Properties

Label 261.2.c.b.28.4
Level $261$
Weight $2$
Character 261.28
Analytic conductor $2.084$
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] = [261,2,Mod(28,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.c (of order \(2\), degree \(1\), 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: \(2.08409549276\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 28.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 261.28
Dual form 261.2.c.b.28.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803i q^{2} -0.618034 q^{4} -3.23607 q^{5} -4.23607 q^{7} +2.23607i q^{8} +O(q^{10})\) \(q+1.61803i q^{2} -0.618034 q^{4} -3.23607 q^{5} -4.23607 q^{7} +2.23607i q^{8} -5.23607i q^{10} -2.23607i q^{11} -1.00000 q^{13} -6.85410i q^{14} -4.85410 q^{16} +3.00000i q^{17} +7.70820i q^{19} +2.00000 q^{20} +3.61803 q^{22} -1.23607 q^{23} +5.47214 q^{25} -1.61803i q^{26} +2.61803 q^{28} +(4.47214 - 3.00000i) q^{29} -7.23607i q^{31} -3.38197i q^{32} -4.85410 q^{34} +13.7082 q^{35} +4.76393i q^{37} -12.4721 q^{38} -7.23607i q^{40} +4.47214i q^{41} -0.472136i q^{43} +1.38197i q^{44} -2.00000i q^{46} +10.2361i q^{47} +10.9443 q^{49} +8.85410i q^{50} +0.618034 q^{52} -6.76393 q^{53} +7.23607i q^{55} -9.47214i q^{56} +(4.85410 + 7.23607i) q^{58} -8.94427 q^{59} +2.76393i q^{61} +11.7082 q^{62} -4.23607 q^{64} +3.23607 q^{65} +4.70820 q^{67} -1.85410i q^{68} +22.1803i q^{70} -4.76393 q^{71} -7.70820i q^{73} -7.70820 q^{74} -4.76393i q^{76} +9.47214i q^{77} -2.29180i q^{79} +15.7082 q^{80} -7.23607 q^{82} +6.00000 q^{83} -9.70820i q^{85} +0.763932 q^{86} +5.00000 q^{88} +0.0557281i q^{89} +4.23607 q^{91} +0.763932 q^{92} -16.5623 q^{94} -24.9443i q^{95} +18.1803i q^{97} +17.7082i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{5} - 8 q^{7} - 4 q^{13} - 6 q^{16} + 8 q^{20} + 10 q^{22} + 4 q^{23} + 4 q^{25} + 6 q^{28} - 6 q^{34} + 28 q^{35} - 32 q^{38} + 8 q^{49} - 2 q^{52} - 36 q^{53} + 6 q^{58} + 20 q^{62} - 8 q^{64} + 4 q^{65} - 8 q^{67} - 28 q^{71} - 4 q^{74} + 36 q^{80} - 20 q^{82} + 24 q^{83} + 12 q^{86} + 20 q^{88} + 8 q^{91} + 12 q^{92} - 26 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803i 1.14412i 0.820211 + 0.572061i \(0.193856\pi\)
−0.820211 + 0.572061i \(0.806144\pi\)
\(3\) 0 0
\(4\) −0.618034 −0.309017
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 2.23607i 0.790569i
\(9\) 0 0
\(10\) 5.23607i 1.65579i
\(11\) 2.23607i 0.674200i −0.941469 0.337100i \(-0.890554\pi\)
0.941469 0.337100i \(-0.109446\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 6.85410i 1.83184i
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 7.70820i 1.76838i 0.467124 + 0.884192i \(0.345290\pi\)
−0.467124 + 0.884192i \(0.654710\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 3.61803 0.771367
\(23\) −1.23607 −0.257738 −0.128869 0.991662i \(-0.541135\pi\)
−0.128869 + 0.991662i \(0.541135\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 1.61803i 0.317323i
\(27\) 0 0
\(28\) 2.61803 0.494762
\(29\) 4.47214 3.00000i 0.830455 0.557086i
\(30\) 0 0
\(31\) 7.23607i 1.29964i −0.760090 0.649818i \(-0.774845\pi\)
0.760090 0.649818i \(-0.225155\pi\)
\(32\) 3.38197i 0.597853i
\(33\) 0 0
\(34\) −4.85410 −0.832472
\(35\) 13.7082 2.31711
\(36\) 0 0
\(37\) 4.76393i 0.783186i 0.920139 + 0.391593i \(0.128076\pi\)
−0.920139 + 0.391593i \(0.871924\pi\)
\(38\) −12.4721 −2.02325
\(39\) 0 0
\(40\) 7.23607i 1.14412i
\(41\) 4.47214i 0.698430i 0.937043 + 0.349215i \(0.113552\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 0.472136i 0.0720001i −0.999352 0.0360000i \(-0.988538\pi\)
0.999352 0.0360000i \(-0.0114616\pi\)
\(44\) 1.38197i 0.208339i
\(45\) 0 0
\(46\) 2.00000i 0.294884i
\(47\) 10.2361i 1.49308i 0.665338 + 0.746542i \(0.268287\pi\)
−0.665338 + 0.746542i \(0.731713\pi\)
\(48\) 0 0
\(49\) 10.9443 1.56347
\(50\) 8.85410i 1.25216i
\(51\) 0 0
\(52\) 0.618034 0.0857059
\(53\) −6.76393 −0.929098 −0.464549 0.885548i \(-0.653783\pi\)
−0.464549 + 0.885548i \(0.653783\pi\)
\(54\) 0 0
\(55\) 7.23607i 0.975711i
\(56\) 9.47214i 1.26577i
\(57\) 0 0
\(58\) 4.85410 + 7.23607i 0.637375 + 0.950142i
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 2.76393i 0.353885i 0.984221 + 0.176943i \(0.0566207\pi\)
−0.984221 + 0.176943i \(0.943379\pi\)
\(62\) 11.7082 1.48694
\(63\) 0 0
\(64\) −4.23607 −0.529508
\(65\) 3.23607 0.401385
\(66\) 0 0
\(67\) 4.70820 0.575199 0.287599 0.957751i \(-0.407143\pi\)
0.287599 + 0.957751i \(0.407143\pi\)
\(68\) 1.85410i 0.224843i
\(69\) 0 0
\(70\) 22.1803i 2.65106i
\(71\) −4.76393 −0.565375 −0.282687 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(72\) 0 0
\(73\) 7.70820i 0.902177i −0.892479 0.451089i \(-0.851036\pi\)
0.892479 0.451089i \(-0.148964\pi\)
\(74\) −7.70820 −0.896061
\(75\) 0 0
\(76\) 4.76393i 0.546460i
\(77\) 9.47214i 1.07945i
\(78\) 0 0
\(79\) 2.29180i 0.257847i −0.991655 0.128924i \(-0.958848\pi\)
0.991655 0.128924i \(-0.0411522\pi\)
\(80\) 15.7082 1.75623
\(81\) 0 0
\(82\) −7.23607 −0.799090
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 9.70820i 1.05300i
\(86\) 0.763932 0.0823769
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) 0.0557281i 0.00590717i 0.999996 + 0.00295358i \(0.000940156\pi\)
−0.999996 + 0.00295358i \(0.999060\pi\)
\(90\) 0 0
\(91\) 4.23607 0.444061
\(92\) 0.763932 0.0796454
\(93\) 0 0
\(94\) −16.5623 −1.70827
\(95\) 24.9443i 2.55923i
\(96\) 0 0
\(97\) 18.1803i 1.84593i 0.384879 + 0.922967i \(0.374243\pi\)
−0.384879 + 0.922967i \(0.625757\pi\)
\(98\) 17.7082i 1.78880i
\(99\) 0 0
\(100\) −3.38197 −0.338197
\(101\) 3.94427i 0.392470i 0.980557 + 0.196235i \(0.0628715\pi\)
−0.980557 + 0.196235i \(0.937129\pi\)
\(102\) 0 0
\(103\) −19.4164 −1.91316 −0.956578 0.291477i \(-0.905853\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(104\) 2.23607i 0.219265i
\(105\) 0 0
\(106\) 10.9443i 1.06300i
\(107\) −0.763932 −0.0738521 −0.0369260 0.999318i \(-0.511757\pi\)
−0.0369260 + 0.999318i \(0.511757\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) −11.7082 −1.11633
\(111\) 0 0
\(112\) 20.5623 1.94296
\(113\) 4.52786i 0.425946i −0.977058 0.212973i \(-0.931685\pi\)
0.977058 0.212973i \(-0.0683146\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −2.76393 + 1.85410i −0.256625 + 0.172149i
\(117\) 0 0
\(118\) 14.4721i 1.33227i
\(119\) 12.7082i 1.16496i
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) −4.47214 −0.404888
\(123\) 0 0
\(124\) 4.47214i 0.401610i
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 12.4721i 1.10672i −0.832941 0.553362i \(-0.813345\pi\)
0.832941 0.553362i \(-0.186655\pi\)
\(128\) 13.6180i 1.20368i
\(129\) 0 0
\(130\) 5.23607i 0.459234i
\(131\) 12.2361i 1.06907i −0.845146 0.534535i \(-0.820487\pi\)
0.845146 0.534535i \(-0.179513\pi\)
\(132\) 0 0
\(133\) 32.6525i 2.83133i
\(134\) 7.61803i 0.658098i
\(135\) 0 0
\(136\) −6.70820 −0.575224
\(137\) 3.52786i 0.301406i 0.988579 + 0.150703i \(0.0481537\pi\)
−0.988579 + 0.150703i \(0.951846\pi\)
\(138\) 0 0
\(139\) −6.70820 −0.568982 −0.284491 0.958679i \(-0.591825\pi\)
−0.284491 + 0.958679i \(0.591825\pi\)
\(140\) −8.47214 −0.716026
\(141\) 0 0
\(142\) 7.70820i 0.646858i
\(143\) 2.23607i 0.186989i
\(144\) 0 0
\(145\) −14.4721 + 9.70820i −1.20185 + 0.806222i
\(146\) 12.4721 1.03220
\(147\) 0 0
\(148\) 2.94427i 0.242018i
\(149\) 5.52786 0.452860 0.226430 0.974027i \(-0.427295\pi\)
0.226430 + 0.974027i \(0.427295\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −17.2361 −1.39803
\(153\) 0 0
\(154\) −15.3262 −1.23502
\(155\) 23.4164i 1.88085i
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 3.70820 0.295009
\(159\) 0 0
\(160\) 10.9443i 0.865221i
\(161\) 5.23607 0.412660
\(162\) 0 0
\(163\) 7.41641i 0.580898i 0.956890 + 0.290449i \(0.0938046\pi\)
−0.956890 + 0.290449i \(0.906195\pi\)
\(164\) 2.76393i 0.215827i
\(165\) 0 0
\(166\) 9.70820i 0.753503i
\(167\) −15.8885 −1.22949 −0.614746 0.788725i \(-0.710742\pi\)
−0.614746 + 0.788725i \(0.710742\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 15.7082 1.20476
\(171\) 0 0
\(172\) 0.291796i 0.0222492i
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −23.1803 −1.75227
\(176\) 10.8541i 0.818159i
\(177\) 0 0
\(178\) −0.0901699 −0.00675852
\(179\) 17.2361 1.28828 0.644142 0.764906i \(-0.277215\pi\)
0.644142 + 0.764906i \(0.277215\pi\)
\(180\) 0 0
\(181\) 20.4164 1.51754 0.758770 0.651359i \(-0.225801\pi\)
0.758770 + 0.651359i \(0.225801\pi\)
\(182\) 6.85410i 0.508060i
\(183\) 0 0
\(184\) 2.76393i 0.203760i
\(185\) 15.4164i 1.13344i
\(186\) 0 0
\(187\) 6.70820 0.490552
\(188\) 6.32624i 0.461388i
\(189\) 0 0
\(190\) 40.3607 2.92807
\(191\) 8.94427i 0.647185i −0.946197 0.323592i \(-0.895109\pi\)
0.946197 0.323592i \(-0.104891\pi\)
\(192\) 0 0
\(193\) 17.4164i 1.25366i 0.779156 + 0.626830i \(0.215648\pi\)
−0.779156 + 0.626830i \(0.784352\pi\)
\(194\) −29.4164 −2.11198
\(195\) 0 0
\(196\) −6.76393 −0.483138
\(197\) 3.70820 0.264199 0.132099 0.991236i \(-0.457828\pi\)
0.132099 + 0.991236i \(0.457828\pi\)
\(198\) 0 0
\(199\) −20.1246 −1.42660 −0.713298 0.700861i \(-0.752799\pi\)
−0.713298 + 0.700861i \(0.752799\pi\)
\(200\) 12.2361i 0.865221i
\(201\) 0 0
\(202\) −6.38197 −0.449034
\(203\) −18.9443 + 12.7082i −1.32963 + 0.891941i
\(204\) 0 0
\(205\) 14.4721i 1.01078i
\(206\) 31.4164i 2.18888i
\(207\) 0 0
\(208\) 4.85410 0.336571
\(209\) 17.2361 1.19224
\(210\) 0 0
\(211\) 10.0000i 0.688428i 0.938891 + 0.344214i \(0.111855\pi\)
−0.938891 + 0.344214i \(0.888145\pi\)
\(212\) 4.18034 0.287107
\(213\) 0 0
\(214\) 1.23607i 0.0844959i
\(215\) 1.52786i 0.104199i
\(216\) 0 0
\(217\) 30.6525i 2.08083i
\(218\) 8.09017i 0.547935i
\(219\) 0 0
\(220\) 4.47214i 0.301511i
\(221\) 3.00000i 0.201802i
\(222\) 0 0
\(223\) −12.7082 −0.851004 −0.425502 0.904957i \(-0.639903\pi\)
−0.425502 + 0.904957i \(0.639903\pi\)
\(224\) 14.3262i 0.957212i
\(225\) 0 0
\(226\) 7.32624 0.487334
\(227\) −9.05573 −0.601050 −0.300525 0.953774i \(-0.597162\pi\)
−0.300525 + 0.953774i \(0.597162\pi\)
\(228\) 0 0
\(229\) 22.1803i 1.46572i 0.680380 + 0.732859i \(0.261815\pi\)
−0.680380 + 0.732859i \(0.738185\pi\)
\(230\) 6.47214i 0.426760i
\(231\) 0 0
\(232\) 6.70820 + 10.0000i 0.440415 + 0.656532i
\(233\) 10.4721 0.686052 0.343026 0.939326i \(-0.388548\pi\)
0.343026 + 0.939326i \(0.388548\pi\)
\(234\) 0 0
\(235\) 33.1246i 2.16081i
\(236\) 5.52786 0.359833
\(237\) 0 0
\(238\) 20.5623 1.33286
\(239\) −26.1803 −1.69347 −0.846733 0.532019i \(-0.821434\pi\)
−0.846733 + 0.532019i \(0.821434\pi\)
\(240\) 0 0
\(241\) −3.00000 −0.193247 −0.0966235 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) 9.70820i 0.624067i
\(243\) 0 0
\(244\) 1.70820i 0.109357i
\(245\) −35.4164 −2.26267
\(246\) 0 0
\(247\) 7.70820i 0.490461i
\(248\) 16.1803 1.02745
\(249\) 0 0
\(250\) 2.47214i 0.156352i
\(251\) 12.2361i 0.772334i 0.922429 + 0.386167i \(0.126201\pi\)
−0.922429 + 0.386167i \(0.873799\pi\)
\(252\) 0 0
\(253\) 2.76393i 0.173767i
\(254\) 20.1803 1.26623
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 26.4721 1.65129 0.825643 0.564193i \(-0.190812\pi\)
0.825643 + 0.564193i \(0.190812\pi\)
\(258\) 0 0
\(259\) 20.1803i 1.25395i
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 19.7984 1.22315
\(263\) 4.94427i 0.304877i 0.988313 + 0.152438i \(0.0487126\pi\)
−0.988313 + 0.152438i \(0.951287\pi\)
\(264\) 0 0
\(265\) 21.8885 1.34460
\(266\) 52.8328 3.23939
\(267\) 0 0
\(268\) −2.90983 −0.177746
\(269\) 21.3607i 1.30238i 0.758913 + 0.651192i \(0.225731\pi\)
−0.758913 + 0.651192i \(0.774269\pi\)
\(270\) 0 0
\(271\) 23.4164i 1.42245i −0.702967 0.711223i \(-0.748142\pi\)
0.702967 0.711223i \(-0.251858\pi\)
\(272\) 14.5623i 0.882969i
\(273\) 0 0
\(274\) −5.70820 −0.344845
\(275\) 12.2361i 0.737863i
\(276\) 0 0
\(277\) 6.41641 0.385525 0.192762 0.981245i \(-0.438255\pi\)
0.192762 + 0.981245i \(0.438255\pi\)
\(278\) 10.8541i 0.650986i
\(279\) 0 0
\(280\) 30.6525i 1.83184i
\(281\) 1.41641 0.0844958 0.0422479 0.999107i \(-0.486548\pi\)
0.0422479 + 0.999107i \(0.486548\pi\)
\(282\) 0 0
\(283\) −13.8885 −0.825588 −0.412794 0.910824i \(-0.635447\pi\)
−0.412794 + 0.910824i \(0.635447\pi\)
\(284\) 2.94427 0.174710
\(285\) 0 0
\(286\) −3.61803 −0.213939
\(287\) 18.9443i 1.11825i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) −15.7082 23.4164i −0.922417 1.37506i
\(291\) 0 0
\(292\) 4.76393i 0.278788i
\(293\) 29.9443i 1.74936i 0.484698 + 0.874682i \(0.338929\pi\)
−0.484698 + 0.874682i \(0.661071\pi\)
\(294\) 0 0
\(295\) 28.9443 1.68520
\(296\) −10.6525 −0.619163
\(297\) 0 0
\(298\) 8.94427i 0.518128i
\(299\) 1.23607 0.0714837
\(300\) 0 0
\(301\) 2.00000i 0.115278i
\(302\) 19.4164i 1.11729i
\(303\) 0 0
\(304\) 37.4164i 2.14598i
\(305\) 8.94427i 0.512148i
\(306\) 0 0
\(307\) 28.1803i 1.60834i 0.594401 + 0.804168i \(0.297389\pi\)
−0.594401 + 0.804168i \(0.702611\pi\)
\(308\) 5.85410i 0.333568i
\(309\) 0 0
\(310\) −37.8885 −2.15192
\(311\) 15.6525i 0.887570i 0.896133 + 0.443785i \(0.146365\pi\)
−0.896133 + 0.443785i \(0.853635\pi\)
\(312\) 0 0
\(313\) −5.47214 −0.309303 −0.154652 0.987969i \(-0.549426\pi\)
−0.154652 + 0.987969i \(0.549426\pi\)
\(314\) −3.23607 −0.182622
\(315\) 0 0
\(316\) 1.41641i 0.0796792i
\(317\) 23.8328i 1.33858i −0.742999 0.669292i \(-0.766597\pi\)
0.742999 0.669292i \(-0.233403\pi\)
\(318\) 0 0
\(319\) −6.70820 10.0000i −0.375587 0.559893i
\(320\) 13.7082 0.766312
\(321\) 0 0
\(322\) 8.47214i 0.472134i
\(323\) −23.1246 −1.28669
\(324\) 0 0
\(325\) −5.47214 −0.303539
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 43.3607i 2.39055i
\(330\) 0 0
\(331\) 8.29180i 0.455758i −0.973689 0.227879i \(-0.926821\pi\)
0.973689 0.227879i \(-0.0731791\pi\)
\(332\) −3.70820 −0.203514
\(333\) 0 0
\(334\) 25.7082i 1.40669i
\(335\) −15.2361 −0.832435
\(336\) 0 0
\(337\) 19.2361i 1.04786i 0.851763 + 0.523928i \(0.175534\pi\)
−0.851763 + 0.523928i \(0.824466\pi\)
\(338\) 19.4164i 1.05611i
\(339\) 0 0
\(340\) 6.00000i 0.325396i
\(341\) −16.1803 −0.876215
\(342\) 0 0
\(343\) −16.7082 −0.902158
\(344\) 1.05573 0.0569210
\(345\) 0 0
\(346\) 9.70820i 0.521916i
\(347\) 22.6525 1.21605 0.608024 0.793918i \(-0.291962\pi\)
0.608024 + 0.793918i \(0.291962\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 37.5066i 2.00481i
\(351\) 0 0
\(352\) −7.56231 −0.403072
\(353\) −4.65248 −0.247626 −0.123813 0.992306i \(-0.539512\pi\)
−0.123813 + 0.992306i \(0.539512\pi\)
\(354\) 0 0
\(355\) 15.4164 0.818218
\(356\) 0.0344419i 0.00182541i
\(357\) 0 0
\(358\) 27.8885i 1.47396i
\(359\) 25.3050i 1.33554i 0.744366 + 0.667772i \(0.232752\pi\)
−0.744366 + 0.667772i \(0.767248\pi\)
\(360\) 0 0
\(361\) −40.4164 −2.12718
\(362\) 33.0344i 1.73625i
\(363\) 0 0
\(364\) −2.61803 −0.137222
\(365\) 24.9443i 1.30564i
\(366\) 0 0
\(367\) 18.0000i 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 24.9443 1.29679
\(371\) 28.6525 1.48756
\(372\) 0 0
\(373\) −3.88854 −0.201341 −0.100671 0.994920i \(-0.532099\pi\)
−0.100671 + 0.994920i \(0.532099\pi\)
\(374\) 10.8541i 0.561252i
\(375\) 0 0
\(376\) −22.8885 −1.18039
\(377\) −4.47214 + 3.00000i −0.230327 + 0.154508i
\(378\) 0 0
\(379\) 14.2918i 0.734120i 0.930197 + 0.367060i \(0.119636\pi\)
−0.930197 + 0.367060i \(0.880364\pi\)
\(380\) 15.4164i 0.790845i
\(381\) 0 0
\(382\) 14.4721 0.740459
\(383\) 5.34752 0.273246 0.136623 0.990623i \(-0.456375\pi\)
0.136623 + 0.990623i \(0.456375\pi\)
\(384\) 0 0
\(385\) 30.6525i 1.56219i
\(386\) −28.1803 −1.43434
\(387\) 0 0
\(388\) 11.2361i 0.570425i
\(389\) 13.4721i 0.683064i 0.939870 + 0.341532i \(0.110946\pi\)
−0.939870 + 0.341532i \(0.889054\pi\)
\(390\) 0 0
\(391\) 3.70820i 0.187532i
\(392\) 24.4721i 1.23603i
\(393\) 0 0
\(394\) 6.00000i 0.302276i
\(395\) 7.41641i 0.373160i
\(396\) 0 0
\(397\) 6.94427 0.348523 0.174262 0.984699i \(-0.444246\pi\)
0.174262 + 0.984699i \(0.444246\pi\)
\(398\) 32.5623i 1.63220i
\(399\) 0 0
\(400\) −26.5623 −1.32812
\(401\) −33.7082 −1.68331 −0.841654 0.540018i \(-0.818418\pi\)
−0.841654 + 0.540018i \(0.818418\pi\)
\(402\) 0 0
\(403\) 7.23607i 0.360454i
\(404\) 2.43769i 0.121280i
\(405\) 0 0
\(406\) −20.5623 30.6525i −1.02049 1.52126i
\(407\) 10.6525 0.528024
\(408\) 0 0
\(409\) 10.5836i 0.523325i −0.965159 0.261662i \(-0.915729\pi\)
0.965159 0.261662i \(-0.0842707\pi\)
\(410\) 23.4164 1.15645
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) 37.8885 1.86437
\(414\) 0 0
\(415\) −19.4164 −0.953114
\(416\) 3.38197i 0.165815i
\(417\) 0 0
\(418\) 27.8885i 1.36407i
\(419\) −3.81966 −0.186603 −0.0933013 0.995638i \(-0.529742\pi\)
−0.0933013 + 0.995638i \(0.529742\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i 0.873195 + 0.487370i \(0.162044\pi\)
−0.873195 + 0.487370i \(0.837956\pi\)
\(422\) −16.1803 −0.787647
\(423\) 0 0
\(424\) 15.1246i 0.734516i
\(425\) 16.4164i 0.796313i
\(426\) 0 0
\(427\) 11.7082i 0.566600i
\(428\) 0.472136 0.0228216
\(429\) 0 0
\(430\) −2.47214 −0.119217
\(431\) −14.3607 −0.691730 −0.345865 0.938284i \(-0.612414\pi\)
−0.345865 + 0.938284i \(0.612414\pi\)
\(432\) 0 0
\(433\) 14.2918i 0.686820i −0.939186 0.343410i \(-0.888418\pi\)
0.939186 0.343410i \(-0.111582\pi\)
\(434\) −49.5967 −2.38072
\(435\) 0 0
\(436\) −3.09017 −0.147992
\(437\) 9.52786i 0.455780i
\(438\) 0 0
\(439\) 13.2918 0.634383 0.317191 0.948362i \(-0.397260\pi\)
0.317191 + 0.948362i \(0.397260\pi\)
\(440\) −16.1803 −0.771367
\(441\) 0 0
\(442\) 4.85410 0.230886
\(443\) 9.65248i 0.458603i −0.973355 0.229301i \(-0.926356\pi\)
0.973355 0.229301i \(-0.0736442\pi\)
\(444\) 0 0
\(445\) 0.180340i 0.00854893i
\(446\) 20.5623i 0.973653i
\(447\) 0 0
\(448\) 17.9443 0.847787
\(449\) 18.8885i 0.891405i −0.895181 0.445703i \(-0.852954\pi\)
0.895181 0.445703i \(-0.147046\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 2.79837i 0.131624i
\(453\) 0 0
\(454\) 14.6525i 0.687675i
\(455\) −13.7082 −0.642651
\(456\) 0 0
\(457\) 6.41641 0.300147 0.150073 0.988675i \(-0.452049\pi\)
0.150073 + 0.988675i \(0.452049\pi\)
\(458\) −35.8885 −1.67696
\(459\) 0 0
\(460\) −2.47214 −0.115264
\(461\) 18.9443i 0.882323i 0.897428 + 0.441161i \(0.145433\pi\)
−0.897428 + 0.441161i \(0.854567\pi\)
\(462\) 0 0
\(463\) 0.708204 0.0329130 0.0164565 0.999865i \(-0.494761\pi\)
0.0164565 + 0.999865i \(0.494761\pi\)
\(464\) −21.7082 + 14.5623i −1.00778 + 0.676038i
\(465\) 0 0
\(466\) 16.9443i 0.784928i
\(467\) 13.5279i 0.625995i 0.949754 + 0.312997i \(0.101333\pi\)
−0.949754 + 0.312997i \(0.898667\pi\)
\(468\) 0 0
\(469\) −19.9443 −0.920941
\(470\) 53.5967 2.47223
\(471\) 0 0
\(472\) 20.0000i 0.920575i
\(473\) −1.05573 −0.0485424
\(474\) 0 0
\(475\) 42.1803i 1.93537i
\(476\) 7.85410i 0.359992i
\(477\) 0 0
\(478\) 42.3607i 1.93753i
\(479\) 10.4721i 0.478484i −0.970960 0.239242i \(-0.923101\pi\)
0.970960 0.239242i \(-0.0768989\pi\)
\(480\) 0 0
\(481\) 4.76393i 0.217217i
\(482\) 4.85410i 0.221098i
\(483\) 0 0
\(484\) −3.70820 −0.168555
\(485\) 58.8328i 2.67146i
\(486\) 0 0
\(487\) 20.3607 0.922630 0.461315 0.887236i \(-0.347378\pi\)
0.461315 + 0.887236i \(0.347378\pi\)
\(488\) −6.18034 −0.279771
\(489\) 0 0
\(490\) 57.3050i 2.58877i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 9.00000 + 13.4164i 0.405340 + 0.604245i
\(494\) 12.4721 0.561148
\(495\) 0 0
\(496\) 35.1246i 1.57714i
\(497\) 20.1803 0.905212
\(498\) 0 0
\(499\) 21.1803 0.948162 0.474081 0.880481i \(-0.342780\pi\)
0.474081 + 0.880481i \(0.342780\pi\)
\(500\) 0.944272 0.0422291
\(501\) 0 0
\(502\) −19.7984 −0.883645
\(503\) 25.1803i 1.12274i −0.827566 0.561368i \(-0.810275\pi\)
0.827566 0.561368i \(-0.189725\pi\)
\(504\) 0 0
\(505\) 12.7639i 0.567988i
\(506\) −4.47214 −0.198811
\(507\) 0 0
\(508\) 7.70820i 0.341996i
\(509\) 26.8328 1.18934 0.594672 0.803969i \(-0.297282\pi\)
0.594672 + 0.803969i \(0.297282\pi\)
\(510\) 0 0
\(511\) 32.6525i 1.44446i
\(512\) 5.29180i 0.233867i
\(513\) 0 0
\(514\) 42.8328i 1.88927i
\(515\) 62.8328 2.76874
\(516\) 0 0
\(517\) 22.8885 1.00664
\(518\) 32.6525 1.43467
\(519\) 0 0
\(520\) 7.23607i 0.317323i
\(521\) 3.52786 0.154559 0.0772793 0.997009i \(-0.475377\pi\)
0.0772793 + 0.997009i \(0.475377\pi\)
\(522\) 0 0
\(523\) 33.0689 1.44600 0.723001 0.690847i \(-0.242762\pi\)
0.723001 + 0.690847i \(0.242762\pi\)
\(524\) 7.56231i 0.330361i
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 21.7082 0.945624
\(528\) 0 0
\(529\) −21.4721 −0.933571
\(530\) 35.4164i 1.53839i
\(531\) 0 0
\(532\) 20.1803i 0.874929i
\(533\) 4.47214i 0.193710i
\(534\) 0 0
\(535\) 2.47214 0.106880
\(536\) 10.5279i 0.454734i
\(537\) 0 0
\(538\) −34.5623 −1.49009
\(539\) 24.4721i 1.05409i
\(540\) 0 0
\(541\) 10.6525i 0.457986i −0.973428 0.228993i \(-0.926457\pi\)
0.973428 0.228993i \(-0.0735432\pi\)
\(542\) 37.8885 1.62745
\(543\) 0 0
\(544\) 10.1459 0.435002
\(545\) −16.1803 −0.693090
\(546\) 0 0
\(547\) −6.34752 −0.271401 −0.135700 0.990750i \(-0.543328\pi\)
−0.135700 + 0.990750i \(0.543328\pi\)
\(548\) 2.18034i 0.0931395i
\(549\) 0 0
\(550\) 19.7984 0.844205
\(551\) 23.1246 + 34.4721i 0.985142 + 1.46856i
\(552\) 0 0
\(553\) 9.70820i 0.412835i
\(554\) 10.3820i 0.441087i
\(555\) 0 0
\(556\) 4.14590 0.175825
\(557\) 16.4721 0.697947 0.348973 0.937133i \(-0.386530\pi\)
0.348973 + 0.937133i \(0.386530\pi\)
\(558\) 0 0
\(559\) 0.472136i 0.0199692i
\(560\) −66.5410 −2.81187
\(561\) 0 0
\(562\) 2.29180i 0.0966736i
\(563\) 13.0689i 0.550788i −0.961331 0.275394i \(-0.911192\pi\)
0.961331 0.275394i \(-0.0888083\pi\)
\(564\) 0 0
\(565\) 14.6525i 0.616434i
\(566\) 22.4721i 0.944574i
\(567\) 0 0
\(568\) 10.6525i 0.446968i
\(569\) 21.0000i 0.880366i −0.897908 0.440183i \(-0.854914\pi\)
0.897908 0.440183i \(-0.145086\pi\)
\(570\) 0 0
\(571\) 18.8328 0.788129 0.394064 0.919083i \(-0.371069\pi\)
0.394064 + 0.919083i \(0.371069\pi\)
\(572\) 1.38197i 0.0577829i
\(573\) 0 0
\(574\) 30.6525 1.27941
\(575\) −6.76393 −0.282075
\(576\) 0 0
\(577\) 24.8328i 1.03380i −0.856045 0.516902i \(-0.827085\pi\)
0.856045 0.516902i \(-0.172915\pi\)
\(578\) 12.9443i 0.538411i
\(579\) 0 0
\(580\) 8.94427 6.00000i 0.371391 0.249136i
\(581\) −25.4164 −1.05445
\(582\) 0 0
\(583\) 15.1246i 0.626397i
\(584\) 17.2361 0.713234
\(585\) 0 0
\(586\) −48.4508 −2.00149
\(587\) 0.944272 0.0389743 0.0194871 0.999810i \(-0.493797\pi\)
0.0194871 + 0.999810i \(0.493797\pi\)
\(588\) 0 0
\(589\) 55.7771 2.29825
\(590\) 46.8328i 1.92808i
\(591\) 0 0
\(592\) 23.1246i 0.950416i
\(593\) −40.1803 −1.65001 −0.825004 0.565126i \(-0.808827\pi\)
−0.825004 + 0.565126i \(0.808827\pi\)
\(594\) 0 0
\(595\) 41.1246i 1.68594i
\(596\) −3.41641 −0.139942
\(597\) 0 0
\(598\) 2.00000i 0.0817861i
\(599\) 1.76393i 0.0720723i 0.999350 + 0.0360362i \(0.0114731\pi\)
−0.999350 + 0.0360362i \(0.988527\pi\)
\(600\) 0 0
\(601\) 18.5410i 0.756304i −0.925744 0.378152i \(-0.876560\pi\)
0.925744 0.378152i \(-0.123440\pi\)
\(602\) −3.23607 −0.131892
\(603\) 0 0
\(604\) −7.41641 −0.301769
\(605\) −19.4164 −0.789389
\(606\) 0 0
\(607\) 25.4164i 1.03162i 0.856703 + 0.515810i \(0.172509\pi\)
−0.856703 + 0.515810i \(0.827491\pi\)
\(608\) 26.0689 1.05723
\(609\) 0 0
\(610\) 14.4721 0.585960
\(611\) 10.2361i 0.414107i
\(612\) 0 0
\(613\) −24.4164 −0.986169 −0.493085 0.869981i \(-0.664131\pi\)
−0.493085 + 0.869981i \(0.664131\pi\)
\(614\) −45.5967 −1.84013
\(615\) 0 0
\(616\) −21.1803 −0.853380
\(617\) 19.8885i 0.800683i −0.916366 0.400341i \(-0.868892\pi\)
0.916366 0.400341i \(-0.131108\pi\)
\(618\) 0 0
\(619\) 32.5410i 1.30793i −0.756523 0.653967i \(-0.773104\pi\)
0.756523 0.653967i \(-0.226896\pi\)
\(620\) 14.4721i 0.581215i
\(621\) 0 0
\(622\) −25.3262 −1.01549
\(623\) 0.236068i 0.00945786i
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 8.85410i 0.353881i
\(627\) 0 0
\(628\) 1.23607i 0.0493245i
\(629\) −14.2918 −0.569851
\(630\) 0 0
\(631\) 28.7082 1.14286 0.571428 0.820652i \(-0.306390\pi\)
0.571428 + 0.820652i \(0.306390\pi\)
\(632\) 5.12461 0.203846
\(633\) 0 0
\(634\) 38.5623 1.53150
\(635\) 40.3607i 1.60166i
\(636\) 0 0
\(637\) −10.9443 −0.433628
\(638\) 16.1803 10.8541i 0.640586 0.429718i
\(639\) 0 0
\(640\) 44.0689i 1.74198i
\(641\) 19.4721i 0.769103i 0.923104 + 0.384552i \(0.125644\pi\)
−0.923104 + 0.384552i \(0.874356\pi\)
\(642\) 0 0
\(643\) 0.708204 0.0279288 0.0139644 0.999902i \(-0.495555\pi\)
0.0139644 + 0.999902i \(0.495555\pi\)
\(644\) −3.23607 −0.127519
\(645\) 0 0
\(646\) 37.4164i 1.47213i
\(647\) −17.5967 −0.691800 −0.345900 0.938271i \(-0.612426\pi\)
−0.345900 + 0.938271i \(0.612426\pi\)
\(648\) 0 0
\(649\) 20.0000i 0.785069i
\(650\) 8.85410i 0.347286i
\(651\) 0 0
\(652\) 4.58359i 0.179507i
\(653\) 32.3050i 1.26419i 0.774891 + 0.632095i \(0.217805\pi\)
−0.774891 + 0.632095i \(0.782195\pi\)
\(654\) 0 0
\(655\) 39.5967i 1.54717i
\(656\) 21.7082i 0.847563i
\(657\) 0 0
\(658\) 70.1591 2.73508
\(659\) 26.2361i 1.02201i 0.859577 + 0.511006i \(0.170727\pi\)
−0.859577 + 0.511006i \(0.829273\pi\)
\(660\) 0 0
\(661\) −29.8328 −1.16036 −0.580181 0.814488i \(-0.697018\pi\)
−0.580181 + 0.814488i \(0.697018\pi\)
\(662\) 13.4164 0.521443
\(663\) 0 0
\(664\) 13.4164i 0.520658i
\(665\) 105.666i 4.09754i
\(666\) 0 0
\(667\) −5.52786 + 3.70820i −0.214040 + 0.143582i
\(668\) 9.81966 0.379934
\(669\) 0 0
\(670\) 24.6525i 0.952408i
\(671\) 6.18034 0.238589
\(672\) 0 0
\(673\) −4.41641 −0.170240 −0.0851200 0.996371i \(-0.527127\pi\)
−0.0851200 + 0.996371i \(0.527127\pi\)
\(674\) −31.1246 −1.19888
\(675\) 0 0
\(676\) 7.41641 0.285246
\(677\) 18.0557i 0.693938i −0.937877 0.346969i \(-0.887211\pi\)
0.937877 0.346969i \(-0.112789\pi\)
\(678\) 0 0
\(679\) 77.0132i 2.95549i
\(680\) 21.7082 0.832472
\(681\) 0 0
\(682\) 26.1803i 1.00250i
\(683\) −32.9443 −1.26058 −0.630289 0.776361i \(-0.717064\pi\)
−0.630289 + 0.776361i \(0.717064\pi\)
\(684\) 0 0
\(685\) 11.4164i 0.436199i
\(686\) 27.0344i 1.03218i
\(687\) 0 0
\(688\) 2.29180i 0.0873739i
\(689\) 6.76393 0.257685
\(690\) 0 0
\(691\) −30.2361 −1.15023 −0.575117 0.818071i \(-0.695044\pi\)
−0.575117 + 0.818071i \(0.695044\pi\)
\(692\) −3.70820 −0.140965
\(693\) 0 0
\(694\) 36.6525i 1.39131i
\(695\) 21.7082 0.823439
\(696\) 0 0
\(697\) −13.4164 −0.508183
\(698\) 16.1803i 0.612435i
\(699\) 0 0
\(700\) 14.3262 0.541481
\(701\) −8.18034 −0.308967 −0.154484 0.987995i \(-0.549371\pi\)
−0.154484 + 0.987995i \(0.549371\pi\)
\(702\) 0 0
\(703\) −36.7214 −1.38497
\(704\) 9.47214i 0.356995i
\(705\) 0 0
\(706\) 7.52786i 0.283315i
\(707\) 16.7082i 0.628377i
\(708\) 0 0
\(709\) −13.4164 −0.503864 −0.251932 0.967745i \(-0.581066\pi\)
−0.251932 + 0.967745i \(0.581066\pi\)
\(710\) 24.9443i 0.936142i
\(711\) 0 0
\(712\) −0.124612 −0.00467002
\(713\) 8.94427i 0.334966i
\(714\) 0 0
\(715\) 7.23607i 0.270614i
\(716\) −10.6525 −0.398102
\(717\) 0 0
\(718\) −40.9443 −1.52803
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 82.2492 3.06312
\(722\) 65.3951i 2.43375i
\(723\) 0 0
\(724\) −12.6180 −0.468946
\(725\) 24.4721 16.4164i 0.908872 0.609690i
\(726\) 0 0
\(727\) 5.81966i 0.215839i 0.994160 + 0.107920i \(0.0344189\pi\)
−0.994160 + 0.107920i \(0.965581\pi\)
\(728\) 9.47214i 0.351061i
\(729\) 0 0
\(730\) −40.3607 −1.49382
\(731\) 1.41641 0.0523877
\(732\) 0 0
\(733\) 8.47214i 0.312925i 0.987684 + 0.156463i \(0.0500091\pi\)
−0.987684 + 0.156463i \(0.949991\pi\)
\(734\) 29.1246 1.07501
\(735\) 0 0
\(736\) 4.18034i 0.154089i
\(737\) 10.5279i 0.387799i
\(738\) 0 0
\(739\) 12.1803i 0.448061i 0.974582 + 0.224031i \(0.0719215\pi\)
−0.974582 + 0.224031i \(0.928078\pi\)
\(740\) 9.52786i 0.350251i
\(741\) 0 0
\(742\) 46.3607i 1.70195i
\(743\) 43.0689i 1.58004i −0.613078 0.790022i \(-0.710069\pi\)
0.613078 0.790022i \(-0.289931\pi\)
\(744\) 0 0
\(745\) −17.8885 −0.655386
\(746\) 6.29180i 0.230359i
\(747\) 0 0
\(748\) −4.14590 −0.151589
\(749\) 3.23607 0.118243
\(750\) 0 0
\(751\) 26.8328i 0.979143i 0.871963 + 0.489572i \(0.162847\pi\)
−0.871963 + 0.489572i \(0.837153\pi\)
\(752\) 49.6869i 1.81190i
\(753\) 0 0
\(754\) −4.85410 7.23607i −0.176776 0.263522i
\(755\) −38.8328 −1.41327
\(756\) 0 0
\(757\) 38.0000i 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) −23.1246 −0.839924
\(759\) 0 0
\(760\) 55.7771 2.02325
\(761\) −2.65248 −0.0961522 −0.0480761 0.998844i \(-0.515309\pi\)
−0.0480761 + 0.998844i \(0.515309\pi\)
\(762\) 0 0
\(763\) −21.1803 −0.766780
\(764\) 5.52786i 0.199991i
\(765\) 0 0
\(766\) 8.65248i 0.312627i
\(767\) 8.94427 0.322959
\(768\) 0 0
\(769\) 40.1803i 1.44894i −0.689306 0.724470i \(-0.742085\pi\)
0.689306 0.724470i \(-0.257915\pi\)
\(770\) 49.5967 1.78734
\(771\) 0 0
\(772\) 10.7639i 0.387402i
\(773\) 31.8885i 1.14695i −0.819223 0.573476i \(-0.805595\pi\)
0.819223 0.573476i \(-0.194405\pi\)
\(774\) 0 0
\(775\) 39.5967i 1.42236i
\(776\) −40.6525 −1.45934
\(777\) 0 0
\(778\) −21.7984 −0.781510
\(779\) −34.4721 −1.23509
\(780\) 0 0
\(781\) 10.6525i 0.381176i
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) −53.1246 −1.89731
\(785\) 6.47214i 0.231000i
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −2.29180 −0.0816419
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) 19.1803i 0.681974i
\(792\) 0 0
\(793\) 2.76393i 0.0981501i
\(794\) 11.2361i 0.398753i
\(795\) 0 0
\(796\) 12.4377 0.440842
\(797\) 44.8328i 1.58806i 0.607879 + 0.794030i \(0.292021\pi\)
−0.607879 + 0.794030i \(0.707979\pi\)
\(798\) 0 0
\(799\) −30.7082 −1.08638
\(800\) 18.5066i 0.654306i
\(801\) 0 0
\(802\) 54.5410i 1.92591i
\(803\) −17.2361 −0.608248
\(804\) 0 0
\(805\) −16.9443 −0.597207
\(806\) −11.7082 −0.412404
\(807\) 0 0
\(808\) −8.81966 −0.310275
\(809\) 43.3607i 1.52448i −0.647294 0.762240i \(-0.724100\pi\)
0.647294 0.762240i \(-0.275900\pi\)
\(810\) 0 0
\(811\) 46.5967 1.63623 0.818117 0.575052i \(-0.195018\pi\)
0.818117 + 0.575052i \(0.195018\pi\)
\(812\) 11.7082 7.85410i 0.410877 0.275625i
\(813\) 0 0
\(814\) 17.2361i 0.604124i
\(815\) 24.0000i 0.840683i
\(816\) 0 0
\(817\) 3.63932 0.127324
\(818\) 17.1246 0.598748
\(819\) 0 0
\(820\) 8.94427i 0.312348i
\(821\) 12.4721 0.435281 0.217640 0.976029i \(-0.430164\pi\)
0.217640 + 0.976029i \(0.430164\pi\)
\(822\) 0 0
\(823\) 26.3607i 0.918876i 0.888210 + 0.459438i \(0.151949\pi\)
−0.888210 + 0.459438i \(0.848051\pi\)
\(824\) 43.4164i 1.51248i
\(825\) 0 0
\(826\) 61.3050i 2.13307i
\(827\) 29.8885i 1.03933i −0.854371 0.519663i \(-0.826057\pi\)
0.854371 0.519663i \(-0.173943\pi\)
\(828\) 0 0
\(829\) 27.3050i 0.948340i 0.880433 + 0.474170i \(0.157252\pi\)
−0.880433 + 0.474170i \(0.842748\pi\)
\(830\) 31.4164i 1.09048i
\(831\) 0 0
\(832\) 4.23607 0.146859
\(833\) 32.8328i 1.13759i
\(834\) 0 0
\(835\) 51.4164 1.77934
\(836\) −10.6525 −0.368424
\(837\) 0 0
\(838\) 6.18034i 0.213496i
\(839\) 0.708204i 0.0244499i 0.999925 + 0.0122250i \(0.00389142\pi\)
−0.999925 + 0.0122250i \(0.996109\pi\)
\(840\) 0 0
\(841\) 11.0000 26.8328i 0.379310 0.925270i
\(842\) −32.3607 −1.11522
\(843\) 0 0
\(844\) 6.18034i 0.212736i
\(845\) 38.8328 1.33589
\(846\) 0 0
\(847\) −25.4164 −0.873318
\(848\) 32.8328 1.12748
\(849\) 0 0
\(850\) −26.5623 −0.911080
\(851\) 5.88854i 0.201857i
\(852\) 0 0
\(853\) 45.1935i 1.54740i −0.633555 0.773698i \(-0.718405\pi\)
0.633555 0.773698i \(-0.281595\pi\)
\(854\) 18.9443 0.648260
\(855\) 0 0
\(856\) 1.70820i 0.0583852i
\(857\) −37.5967 −1.28428 −0.642140 0.766587i \(-0.721953\pi\)
−0.642140 + 0.766587i \(0.721953\pi\)
\(858\) 0 0
\(859\) 36.3607i 1.24061i −0.784361 0.620305i \(-0.787009\pi\)
0.784361 0.620305i \(-0.212991\pi\)
\(860\) 0.944272i 0.0321994i
\(861\) 0 0
\(862\) 23.2361i 0.791424i
\(863\) 46.6525 1.58807 0.794034 0.607873i \(-0.207977\pi\)
0.794034 + 0.607873i \(0.207977\pi\)
\(864\) 0 0
\(865\) −19.4164 −0.660178
\(866\) 23.1246 0.785806
\(867\) 0 0
\(868\) 18.9443i 0.643010i
\(869\) −5.12461 −0.173841
\(870\) 0 0
\(871\) −4.70820 −0.159531
\(872\) 11.1803i 0.378614i
\(873\) 0 0
\(874\) 15.4164 0.521468
\(875\) 6.47214 0.218798
\(876\) 0 0
\(877\) 32.4721 1.09651 0.548253 0.836312i \(-0.315293\pi\)
0.548253 + 0.836312i \(0.315293\pi\)
\(878\) 21.5066i 0.725812i
\(879\) 0 0
\(880\) 35.1246i 1.18405i
\(881\) 55.2492i 1.86139i 0.365792 + 0.930697i \(0.380798\pi\)
−0.365792 + 0.930697i \(0.619202\pi\)
\(882\) 0 0
\(883\) 34.2492 1.15258 0.576289 0.817246i \(-0.304500\pi\)
0.576289 + 0.817246i \(0.304500\pi\)
\(884\) 1.85410i 0.0623602i
\(885\) 0 0
\(886\) 15.6180 0.524698
\(887\) 17.0689i 0.573117i 0.958063 + 0.286559i \(0.0925113\pi\)
−0.958063 + 0.286559i \(0.907489\pi\)
\(888\) 0 0
\(889\) 52.8328i 1.77196i
\(890\) 0.291796 0.00978103
\(891\) 0 0
\(892\) 7.85410 0.262975
\(893\) −78.9017 −2.64034
\(894\) 0 0
\(895\) −55.7771 −1.86442
\(896\) 57.6869i 1.92718i
\(897\) 0 0
\(898\) 30.5623 1.01988
\(899\) −21.7082 32.3607i −0.724009 1.07929i
\(900\) 0 0
\(901\) 20.2918i 0.676018i
\(902\) 16.1803i 0.538746i
\(903\) 0 0
\(904\) 10.1246 0.336740
\(905\) −66.0689 −2.19620
\(906\) 0 0
\(907\) 26.4721i 0.878993i 0.898244 + 0.439496i \(0.144843\pi\)
−0.898244 + 0.439496i \(0.855157\pi\)
\(908\) 5.59675 0.185735
\(909\) 0 0
\(910\) 22.1803i 0.735271i
\(911\) 51.1803i 1.69568i 0.530252 + 0.847840i \(0.322097\pi\)
−0.530252 + 0.847840i \(0.677903\pi\)
\(912\) 0 0
\(913\) 13.4164i 0.444018i
\(914\) 10.3820i 0.343405i
\(915\) 0 0
\(916\) 13.7082i 0.452932i
\(917\) 51.8328i 1.71167i
\(918\) 0 0
\(919\) −25.6525 −0.846197 −0.423099 0.906084i \(-0.639058\pi\)
−0.423099 + 0.906084i \(0.639058\pi\)
\(920\) 8.94427i 0.294884i
\(921\) 0 0
\(922\) −30.6525 −1.00949
\(923\) 4.76393 0.156807
\(924\) 0 0
\(925\) 26.0689i 0.857140i
\(926\) 1.14590i 0.0376565i
\(927\) 0 0
\(928\) −10.1459 15.1246i −0.333055 0.496490i
\(929\) 32.3607 1.06172 0.530860 0.847460i \(-0.321869\pi\)
0.530860 + 0.847460i \(0.321869\pi\)
\(930\) 0 0
\(931\) 84.3607i 2.76481i
\(932\) −6.47214 −0.212002
\(933\) 0 0
\(934\) −21.8885 −0.716215
\(935\) −21.7082 −0.709934
\(936\) 0 0
\(937\) 33.0000 1.07806 0.539032 0.842286i \(-0.318790\pi\)
0.539032 + 0.842286i \(0.318790\pi\)
\(938\) 32.2705i 1.05367i
\(939\) 0 0
\(940\) 20.4721i 0.667727i
\(941\) 44.8328 1.46151 0.730754 0.682641i \(-0.239169\pi\)
0.730754 + 0.682641i \(0.239169\pi\)
\(942\) 0 0
\(943\) 5.52786i 0.180012i
\(944\) 43.4164 1.41308
\(945\) 0 0
\(946\) 1.70820i 0.0555385i
\(947\) 16.8197i 0.546566i 0.961934 + 0.273283i \(0.0881095\pi\)
−0.961934 + 0.273283i \(0.911891\pi\)
\(948\) 0 0
\(949\) 7.70820i 0.250219i
\(950\) −68.2492 −2.21430
\(951\) 0 0
\(952\) 28.4164 0.920981
\(953\) 59.6656 1.93276 0.966380 0.257119i \(-0.0827733\pi\)
0.966380 + 0.257119i \(0.0827733\pi\)
\(954\) 0 0
\(955\) 28.9443i 0.936615i
\(956\) 16.1803 0.523310
\(957\) 0 0
\(958\) 16.9443 0.547445
\(959\) 14.9443i 0.482576i
\(960\) 0 0
\(961\) −21.3607 −0.689054
\(962\) 7.70820 0.248522
\(963\) 0 0
\(964\) 1.85410 0.0597166
\(965\) 56.3607i 1.81431i
\(966\) 0 0
\(967\) 27.1246i 0.872269i 0.899882 + 0.436134i \(0.143653\pi\)
−0.899882 + 0.436134i \(0.856347\pi\)
\(968\) 13.4164i 0.431220i
\(969\) 0 0
\(970\) 95.1935 3.05648
\(971\) 16.5836i 0.532193i −0.963946 0.266096i \(-0.914266\pi\)
0.963946 0.266096i \(-0.0857340\pi\)
\(972\) 0 0
\(973\) 28.4164 0.910988
\(974\) 32.9443i 1.05560i
\(975\) 0 0
\(976\) 13.4164i 0.429449i
\(977\) −51.0132 −1.63206 −0.816028 0.578013i \(-0.803828\pi\)
−0.816028 + 0.578013i \(0.803828\pi\)
\(978\) 0 0
\(979\) 0.124612 0.00398261
\(980\) 21.8885 0.699204
\(981\) 0 0
\(982\) 0 0
\(983\) 13.8885i 0.442976i 0.975163 + 0.221488i \(0.0710913\pi\)
−0.975163 + 0.221488i \(0.928909\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) −21.7082 + 14.5623i −0.691330 + 0.463758i
\(987\) 0 0
\(988\) 4.76393i 0.151561i
\(989\) 0.583592i 0.0185572i
\(990\) 0 0
\(991\) −21.2918 −0.676356 −0.338178 0.941082i \(-0.609811\pi\)
−0.338178 + 0.941082i \(0.609811\pi\)
\(992\) −24.4721 −0.776991
\(993\) 0 0
\(994\) 32.6525i 1.03567i
\(995\) 65.1246 2.06459
\(996\) 0 0
\(997\) 24.5836i 0.778570i −0.921117 0.389285i \(-0.872722\pi\)
0.921117 0.389285i \(-0.127278\pi\)
\(998\) 34.2705i 1.08481i
\(999\) 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 261.2.c.b.28.4 4
3.2 odd 2 87.2.c.a.28.1 4
4.3 odd 2 4176.2.o.l.289.2 4
12.11 even 2 1392.2.o.i.289.2 4
15.2 even 4 2175.2.f.b.724.3 4
15.8 even 4 2175.2.f.a.724.2 4
15.14 odd 2 2175.2.d.e.376.4 4
29.12 odd 4 7569.2.a.f.1.1 2
29.17 odd 4 7569.2.a.n.1.2 2
29.28 even 2 inner 261.2.c.b.28.1 4
87.17 even 4 2523.2.a.d.1.1 2
87.41 even 4 2523.2.a.e.1.2 2
87.86 odd 2 87.2.c.a.28.4 yes 4
116.115 odd 2 4176.2.o.l.289.1 4
348.347 even 2 1392.2.o.i.289.4 4
435.173 even 4 2175.2.f.b.724.4 4
435.347 even 4 2175.2.f.a.724.1 4
435.434 odd 2 2175.2.d.e.376.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.c.a.28.1 4 3.2 odd 2
87.2.c.a.28.4 yes 4 87.86 odd 2
261.2.c.b.28.1 4 29.28 even 2 inner
261.2.c.b.28.4 4 1.1 even 1 trivial
1392.2.o.i.289.2 4 12.11 even 2
1392.2.o.i.289.4 4 348.347 even 2
2175.2.d.e.376.1 4 435.434 odd 2
2175.2.d.e.376.4 4 15.14 odd 2
2175.2.f.a.724.1 4 435.347 even 4
2175.2.f.a.724.2 4 15.8 even 4
2175.2.f.b.724.3 4 15.2 even 4
2175.2.f.b.724.4 4 435.173 even 4
2523.2.a.d.1.1 2 87.17 even 4
2523.2.a.e.1.2 2 87.41 even 4
4176.2.o.l.289.1 4 116.115 odd 2
4176.2.o.l.289.2 4 4.3 odd 2
7569.2.a.f.1.1 2 29.12 odd 4
7569.2.a.n.1.2 2 29.17 odd 4