − 0.763932i − 0.0428392i
\(319\) −26.3607 −1.47591
\(320\) 0 0
\(321\) 29.4164 1.64186
\(322\) − 0.618034i − 0.0344417i
\(323\) − 11.2918i − 0.628292i
\(324\) 7.70820 0.428234
\(325\) 0 0
\(326\) 1.38197 0.0765400
\(327\) − 18.7082i − 1.03457i
\(328\) − 3.32624i − 0.183661i
\(329\) −2.29180 −0.126351
\(330\) 0 0
\(331\) 19.2361 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(332\) − 6.94427i − 0.381116i
\(333\) 2.47214i 0.135472i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 13.6738i 0.744857i 0.928061 + 0.372429i \(0.121475\pi\)
−0.928061 + 0.372429i \(0.878525\pi\)
\(338\) − 37.2705i − 2.02725i
\(339\) −1.70820 −0.0927769
\(340\) 0 0
\(341\) −25.9443 −1.40496
\(342\) − 0.708204i − 0.0382953i
\(343\) − 8.41641i − 0.454443i
\(344\) 0 0
\(345\) 0 0
\(346\) −1.43769 −0.0772909
\(347\) − 6.38197i − 0.342602i −0.985219 0.171301i \(-0.945203\pi\)
0.985219 0.171301i \(-0.0547970\pi\)
\(348\) 14.9443i 0.801097i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 38.7984 2.07090
\(352\) − 2.85410i − 0.152124i
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) −2.76393 −0.146901
\(355\) 0 0
\(356\) −10.4721 −0.555022
\(357\) − 6.09017i − 0.322326i
\(358\) − 2.18034i − 0.115235i
\(359\) −26.3607 −1.39126 −0.695632 0.718399i \(-0.744875\pi\)
−0.695632 + 0.718399i \(0.744875\pi\)
\(360\) 0 0
\(361\) −15.5623 −0.819069
\(362\) − 12.1459i − 0.638374i
\(363\) 4.61803i 0.242384i
\(364\) −4.38197 −0.229677
\(365\) 0 0
\(366\) −15.0902 −0.788776
\(367\) 6.47214i 0.337843i 0.985630 + 0.168921i \(0.0540284\pi\)
−0.985630 + 0.168921i \(0.945972\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 1.27051 0.0661401
\(370\) 0 0
\(371\) −0.291796 −0.0151493
\(372\) 14.7082i 0.762585i
\(373\) − 20.1803i − 1.04490i −0.852670 0.522449i \(-0.825018\pi\)
0.852670 0.522449i \(-0.174982\pi\)
\(374\) 17.3820 0.898800
\(375\) 0 0
\(376\) −3.70820 −0.191236
\(377\) 65.4853i 3.37266i
\(378\) − 3.38197i − 0.173950i
\(379\) 22.4508 1.15322 0.576611 0.817019i \(-0.304375\pi\)
0.576611 + 0.817019i \(0.304375\pi\)
\(380\) 0 0
\(381\) 26.1803 1.34126
\(382\) − 13.7082i − 0.701373i
\(383\) 17.8885i 0.914062i 0.889451 + 0.457031i \(0.151087\pi\)
−0.889451 + 0.457031i \(0.848913\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 0 0
\(386\) 0.763932 0.0388831
\(387\) 0 0
\(388\) − 12.3820i − 0.628599i
\(389\) −21.3262 −1.08128 −0.540642 0.841253i \(-0.681818\pi\)
−0.540642 + 0.841253i \(0.681818\pi\)
\(390\) 0 0
\(391\) 6.09017 0.307993
\(392\) − 6.61803i − 0.334261i
\(393\) − 4.76393i − 0.240309i
\(394\) 22.5623 1.13667
\(395\) 0 0
\(396\) 1.09017 0.0547831
\(397\) 7.32624i 0.367693i 0.982955 + 0.183847i \(0.0588550\pi\)
−0.982955 + 0.183847i \(0.941145\pi\)
\(398\) − 2.00000i − 0.100251i
\(399\) 1.85410 0.0928212
\(400\) 0 0
\(401\) −1.70820 −0.0853036 −0.0426518 0.999090i \(-0.513581\pi\)
−0.0426518 + 0.999090i \(0.513581\pi\)
\(402\) 23.4164i 1.16790i
\(403\) 64.4508i 3.21053i
\(404\) 0.291796 0.0145174
\(405\) 0 0
\(406\) 5.70820 0.283293
\(407\) − 18.4721i − 0.915630i
\(408\) − 9.85410i − 0.487851i
\(409\) 30.2148 1.49402 0.747012 0.664810i \(-0.231488\pi\)
0.747012 + 0.664810i \(0.231488\pi\)
\(410\) 0 0
\(411\) −16.7082 −0.824155
\(412\) 16.5623i 0.815966i
\(413\) 1.05573i 0.0519490i
\(414\) 0.381966 0.0187726
\(415\) 0 0
\(416\) −7.09017 −0.347624
\(417\) 20.6525i 1.01136i
\(418\) 5.29180i 0.258830i
\(419\) −14.4721 −0.707010 −0.353505 0.935433i \(-0.615010\pi\)
−0.353505 + 0.935433i \(0.615010\pi\)
\(420\) 0 0
\(421\) 13.7426 0.669776 0.334888 0.942258i \(-0.391302\pi\)
0.334888 + 0.942258i \(0.391302\pi\)
\(422\) 14.0000i 0.681509i
\(423\) − 1.41641i − 0.0688681i
\(424\) −0.472136 −0.0229289
\(425\) 0 0
\(426\) −6.61803 −0.320645
\(427\) 5.76393i 0.278936i
\(428\) − 18.1803i − 0.878780i
\(429\) −32.7426 −1.58083
\(430\) 0 0
\(431\) 3.34752 0.161245 0.0806223 0.996745i \(-0.474309\pi\)
0.0806223 + 0.996745i \(0.474309\pi\)
\(432\) − 5.47214i − 0.263278i
\(433\) − 8.50658i − 0.408800i −0.978887 0.204400i \(-0.934476\pi\)
0.978887 0.204400i \(-0.0655243\pi\)
\(434\) 5.61803 0.269674
\(435\) 0 0
\(436\) −11.5623 −0.553734
\(437\) 1.85410i 0.0886937i
\(438\) − 5.23607i − 0.250189i
\(439\) 13.3820 0.638686 0.319343 0.947639i \(-0.396538\pi\)
0.319343 + 0.947639i \(0.396538\pi\)
\(440\) 0 0
\(441\) 2.52786 0.120374
\(442\) − 43.1803i − 2.05388i
\(443\) − 25.0902i − 1.19207i −0.802958 0.596035i \(-0.796742\pi\)
0.802958 0.596035i \(-0.203258\pi\)
\(444\) −10.4721 −0.496986
\(445\) 0 0
\(446\) −20.9443 −0.991740
\(447\) − 12.7082i − 0.601077i
\(448\) 0.618034i 0.0291994i
\(449\) 1.56231 0.0737298 0.0368649 0.999320i \(-0.488263\pi\)
0.0368649 + 0.999320i \(0.488263\pi\)
\(450\) 0 0
\(451\) −9.49342 −0.447028
\(452\) 1.05573i 0.0496573i
\(453\) 4.14590i 0.194791i
\(454\) 18.7639 0.880635
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) − 37.7771i − 1.76714i −0.468301 0.883569i \(-0.655134\pi\)
0.468301 0.883569i \(-0.344866\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 33.3262 1.55554
\(460\) 0 0
\(461\) −39.2361 −1.82741 −0.913703 0.406383i \(-0.866790\pi\)
−0.913703 + 0.406383i \(0.866790\pi\)
\(462\) 2.85410i 0.132785i
\(463\) − 2.00000i − 0.0929479i −0.998920 0.0464739i \(-0.985202\pi\)
0.998920 0.0464739i \(-0.0147984\pi\)
\(464\) 9.23607 0.428774
\(465\) 0 0
\(466\) 6.29180 0.291462
\(467\) − 17.1246i − 0.792433i −0.918157 0.396216i \(-0.870323\pi\)
0.918157 0.396216i \(-0.129677\pi\)
\(468\) − 2.70820i − 0.125187i
\(469\) 8.94427 0.413008
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 1.70820i 0.0786265i
\(473\) 0 0
\(474\) −2.47214 −0.113549
\(475\) 0 0
\(476\) −3.76393 −0.172520
\(477\) − 0.180340i − 0.00825720i
\(478\) 20.3607i 0.931276i
\(479\) 31.8885 1.45702 0.728512 0.685033i \(-0.240212\pi\)
0.728512 + 0.685033i \(0.240212\pi\)
\(480\) 0 0
\(481\) −45.8885 −2.09234
\(482\) 0 0
\(483\) 1.00000i 0.0455016i
\(484\) 2.85410 0.129732
\(485\) 0 0
\(486\) 3.94427 0.178916
\(487\) − 19.8197i − 0.898115i −0.893503 0.449057i \(-0.851760\pi\)
0.893503 0.449057i \(-0.148240\pi\)
\(488\) 9.32624i 0.422179i
\(489\) −2.23607 −0.101118
\(490\) 0 0
\(491\) 6.18034 0.278915 0.139457 0.990228i \(-0.455464\pi\)
0.139457 + 0.990228i \(0.455464\pi\)
\(492\) 5.38197i 0.242638i
\(493\) 56.2492i 2.53334i
\(494\) 13.1459 0.591462
\(495\) 0 0
\(496\) 9.09017 0.408161
\(497\) 2.52786i 0.113390i
\(498\) 11.2361i 0.503500i
\(499\) −12.3607 −0.553340 −0.276670 0.960965i \(-0.589231\pi\)
−0.276670 + 0.960965i \(0.589231\pi\)
\(500\) 0 0
\(501\) −12.9443 −0.578307
\(502\) 6.14590i 0.274305i
\(503\) − 36.3262i − 1.61971i −0.586632 0.809853i \(-0.699547\pi\)
0.586632 0.809853i \(-0.300453\pi\)
\(504\) −0.236068 −0.0105153
\(505\) 0 0
\(506\) −2.85410 −0.126880
\(507\) 60.3050i 2.67824i
\(508\) − 16.1803i − 0.717886i
\(509\) −36.6525 −1.62459 −0.812296 0.583245i \(-0.801783\pi\)
−0.812296 + 0.583245i \(0.801783\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 1.00000i 0.0441942i
\(513\) 10.1459i 0.447952i
\(514\) 7.81966 0.344910
\(515\) 0 0
\(516\) 0 0
\(517\) 10.5836i 0.465466i
\(518\) 4.00000i 0.175750i
\(519\) 2.32624 0.102111
\(520\) 0 0
\(521\) −15.5279 −0.680288 −0.340144 0.940373i \(-0.610476\pi\)
−0.340144 + 0.940373i \(0.610476\pi\)
\(522\) 3.52786i 0.154410i
\(523\) 26.0000i 1.13690i 0.822718 + 0.568450i \(0.192457\pi\)
−0.822718 + 0.568450i \(0.807543\pi\)
\(524\) −2.94427 −0.128621
\(525\) 0 0
\(526\) 20.7426 0.904422
\(527\) 55.3607i 2.41155i
\(528\) 4.61803i 0.200974i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −0.652476 −0.0283150
\(532\) − 1.14590i − 0.0496810i
\(533\) 23.5836i 1.02152i
\(534\) 16.9443 0.733250
\(535\) 0 0
\(536\) 14.4721 0.625101
\(537\) 3.52786i 0.152239i
\(538\) 14.1803i 0.611358i
\(539\) −18.8885 −0.813587
\(540\) 0 0
\(541\) 22.8328 0.981659 0.490830 0.871256i \(-0.336694\pi\)
0.490830 + 0.871256i \(0.336694\pi\)
\(542\) − 30.3262i − 1.30262i
\(543\) 19.6525i 0.843368i
\(544\) −6.09017 −0.261114
\(545\) 0 0
\(546\) 7.09017 0.303431
\(547\) 27.9230i 1.19390i 0.802278 + 0.596950i \(0.203621\pi\)
−0.802278 + 0.596950i \(0.796379\pi\)
\(548\) 10.3262i 0.441115i
\(549\) −3.56231 −0.152036
\(550\) 0 0
\(551\) −17.1246 −0.729533
\(552\) 1.61803i 0.0688681i
\(553\) 0.944272i 0.0401545i
\(554\) −29.4164 −1.24978
\(555\) 0 0
\(556\) 12.7639 0.541311
\(557\) − 22.8328i − 0.967457i −0.875218 0.483729i \(-0.839282\pi\)
0.875218 0.483729i \(-0.160718\pi\)
\(558\) 3.47214i 0.146987i
\(559\) 0 0
\(560\) 0 0
\(561\) −28.1246 −1.18742
\(562\) 22.7639i 0.960239i
\(563\) 13.8885i 0.585332i 0.956215 + 0.292666i \(0.0945425\pi\)
−0.956215 + 0.292666i \(0.905458\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −26.9443 −1.13255
\(567\) 4.76393i 0.200066i
\(568\) 4.09017i 0.171620i
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 15.9787 0.668688 0.334344 0.942451i \(-0.391485\pi\)
0.334344 + 0.942451i \(0.391485\pi\)
\(572\) 20.2361i 0.846113i
\(573\) 22.1803i 0.926597i
\(574\) 2.05573 0.0858044
\(575\) 0 0
\(576\) −0.381966 −0.0159153
\(577\) − 3.52786i − 0.146867i −0.997300 0.0734335i \(-0.976604\pi\)
0.997300 0.0734335i \(-0.0233957\pi\)
\(578\) − 20.0902i − 0.835641i
\(579\) −1.23607 −0.0513692
\(580\) 0 0
\(581\) 4.29180 0.178054
\(582\) 20.0344i 0.830454i
\(583\) 1.34752i 0.0558087i
\(584\) −3.23607 −0.133909
\(585\) 0 0
\(586\) −19.8885 −0.821588
\(587\) 13.6180i 0.562076i 0.959697 + 0.281038i \(0.0906788\pi\)
−0.959697 + 0.281038i \(0.909321\pi\)
\(588\) 10.7082i 0.441599i
\(589\) −16.8541 −0.694461
\(590\) 0 0
\(591\) −36.5066 −1.50168
\(592\) 6.47214i 0.266003i
\(593\) − 39.2361i − 1.61123i −0.592438 0.805616i \(-0.701834\pi\)
0.592438 0.805616i \(-0.298166\pi\)
\(594\) −15.6180 −0.640816
\(595\) 0 0
\(596\) −7.85410 −0.321717
\(597\) 3.23607i 0.132443i
\(598\) 7.09017i 0.289939i
\(599\) 18.3820 0.751067 0.375533 0.926809i \(-0.377460\pi\)
0.375533 + 0.926809i \(0.377460\pi\)
\(600\) 0 0
\(601\) −33.2705 −1.35713 −0.678566 0.734539i \(-0.737398\pi\)
−0.678566 + 0.734539i \(0.737398\pi\)
\(602\) 0 0
\(603\) 5.52786i 0.225112i
\(604\) 2.56231 0.104259
\(605\) 0 0
\(606\) −0.472136 −0.0191792
\(607\) 26.4721i 1.07447i 0.843432 + 0.537235i \(0.180531\pi\)
−0.843432 + 0.537235i \(0.819469\pi\)
\(608\) − 1.85410i − 0.0751938i
\(609\) −9.23607 −0.374264
\(610\) 0 0
\(611\) 26.2918 1.06365
\(612\) − 2.32624i − 0.0940326i
\(613\) − 19.3050i − 0.779720i −0.920874 0.389860i \(-0.872523\pi\)
0.920874 0.389860i \(-0.127477\pi\)
\(614\) 28.4508 1.14818
\(615\) 0 0
\(616\) 1.76393 0.0710708
\(617\) 34.0902i 1.37242i 0.727404 + 0.686209i \(0.240727\pi\)
−0.727404 + 0.686209i \(0.759273\pi\)
\(618\) − 26.7984i − 1.07799i
\(619\) 2.79837 0.112476 0.0562381 0.998417i \(-0.482089\pi\)
0.0562381 + 0.998417i \(0.482089\pi\)
\(620\) 0 0
\(621\) −5.47214 −0.219589
\(622\) 4.00000i 0.160385i
\(623\) − 6.47214i − 0.259301i
\(624\) 11.4721 0.459253
\(625\) 0 0
\(626\) 12.7984 0.511526
\(627\) − 8.56231i − 0.341946i
\(628\) 3.70820i 0.147973i
\(629\) −39.4164 −1.57164
\(630\) 0 0
\(631\) 42.0689 1.67474 0.837368 0.546640i \(-0.184093\pi\)
0.837368 + 0.546640i \(0.184093\pi\)
\(632\) 1.52786i 0.0607752i
\(633\) − 22.6525i − 0.900355i
\(634\) 11.0902 0.440447
\(635\) 0 0
\(636\) 0.763932 0.0302919
\(637\) 46.9230i 1.85916i
\(638\) − 26.3607i − 1.04363i
\(639\) −1.56231 −0.0618039
\(640\) 0 0
\(641\) 0.360680 0.0142460 0.00712300 0.999975i \(-0.497733\pi\)
0.00712300 + 0.999975i \(0.497733\pi\)
\(642\) 29.4164i 1.16097i
\(643\) 8.29180i 0.326997i 0.986544 + 0.163498i \(0.0522778\pi\)
−0.986544 + 0.163498i \(0.947722\pi\)
\(644\) 0.618034 0.0243540
\(645\) 0 0
\(646\) 11.2918 0.444270
\(647\) − 36.2492i − 1.42510i −0.701619 0.712552i \(-0.747539\pi\)
0.701619 0.712552i \(-0.252461\pi\)
\(648\) 7.70820i 0.302807i
\(649\) 4.87539 0.191376
\(650\) 0 0
\(651\) −9.09017 −0.356272
\(652\) 1.38197i 0.0541220i
\(653\) 8.03444i 0.314412i 0.987566 + 0.157206i \(0.0502487\pi\)
−0.987566 + 0.157206i \(0.949751\pi\)
\(654\) 18.7082 0.731549
\(655\) 0 0
\(656\) 3.32624 0.129868
\(657\) − 1.23607i − 0.0482236i
\(658\) − 2.29180i − 0.0893435i
\(659\) 46.2492 1.80161 0.900807 0.434220i \(-0.142976\pi\)
0.900807 + 0.434220i \(0.142976\pi\)
\(660\) 0 0
\(661\) 18.6738 0.726325 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(662\) 19.2361i 0.747631i
\(663\) 69.8673i 2.71342i
\(664\) 6.94427 0.269490
\(665\) 0 0
\(666\) −2.47214 −0.0957933
\(667\) − 9.23607i − 0.357622i
\(668\) 8.00000i 0.309529i
\(669\) 33.8885 1.31021
\(670\) 0 0
\(671\) 26.6180 1.02758
\(672\) − 1.00000i − 0.0385758i
\(673\) 10.9443i 0.421871i 0.977500 + 0.210935i \(0.0676510\pi\)
−0.977500 + 0.210935i \(0.932349\pi\)
\(674\) −13.6738 −0.526694
\(675\) 0 0
\(676\) 37.2705 1.43348
\(677\) − 50.9443i − 1.95795i −0.203986 0.978974i \(-0.565390\pi\)
0.203986 0.978974i \(-0.434610\pi\)
\(678\) − 1.70820i − 0.0656032i
\(679\) 7.65248 0.293675
\(680\) 0 0
\(681\) −30.3607 −1.16342
\(682\) − 25.9443i − 0.993458i
\(683\) 31.5623i 1.20770i 0.797099 + 0.603849i \(0.206367\pi\)
−0.797099 + 0.603849i \(0.793633\pi\)
\(684\) 0.708204 0.0270789
\(685\) 0 0
\(686\) 8.41641 0.321340
\(687\) 16.1803i 0.617318i
\(688\) 0 0
\(689\) 3.34752 0.127531
\(690\) 0 0
\(691\) −29.2361 −1.11219 −0.556096 0.831118i \(-0.687701\pi\)
−0.556096 + 0.831118i \(0.687701\pi\)
\(692\) − 1.43769i − 0.0546529i
\(693\) 0.673762i 0.0255941i
\(694\) 6.38197 0.242256
\(695\) 0 0
\(696\) −14.9443 −0.566461
\(697\) 20.2574i 0.767302i
\(698\) 2.00000i 0.0757011i
\(699\) −10.1803 −0.385056
\(700\) 0 0
\(701\) 43.3394 1.63691 0.818453 0.574573i \(-0.194832\pi\)
0.818453 + 0.574573i \(0.194832\pi\)
\(702\) 38.7984i 1.46435i
\(703\) − 12.0000i − 0.452589i
\(704\) 2.85410 0.107568
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0.180340i 0.00678238i
\(708\) − 2.76393i − 0.103875i
\(709\) 26.0902 0.979837 0.489918 0.871768i \(-0.337027\pi\)
0.489918 + 0.871768i \(0.337027\pi\)
\(710\) 0 0
\(711\) −0.583592 −0.0218864
\(712\) − 10.4721i − 0.392460i
\(713\) − 9.09017i − 0.340430i
\(714\) 6.09017 0.227919
\(715\) 0 0
\(716\) 2.18034 0.0814831
\(717\) − 32.9443i − 1.23033i
\(718\) − 26.3607i − 0.983772i
\(719\) −35.2705 −1.31537 −0.657684 0.753294i \(-0.728464\pi\)
−0.657684 + 0.753294i \(0.728464\pi\)
\(720\) 0 0
\(721\) −10.2361 −0.381211
\(722\) − 15.5623i − 0.579169i
\(723\) 0 0
\(724\) 12.1459 0.451399
\(725\) 0 0
\(726\) −4.61803 −0.171391
\(727\) 28.2016i 1.04594i 0.852351 + 0.522970i \(0.175176\pi\)
−0.852351 + 0.522970i \(0.824824\pi\)
\(728\) − 4.38197i − 0.162406i
\(729\) −29.5066 −1.09284
\(730\) 0 0
\(731\) 0 0
\(732\) − 15.0902i − 0.557749i
\(733\) 29.4164i 1.08652i 0.839565 + 0.543260i \(0.182810\pi\)
−0.839565 + 0.543260i \(0.817190\pi\)
\(734\) −6.47214 −0.238891
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 41.3050i − 1.52149i
\(738\) 1.27051i 0.0467681i
\(739\) 13.8885 0.510898 0.255449 0.966822i \(-0.417777\pi\)
0.255449 + 0.966822i \(0.417777\pi\)
\(740\) 0 0
\(741\) −21.2705 −0.781392
\(742\) − 0.291796i − 0.0107122i
\(743\) − 33.6312i − 1.23381i −0.787038 0.616904i \(-0.788387\pi\)
0.787038 0.616904i \(-0.211613\pi\)
\(744\) −14.7082 −0.539229
\(745\) 0 0
\(746\) 20.1803 0.738855
\(747\) 2.65248i 0.0970490i
\(748\) 17.3820i 0.635548i
\(749\) 11.2361 0.410557
\(750\) 0 0
\(751\) −47.0132 −1.71553 −0.857767 0.514038i \(-0.828149\pi\)
−0.857767 + 0.514038i \(0.828149\pi\)
\(752\) − 3.70820i − 0.135224i
\(753\) − 9.94427i − 0.362389i
\(754\) −65.4853 −2.38483
\(755\) 0 0
\(756\) 3.38197 0.123001
\(757\) − 17.8885i − 0.650170i −0.945685 0.325085i \(-0.894607\pi\)
0.945685 0.325085i \(-0.105393\pi\)
\(758\) 22.4508i 0.815452i
\(759\) 4.61803 0.167624
\(760\) 0 0
\(761\) 46.8673 1.69894 0.849468 0.527640i \(-0.176923\pi\)
0.849468 + 0.527640i \(0.176923\pi\)
\(762\) 26.1803i 0.948414i
\(763\) − 7.14590i − 0.258699i
\(764\) 13.7082 0.495945
\(765\) 0 0
\(766\) −17.8885 −0.646339
\(767\) − 12.1115i − 0.437319i
\(768\) − 1.61803i − 0.0583858i
\(769\) 6.58359 0.237410 0.118705 0.992930i \(-0.462126\pi\)
0.118705 + 0.992930i \(0.462126\pi\)
\(770\) 0 0
\(771\) −12.6525 −0.455668
\(772\) 0.763932i 0.0274945i
\(773\) − 28.9443i − 1.04105i −0.853845 0.520527i \(-0.825736\pi\)
0.853845 0.520527i \(-0.174264\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.3820 0.444487
\(777\) − 6.47214i − 0.232187i
\(778\) − 21.3262i − 0.764583i
\(779\) −6.16718 −0.220962
\(780\) 0 0
\(781\) 11.6738 0.417720
\(782\) 6.09017i 0.217784i
\(783\) − 50.5410i − 1.80619i
\(784\) 6.61803 0.236358
\(785\) 0 0
\(786\) 4.76393 0.169924
\(787\) − 2.87539i − 0.102497i −0.998686 0.0512483i \(-0.983680\pi\)
0.998686 0.0512483i \(-0.0163200\pi\)
\(788\) 22.5623i 0.803749i
\(789\) −33.5623 −1.19485
\(790\) 0 0
\(791\) −0.652476 −0.0231994
\(792\) 1.09017i 0.0387375i
\(793\) − 66.1246i − 2.34815i
\(794\) −7.32624 −0.259998
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 13.7082i 0.485569i 0.970080 + 0.242785i \(0.0780609\pi\)
−0.970080 + 0.242785i \(0.921939\pi\)
\(798\) 1.85410i 0.0656345i
\(799\) 22.5836 0.798950
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) − 1.70820i − 0.0603188i
\(803\) 9.23607i 0.325934i
\(804\) −23.4164 −0.825833
\(805\) 0 0
\(806\) −64.4508 −2.27018
\(807\) − 22.9443i − 0.807677i
\(808\) 0.291796i 0.0102653i
\(809\) 46.7426 1.64338 0.821692 0.569932i \(-0.193030\pi\)
0.821692 + 0.569932i \(0.193030\pi\)
\(810\) 0 0
\(811\) 21.8197 0.766192 0.383096 0.923709i \(-0.374858\pi\)
0.383096 + 0.923709i \(0.374858\pi\)
\(812\) 5.70820i 0.200319i
\(813\) 49.0689i 1.72092i
\(814\) 18.4721 0.647448
\(815\) 0 0
\(816\) 9.85410 0.344963
\(817\) 0 0
\(818\) 30.2148i 1.05644i
\(819\) 1.67376 0.0584860
\(820\) 0 0
\(821\) −33.0557 −1.15365 −0.576826 0.816867i \(-0.695709\pi\)
−0.576826 + 0.816867i \(0.695709\pi\)
\(822\) − 16.7082i − 0.582766i
\(823\) − 25.4164i − 0.885960i −0.896531 0.442980i \(-0.853921\pi\)
0.896531 0.442980i \(-0.146079\pi\)
\(824\) −16.5623 −0.576975
\(825\) 0 0
\(826\) −1.05573 −0.0367335
\(827\) 21.7082i 0.754868i 0.926036 + 0.377434i \(0.123194\pi\)
−0.926036 + 0.377434i \(0.876806\pi\)
\(828\) 0.381966i 0.0132742i
\(829\) −18.9443 −0.657962 −0.328981 0.944337i \(-0.606705\pi\)
−0.328981 + 0.944337i \(0.606705\pi\)
\(830\) 0 0
\(831\) 47.5967 1.65111
\(832\) − 7.09017i − 0.245807i
\(833\) 40.3050i 1.39648i
\(834\) −20.6525 −0.715137
\(835\) 0 0
\(836\) −5.29180 −0.183021
\(837\) − 49.7426i − 1.71936i
\(838\) − 14.4721i − 0.499932i
\(839\) −33.0132 −1.13974 −0.569870 0.821735i \(-0.693007\pi\)
−0.569870 + 0.821735i \(0.693007\pi\)
\(840\) 0 0
\(841\) 56.3050 1.94155
\(842\) 13.7426i 0.473603i
\(843\) − 36.8328i − 1.26859i
\(844\) −14.0000 −0.481900
\(845\) 0 0
\(846\) 1.41641 0.0486971
\(847\) 1.76393i 0.0606094i
\(848\) − 0.472136i − 0.0162132i
\(849\) 43.5967 1.49624
\(850\) 0 0
\(851\) 6.47214 0.221862
\(852\) − 6.61803i − 0.226730i
\(853\) 10.7984i 0.369729i 0.982764 + 0.184865i \(0.0591847\pi\)
−0.982764 + 0.184865i \(0.940815\pi\)
\(854\) −5.76393 −0.197238
\(855\) 0 0
\(856\) 18.1803 0.621391
\(857\) − 6.58359i − 0.224891i −0.993658 0.112446i \(-0.964132\pi\)
0.993658 0.112446i \(-0.0358684\pi\)
\(858\) − 32.7426i − 1.11782i
\(859\) 24.0689 0.821220 0.410610 0.911811i \(-0.365316\pi\)
0.410610 + 0.911811i \(0.365316\pi\)
\(860\) 0 0
\(861\) −3.32624 −0.113358
\(862\) 3.34752i 0.114017i
\(863\) − 32.7639i − 1.11530i −0.830077 0.557649i \(-0.811704\pi\)
0.830077 0.557649i \(-0.188296\pi\)
\(864\) 5.47214 0.186166
\(865\) 0 0
\(866\) 8.50658 0.289065
\(867\) 32.5066i 1.10398i
\(868\) 5.61803i 0.190688i
\(869\) 4.36068 0.147926
\(870\) 0 0
\(871\) −102.610 −3.47680
\(872\) − 11.5623i − 0.391549i
\(873\) 4.72949i 0.160069i
\(874\) −1.85410 −0.0627159
\(875\) 0 0
\(876\) 5.23607 0.176910
\(877\) 18.7426i 0.632894i 0.948610 + 0.316447i \(0.102490\pi\)
−0.948610 + 0.316447i \(0.897510\pi\)
\(878\) 13.3820i 0.451619i
\(879\) 32.1803 1.08542
\(880\) 0 0
\(881\) 8.58359 0.289189 0.144594 0.989491i \(-0.453812\pi\)
0.144594 + 0.989491i \(0.453812\pi\)
\(882\) 2.52786i 0.0851176i
\(883\) − 15.5623i − 0.523713i −0.965107 0.261857i \(-0.915665\pi\)
0.965107 0.261857i \(-0.0843348\pi\)
\(884\) 43.1803 1.45231
\(885\) 0 0
\(886\) 25.0902 0.842921
\(887\) − 5.16718i − 0.173497i −0.996230 0.0867485i \(-0.972352\pi\)
0.996230 0.0867485i \(-0.0276477\pi\)
\(888\) − 10.4721i − 0.351422i
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) − 20.9443i − 0.701266i
\(893\) 6.87539i 0.230076i
\(894\) 12.7082 0.425026
\(895\) 0 0
\(896\) −0.618034 −0.0206471
\(897\) − 11.4721i − 0.383043i
\(898\) 1.56231i 0.0521348i
\(899\) 83.9574 2.80014
\(900\) 0 0
\(901\) 2.87539 0.0957931
\(902\) − 9.49342i − 0.316096i
\(903\) 0 0
\(904\) −1.05573 −0.0351130
\(905\) 0 0
\(906\) −4.14590 −0.137738
\(907\) 7.12461i 0.236569i 0.992980 + 0.118284i \(0.0377395\pi\)
−0.992980 + 0.118284i \(0.962261\pi\)
\(908\) 18.7639i 0.622703i
\(909\) −0.111456 −0.00369677
\(910\) 0 0
\(911\) 36.0689 1.19502 0.597508 0.801863i \(-0.296158\pi\)
0.597508 + 0.801863i \(0.296158\pi\)
\(912\) 3.00000i 0.0993399i
\(913\) − 19.8197i − 0.655935i
\(914\) 37.7771 1.24955
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) − 1.81966i − 0.0600905i
\(918\) 33.3262i 1.09993i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −46.0344 −1.51689
\(922\) − 39.2361i − 1.29217i
\(923\) − 29.0000i − 0.954547i
\(924\) −2.85410 −0.0938931
\(925\) 0 0
\(926\) 2.00000 0.0657241
\(927\) − 6.32624i − 0.207781i
\(928\) 9.23607i 0.303189i
\(929\) 3.52786 0.115745 0.0578727 0.998324i \(-0.481568\pi\)
0.0578727 + 0.998324i \(0.481568\pi\)
\(930\) 0 0
\(931\) −12.2705 −0.402150
\(932\) 6.29180i 0.206095i
\(933\) − 6.47214i − 0.211888i
\(934\) 17.1246 0.560334
\(935\) 0 0
\(936\) 2.70820 0.0885204
\(937\) − 36.7984i − 1.20215i −0.799192 0.601075i \(-0.794739\pi\)
0.799192 0.601075i \(-0.205261\pi\)
\(938\) 8.94427i 0.292041i
\(939\) −20.7082 −0.675787
\(940\) 0 0
\(941\) −22.4934 −0.733265 −0.366632 0.930366i \(-0.619489\pi\)
−0.366632 + 0.930366i \(0.619489\pi\)
\(942\) − 6.00000i − 0.195491i
\(943\) − 3.32624i − 0.108317i
\(944\) −1.70820 −0.0555973
\(945\) 0 0
\(946\) 0 0
\(947\) − 54.6869i − 1.77709i −0.458793 0.888543i \(-0.651718\pi\)
0.458793 0.888543i \(-0.348282\pi\)
\(948\) − 2.47214i − 0.0802912i
\(949\) 22.9443 0.744803
\(950\) 0 0
\(951\) −17.9443 −0.581883
\(952\) − 3.76393i − 0.121990i
\(953\) − 3.79837i − 0.123041i −0.998106 0.0615207i \(-0.980405\pi\)
0.998106 0.0615207i \(-0.0195950\pi\)
\(954\) 0.180340 0.00583872
\(955\) 0 0
\(956\) −20.3607 −0.658511
\(957\) 42.6525i 1.37876i
\(958\) 31.8885i 1.03027i
\(959\) −6.38197 −0.206084
\(960\) 0 0
\(961\) 51.6312 1.66552
\(962\) − 45.8885i − 1.47951i
\(963\) 6.94427i 0.223776i
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) − 16.5410i − 0.531923i −0.963984 0.265962i \(-0.914311\pi\)
0.963984 0.265962i \(-0.0856895\pi\)
\(968\) 2.85410i 0.0917343i
\(969\) −18.2705 −0.586933
\(970\) 0 0
\(971\) 34.2705 1.09979 0.549896 0.835233i \(-0.314667\pi\)
0.549896 + 0.835233i \(0.314667\pi\)
\(972\) 3.94427i 0.126513i
\(973\) 7.88854i 0.252895i
\(974\) 19.8197 0.635063
\(975\) 0 0
\(976\) −9.32624 −0.298526
\(977\) − 23.5623i − 0.753825i −0.926249 0.376912i \(-0.876986\pi\)
0.926249 0.376912i \(-0.123014\pi\)
\(978\) − 2.23607i − 0.0715016i
\(979\) −29.8885 −0.955242
\(980\) 0 0
\(981\) 4.41641 0.141005
\(982\) 6.18034i 0.197223i
\(983\) 14.2705i 0.455159i 0.973760 + 0.227579i \(0.0730811\pi\)
−0.973760 + 0.227579i \(0.926919\pi\)
\(984\) −5.38197 −0.171571
\(985\) 0 0
\(986\) −56.2492 −1.79134
\(987\) 3.70820i 0.118033i
\(988\) 13.1459i 0.418227i
\(989\) 0 0
\(990\) 0 0
\(991\) 27.5066 0.873775 0.436888 0.899516i \(-0.356081\pi\)
0.436888 + 0.899516i \(0.356081\pi\)
\(992\) 9.09017i 0.288613i
\(993\) − 31.1246i − 0.987710i
\(994\) −2.52786 −0.0801790
\(995\) 0 0
\(996\) −11.2361 −0.356028
\(997\) 57.1935i 1.81134i 0.423987 + 0.905668i \(0.360630\pi\)
−0.423987 + 0.905668i \(0.639370\pi\)
\(998\) − 12.3607i − 0.391270i
\(999\) 35.4164 1.12053
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 1150.2.b.i.599.3 4
5.2 odd 4 1150.2.a.j.1.1 2
5.3 odd 4 230.2.a.c.1.2 2
5.4 even 2 inner 1150.2.b.i.599.2 4
15.8 even 4 2070.2.a.u.1.1 2
20.3 even 4 1840.2.a.l.1.1 2
20.7 even 4 9200.2.a.bu.1.2 2
40.3 even 4 7360.2.a.bn.1.2 2
40.13 odd 4 7360.2.a.bh.1.1 2
115.68 even 4 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 5.3 odd 4
1150.2.a.j.1.1 2 5.2 odd 4
1150.2.b.i.599.2 4 5.4 even 2 inner
1150.2.b.i.599.3 4 1.1 even 1 trivial
1840.2.a.l.1.1 2 20.3 even 4
2070.2.a.u.1.1 2 15.8 even 4
5290.2.a.o.1.2 2 115.68 even 4
7360.2.a.bh.1.1 2 40.13 odd 4
7360.2.a.bn.1.2 2 40.3 even 4
9200.2.a.bu.1.2 2 20.7 even 4