Properties

Label 1183.2.c.g.337.5
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, 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("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11667456256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.5
Root \(0.231361i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.231361i q^{2} -3.32225 q^{3} +1.94647 q^{4} -2.23136i q^{5} -0.768639i q^{6} +1.00000i q^{7} +0.913059i q^{8} +8.03736 q^{9} +O(q^{10})\) \(q+0.231361i q^{2} -3.32225 q^{3} +1.94647 q^{4} -2.23136i q^{5} -0.768639i q^{6} +1.00000i q^{7} +0.913059i q^{8} +8.03736 q^{9} +0.516249 q^{10} -3.32225i q^{11} -6.46667 q^{12} -0.231361 q^{14} +7.41314i q^{15} +3.68170 q^{16} +1.37578 q^{17} +1.85953i q^{18} +3.23531i q^{19} -4.34328i q^{20} -3.32225i q^{21} +0.768639 q^{22} -0.838502 q^{23} -3.03341i q^{24} +0.0210289 q^{25} -16.7354 q^{27} +1.94647i q^{28} -0.607142 q^{29} -1.71511 q^{30} +1.71511i q^{31} +2.67792i q^{32} +11.0374i q^{33} +0.318302i q^{34} +2.23136 q^{35} +15.6445 q^{36} -1.55361i q^{37} -0.748524 q^{38} +2.03736 q^{40} -9.17783i q^{41} +0.768639 q^{42} -1.23136 q^{43} -6.46667i q^{44} -17.9343i q^{45} -0.193997i q^{46} +1.62817i q^{47} -12.2315 q^{48} -1.00000 q^{49} +0.00486525i q^{50} -4.57069 q^{51} +8.39607 q^{53} -3.87192i q^{54} -7.41314 q^{55} -0.913059 q^{56} -10.7485i q^{57} -0.140469i q^{58} -8.82234i q^{59} +14.4295i q^{60} +5.46667 q^{61} -0.396810 q^{62} +8.03736i q^{63} +6.74383 q^{64} -2.55361 q^{66} -10.1857i q^{67} +2.67792 q^{68} +2.78572 q^{69} +0.516249i q^{70} -5.21428i q^{71} +7.33859i q^{72} +3.96355i q^{73} +0.359445 q^{74} -0.0698632 q^{75} +6.29744i q^{76} +3.32225 q^{77} +6.45051 q^{79} -8.21520i q^{80} +31.4871 q^{81} +2.12339 q^{82} -4.64055i q^{83} -6.46667i q^{84} -3.06986i q^{85} -0.284889i q^{86} +2.01708 q^{87} +3.03341 q^{88} -9.12826i q^{89} +4.14929 q^{90} -1.63212 q^{92} -5.69803i q^{93} -0.376695 q^{94} +7.21915 q^{95} -8.89672i q^{96} -15.3589i q^{97} -0.231361i q^{98} -26.7022i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40} + 10 q^{42} - 6 q^{43} + 22 q^{48} - 8 q^{49} - 14 q^{51} + 4 q^{53} - 6 q^{55} - 12 q^{56} + 16 q^{61} + 10 q^{62} - 28 q^{64} + 12 q^{66} - 66 q^{68} + 36 q^{69} + 40 q^{74} + 14 q^{75} - 2 q^{77} - 52 q^{79} + 48 q^{81} + 28 q^{82} + 26 q^{87} - 6 q^{88} - 52 q^{90} + 24 q^{92} + 66 q^{94} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.231361i 0.163597i 0.996649 + 0.0817984i \(0.0260664\pi\)
−0.996649 + 0.0817984i \(0.973934\pi\)
\(3\) −3.32225 −1.91810 −0.959052 0.283231i \(-0.908594\pi\)
−0.959052 + 0.283231i \(0.908594\pi\)
\(4\) 1.94647 0.973236
\(5\) − 2.23136i − 0.997895i −0.866632 0.498947i \(-0.833720\pi\)
0.866632 0.498947i \(-0.166280\pi\)
\(6\) − 0.768639i − 0.313796i
\(7\) 1.00000i 0.377964i
\(8\) 0.913059i 0.322815i
\(9\) 8.03736 2.67912
\(10\) 0.516249 0.163252
\(11\) − 3.32225i − 1.00170i −0.865535 0.500848i \(-0.833021\pi\)
0.865535 0.500848i \(-0.166979\pi\)
\(12\) −6.46667 −1.86677
\(13\) 0 0
\(14\) −0.231361 −0.0618338
\(15\) 7.41314i 1.91407i
\(16\) 3.68170 0.920425
\(17\) 1.37578 0.333676 0.166838 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(18\) 1.85953i 0.438296i
\(19\) 3.23531i 0.742231i 0.928587 + 0.371116i \(0.121025\pi\)
−0.928587 + 0.371116i \(0.878975\pi\)
\(20\) − 4.34328i − 0.971187i
\(21\) − 3.32225i − 0.724975i
\(22\) 0.768639 0.163874
\(23\) −0.838502 −0.174840 −0.0874199 0.996172i \(-0.527862\pi\)
−0.0874199 + 0.996172i \(0.527862\pi\)
\(24\) − 3.03341i − 0.619193i
\(25\) 0.0210289 0.00420577
\(26\) 0 0
\(27\) −16.7354 −3.22073
\(28\) 1.94647i 0.367849i
\(29\) −0.607142 −0.112743 −0.0563717 0.998410i \(-0.517953\pi\)
−0.0563717 + 0.998410i \(0.517953\pi\)
\(30\) −1.71511 −0.313135
\(31\) 1.71511i 0.308043i 0.988067 + 0.154022i \(0.0492225\pi\)
−0.988067 + 0.154022i \(0.950777\pi\)
\(32\) 2.67792i 0.473394i
\(33\) 11.0374i 1.92136i
\(34\) 0.318302i 0.0545883i
\(35\) 2.23136 0.377169
\(36\) 15.6445 2.60742
\(37\) − 1.55361i − 0.255413i −0.991812 0.127706i \(-0.959239\pi\)
0.991812 0.127706i \(-0.0407615\pi\)
\(38\) −0.748524 −0.121427
\(39\) 0 0
\(40\) 2.03736 0.322136
\(41\) − 9.17783i − 1.43334i −0.697414 0.716668i \(-0.745666\pi\)
0.697414 0.716668i \(-0.254334\pi\)
\(42\) 0.768639 0.118604
\(43\) −1.23136 −0.187781 −0.0938904 0.995583i \(-0.529930\pi\)
−0.0938904 + 0.995583i \(0.529930\pi\)
\(44\) − 6.46667i − 0.974888i
\(45\) − 17.9343i − 2.67348i
\(46\) − 0.193997i − 0.0286032i
\(47\) 1.62817i 0.237493i 0.992925 + 0.118747i \(0.0378876\pi\)
−0.992925 + 0.118747i \(0.962112\pi\)
\(48\) −12.2315 −1.76547
\(49\) −1.00000 −0.142857
\(50\) 0.00486525i 0 0.000688051i
\(51\) −4.57069 −0.640025
\(52\) 0 0
\(53\) 8.39607 1.15329 0.576644 0.816995i \(-0.304362\pi\)
0.576644 + 0.816995i \(0.304362\pi\)
\(54\) − 3.87192i − 0.526901i
\(55\) −7.41314 −0.999588
\(56\) −0.913059 −0.122013
\(57\) − 10.7485i − 1.42368i
\(58\) − 0.140469i − 0.0184445i
\(59\) − 8.82234i − 1.14857i −0.818655 0.574285i \(-0.805280\pi\)
0.818655 0.574285i \(-0.194720\pi\)
\(60\) 14.4295i 1.86284i
\(61\) 5.46667 0.699936 0.349968 0.936762i \(-0.386193\pi\)
0.349968 + 0.936762i \(0.386193\pi\)
\(62\) −0.396810 −0.0503949
\(63\) 8.03736i 1.01261i
\(64\) 6.74383 0.842979
\(65\) 0 0
\(66\) −2.55361 −0.314328
\(67\) − 10.1857i − 1.24439i −0.782864 0.622193i \(-0.786242\pi\)
0.782864 0.622193i \(-0.213758\pi\)
\(68\) 2.67792 0.324745
\(69\) 2.78572 0.335361
\(70\) 0.516249i 0.0617036i
\(71\) − 5.21428i − 0.618822i −0.950928 0.309411i \(-0.899868\pi\)
0.950928 0.309411i \(-0.100132\pi\)
\(72\) 7.33859i 0.864861i
\(73\) 3.96355i 0.463898i 0.972728 + 0.231949i \(0.0745103\pi\)
−0.972728 + 0.231949i \(0.925490\pi\)
\(74\) 0.359445 0.0417847
\(75\) −0.0698632 −0.00806711
\(76\) 6.29744i 0.722366i
\(77\) 3.32225 0.378606
\(78\) 0 0
\(79\) 6.45051 0.725739 0.362869 0.931840i \(-0.381797\pi\)
0.362869 + 0.931840i \(0.381797\pi\)
\(80\) − 8.21520i − 0.918487i
\(81\) 31.4871 3.49857
\(82\) 2.12339 0.234489
\(83\) − 4.64055i − 0.509367i −0.967024 0.254684i \(-0.918029\pi\)
0.967024 0.254684i \(-0.0819713\pi\)
\(84\) − 6.46667i − 0.705572i
\(85\) − 3.06986i − 0.332973i
\(86\) − 0.284889i − 0.0307203i
\(87\) 2.01708 0.216253
\(88\) 3.03341 0.323363
\(89\) − 9.12826i − 0.967593i −0.875180 0.483797i \(-0.839257\pi\)
0.875180 0.483797i \(-0.160743\pi\)
\(90\) 4.14929 0.437373
\(91\) 0 0
\(92\) −1.63212 −0.170160
\(93\) − 5.69803i − 0.590859i
\(94\) −0.376695 −0.0388531
\(95\) 7.21915 0.740669
\(96\) − 8.89672i − 0.908018i
\(97\) − 15.3589i − 1.55946i −0.626117 0.779729i \(-0.715357\pi\)
0.626117 0.779729i \(-0.284643\pi\)
\(98\) − 0.231361i − 0.0233710i
\(99\) − 26.7022i − 2.68367i
\(100\) 0.0409321 0.00409321
\(101\) 7.95042 0.791097 0.395548 0.918445i \(-0.370555\pi\)
0.395548 + 0.918445i \(0.370555\pi\)
\(102\) − 1.05748i − 0.104706i
\(103\) −0.694825 −0.0684631 −0.0342316 0.999414i \(-0.510898\pi\)
−0.0342316 + 0.999414i \(0.510898\pi\)
\(104\) 0 0
\(105\) −7.41314 −0.723449
\(106\) 1.94252i 0.188674i
\(107\) 8.94647 0.864888 0.432444 0.901661i \(-0.357651\pi\)
0.432444 + 0.901661i \(0.357651\pi\)
\(108\) −32.5750 −3.13453
\(109\) − 2.27268i − 0.217683i −0.994059 0.108841i \(-0.965286\pi\)
0.994059 0.108841i \(-0.0347141\pi\)
\(110\) − 1.71511i − 0.163529i
\(111\) 5.16150i 0.489908i
\(112\) 3.68170i 0.347888i
\(113\) −9.50478 −0.894134 −0.447067 0.894500i \(-0.647532\pi\)
−0.447067 + 0.894500i \(0.647532\pi\)
\(114\) 2.48679 0.232909
\(115\) 1.87100i 0.174472i
\(116\) −1.18178 −0.109726
\(117\) 0 0
\(118\) 2.04114 0.187902
\(119\) 1.37578i 0.126118i
\(120\) −6.76864 −0.617889
\(121\) −0.0373642 −0.00339675
\(122\) 1.26477i 0.114507i
\(123\) 30.4911i 2.74929i
\(124\) 3.33842i 0.299799i
\(125\) − 11.2037i − 1.00209i
\(126\) −1.85953 −0.165660
\(127\) −18.4334 −1.63570 −0.817851 0.575430i \(-0.804835\pi\)
−0.817851 + 0.575430i \(0.804835\pi\)
\(128\) 6.91610i 0.611302i
\(129\) 4.09089 0.360183
\(130\) 0 0
\(131\) −1.74835 −0.152754 −0.0763771 0.997079i \(-0.524335\pi\)
−0.0763771 + 0.997079i \(0.524335\pi\)
\(132\) 21.4839i 1.86994i
\(133\) −3.23531 −0.280537
\(134\) 2.35658 0.203578
\(135\) 37.3427i 3.21395i
\(136\) 1.25617i 0.107716i
\(137\) 18.0032i 1.53812i 0.639178 + 0.769059i \(0.279275\pi\)
−0.639178 + 0.769059i \(0.720725\pi\)
\(138\) 0.644506i 0.0548640i
\(139\) 13.9179 1.18050 0.590251 0.807219i \(-0.299029\pi\)
0.590251 + 0.807219i \(0.299029\pi\)
\(140\) 4.34328 0.367074
\(141\) − 5.40919i − 0.455536i
\(142\) 1.20638 0.101237
\(143\) 0 0
\(144\) 29.5911 2.46593
\(145\) 1.35475i 0.112506i
\(146\) −0.917010 −0.0758923
\(147\) 3.32225 0.274015
\(148\) − 3.02407i − 0.248577i
\(149\) − 15.9303i − 1.30506i −0.757762 0.652531i \(-0.773707\pi\)
0.757762 0.652531i \(-0.226293\pi\)
\(150\) − 0.0161636i − 0.00131975i
\(151\) 13.9497i 1.13521i 0.823301 + 0.567604i \(0.192130\pi\)
−0.823301 + 0.567604i \(0.807870\pi\)
\(152\) −2.95403 −0.239604
\(153\) 11.0577 0.893958
\(154\) 0.768639i 0.0619387i
\(155\) 3.82703 0.307395
\(156\) 0 0
\(157\) −12.9747 −1.03549 −0.517745 0.855535i \(-0.673229\pi\)
−0.517745 + 0.855535i \(0.673229\pi\)
\(158\) 1.49240i 0.118729i
\(159\) −27.8939 −2.21213
\(160\) 5.97540 0.472397
\(161\) − 0.838502i − 0.0660832i
\(162\) 7.28489i 0.572355i
\(163\) − 18.4085i − 1.44186i −0.693007 0.720931i \(-0.743715\pi\)
0.693007 0.720931i \(-0.256285\pi\)
\(164\) − 17.8644i − 1.39498i
\(165\) 24.6283 1.91731
\(166\) 1.07364 0.0833308
\(167\) 18.4993i 1.43152i 0.698345 + 0.715761i \(0.253920\pi\)
−0.698345 + 0.715761i \(0.746080\pi\)
\(168\) 3.03341 0.234033
\(169\) 0 0
\(170\) 0.710246 0.0544734
\(171\) 26.0034i 1.98853i
\(172\) −2.39681 −0.182755
\(173\) 17.1981 1.30755 0.653774 0.756690i \(-0.273185\pi\)
0.653774 + 0.756690i \(0.273185\pi\)
\(174\) 0.466673i 0.0353784i
\(175\) 0.0210289i 0.00158963i
\(176\) − 12.2315i − 0.921986i
\(177\) 29.3100i 2.20308i
\(178\) 2.11192 0.158295
\(179\) −14.4886 −1.08293 −0.541465 0.840723i \(-0.682130\pi\)
−0.541465 + 0.840723i \(0.682130\pi\)
\(180\) − 34.9085i − 2.60193i
\(181\) −6.85484 −0.509516 −0.254758 0.967005i \(-0.581996\pi\)
−0.254758 + 0.967005i \(0.581996\pi\)
\(182\) 0 0
\(183\) −18.1617 −1.34255
\(184\) − 0.765602i − 0.0564409i
\(185\) −3.46667 −0.254875
\(186\) 1.31830 0.0966626
\(187\) − 4.57069i − 0.334242i
\(188\) 3.16919i 0.231137i
\(189\) − 16.7354i − 1.21732i
\(190\) 1.67023i 0.121171i
\(191\) −2.85163 −0.206337 −0.103168 0.994664i \(-0.532898\pi\)
−0.103168 + 0.994664i \(0.532898\pi\)
\(192\) −22.4047 −1.61692
\(193\) 10.0505i 0.723450i 0.932285 + 0.361725i \(0.117812\pi\)
−0.932285 + 0.361725i \(0.882188\pi\)
\(194\) 3.55344 0.255122
\(195\) 0 0
\(196\) −1.94647 −0.139034
\(197\) − 25.4171i − 1.81089i −0.424460 0.905447i \(-0.639536\pi\)
0.424460 0.905447i \(-0.360464\pi\)
\(198\) 6.17783 0.439039
\(199\) 12.4466 0.882313 0.441157 0.897430i \(-0.354568\pi\)
0.441157 + 0.897430i \(0.354568\pi\)
\(200\) 0.0192006i 0.00135769i
\(201\) 33.8396i 2.38686i
\(202\) 1.83942i 0.129421i
\(203\) − 0.607142i − 0.0426130i
\(204\) −8.89672 −0.622895
\(205\) −20.4791 −1.43032
\(206\) − 0.160755i − 0.0112003i
\(207\) −6.73935 −0.468417
\(208\) 0 0
\(209\) 10.7485 0.743491
\(210\) − 1.71511i − 0.118354i
\(211\) −24.3923 −1.67923 −0.839617 0.543179i \(-0.817221\pi\)
−0.839617 + 0.543179i \(0.817221\pi\)
\(212\) 16.3427 1.12242
\(213\) 17.3232i 1.18696i
\(214\) 2.06986i 0.141493i
\(215\) 2.74761i 0.187385i
\(216\) − 15.2804i − 1.03970i
\(217\) −1.71511 −0.116429
\(218\) 0.525808 0.0356122
\(219\) − 13.1679i − 0.889805i
\(220\) −14.4295 −0.972835
\(221\) 0 0
\(222\) −1.19417 −0.0801473
\(223\) 22.6494i 1.51671i 0.651839 + 0.758357i \(0.273998\pi\)
−0.651839 + 0.758357i \(0.726002\pi\)
\(224\) −2.67792 −0.178926
\(225\) 0.169017 0.0112678
\(226\) − 2.19903i − 0.146278i
\(227\) 1.28506i 0.0852925i 0.999090 + 0.0426462i \(0.0135788\pi\)
−0.999090 + 0.0426462i \(0.986421\pi\)
\(228\) − 20.9217i − 1.38557i
\(229\) 4.64451i 0.306918i 0.988155 + 0.153459i \(0.0490412\pi\)
−0.988155 + 0.153459i \(0.950959\pi\)
\(230\) −0.432876 −0.0285430
\(231\) −11.0374 −0.726205
\(232\) − 0.554356i − 0.0363953i
\(233\) −11.8877 −0.778790 −0.389395 0.921071i \(-0.627316\pi\)
−0.389395 + 0.921071i \(0.627316\pi\)
\(234\) 0 0
\(235\) 3.63304 0.236993
\(236\) − 17.1724i − 1.11783i
\(237\) −21.4302 −1.39204
\(238\) −0.318302 −0.0206324
\(239\) − 4.17783i − 0.270242i −0.990829 0.135121i \(-0.956858\pi\)
0.990829 0.135121i \(-0.0431422\pi\)
\(240\) 27.2930i 1.76175i
\(241\) − 4.03341i − 0.259815i −0.991526 0.129907i \(-0.958532\pi\)
0.991526 0.129907i \(-0.0414680\pi\)
\(242\) − 0.00864462i 0 0.000555697i
\(243\) −54.4020 −3.48989
\(244\) 10.6407 0.681203
\(245\) 2.23136i 0.142556i
\(246\) −7.05444 −0.449775
\(247\) 0 0
\(248\) −1.56600 −0.0994410
\(249\) 15.4171i 0.977019i
\(250\) 2.59210 0.163939
\(251\) −27.8685 −1.75905 −0.879523 0.475857i \(-0.842138\pi\)
−0.879523 + 0.475857i \(0.842138\pi\)
\(252\) 15.6445i 0.985511i
\(253\) 2.78572i 0.175137i
\(254\) − 4.26477i − 0.267596i
\(255\) 10.1989i 0.638678i
\(256\) 11.8875 0.742972
\(257\) 7.14064 0.445421 0.222710 0.974885i \(-0.428510\pi\)
0.222710 + 0.974885i \(0.428510\pi\)
\(258\) 0.946472i 0.0589248i
\(259\) 1.55361 0.0965369
\(260\) 0 0
\(261\) −4.87982 −0.302053
\(262\) − 0.404500i − 0.0249901i
\(263\) 21.3192 1.31460 0.657300 0.753629i \(-0.271699\pi\)
0.657300 + 0.753629i \(0.271699\pi\)
\(264\) −10.0778 −0.620244
\(265\) − 18.7347i − 1.15086i
\(266\) − 0.748524i − 0.0458950i
\(267\) 30.3264i 1.85594i
\(268\) − 19.8263i − 1.21108i
\(269\) 2.78875 0.170033 0.0850167 0.996380i \(-0.472906\pi\)
0.0850167 + 0.996380i \(0.472906\pi\)
\(270\) −8.63964 −0.525792
\(271\) 15.4747i 0.940024i 0.882660 + 0.470012i \(0.155750\pi\)
−0.882660 + 0.470012i \(0.844250\pi\)
\(272\) 5.06521 0.307123
\(273\) 0 0
\(274\) −4.16524 −0.251631
\(275\) − 0.0698632i − 0.00421291i
\(276\) 5.42232 0.326385
\(277\) −5.52955 −0.332238 −0.166119 0.986106i \(-0.553124\pi\)
−0.166119 + 0.986106i \(0.553124\pi\)
\(278\) 3.22006i 0.193127i
\(279\) 13.7850i 0.825285i
\(280\) 2.03736i 0.121756i
\(281\) 31.4871i 1.87836i 0.343419 + 0.939182i \(0.388415\pi\)
−0.343419 + 0.939182i \(0.611585\pi\)
\(282\) 1.25148 0.0745243
\(283\) 7.35118 0.436983 0.218491 0.975839i \(-0.429886\pi\)
0.218491 + 0.975839i \(0.429886\pi\)
\(284\) − 10.1495i − 0.602259i
\(285\) −23.9838 −1.42068
\(286\) 0 0
\(287\) 9.17783 0.541750
\(288\) 21.5234i 1.26828i
\(289\) −15.1072 −0.888660
\(290\) −0.313437 −0.0184056
\(291\) 51.0261i 2.99120i
\(292\) 7.71494i 0.451483i
\(293\) 13.5335i 0.790635i 0.918544 + 0.395318i \(0.129366\pi\)
−0.918544 + 0.395318i \(0.870634\pi\)
\(294\) 0.768639i 0.0448279i
\(295\) −19.6858 −1.14615
\(296\) 1.41854 0.0824510
\(297\) 55.5992i 3.22619i
\(298\) 3.68565 0.213504
\(299\) 0 0
\(300\) −0.135987 −0.00785120
\(301\) − 1.23136i − 0.0709745i
\(302\) −3.22741 −0.185717
\(303\) −26.4133 −1.51741
\(304\) 11.9114i 0.683168i
\(305\) − 12.1981i − 0.698462i
\(306\) 2.55831i 0.146249i
\(307\) 3.30609i 0.188688i 0.995540 + 0.0943442i \(0.0300754\pi\)
−0.995540 + 0.0943442i \(0.969925\pi\)
\(308\) 6.46667 0.368473
\(309\) 2.30838 0.131319
\(310\) 0.885425i 0.0502888i
\(311\) 34.3063 1.94533 0.972665 0.232214i \(-0.0745969\pi\)
0.972665 + 0.232214i \(0.0745969\pi\)
\(312\) 0 0
\(313\) 7.21428 0.407775 0.203888 0.978994i \(-0.434642\pi\)
0.203888 + 0.978994i \(0.434642\pi\)
\(314\) − 3.00183i − 0.169403i
\(315\) 17.9343 1.01048
\(316\) 12.5557 0.706315
\(317\) − 8.04040i − 0.451594i −0.974174 0.225797i \(-0.927501\pi\)
0.974174 0.225797i \(-0.0724986\pi\)
\(318\) − 6.45355i − 0.361897i
\(319\) 2.01708i 0.112935i
\(320\) − 15.0479i − 0.841204i
\(321\) −29.7224 −1.65894
\(322\) 0.193997 0.0108110
\(323\) 4.45108i 0.247665i
\(324\) 61.2888 3.40493
\(325\) 0 0
\(326\) 4.25899 0.235884
\(327\) 7.55040i 0.417538i
\(328\) 8.37990 0.462703
\(329\) −1.62817 −0.0897639
\(330\) 5.69803i 0.313666i
\(331\) 0.893687i 0.0491215i 0.999698 + 0.0245607i \(0.00781871\pi\)
−0.999698 + 0.0245607i \(0.992181\pi\)
\(332\) − 9.03271i − 0.495734i
\(333\) − 12.4870i − 0.684281i
\(334\) −4.28002 −0.234192
\(335\) −22.7281 −1.24177
\(336\) − 12.2315i − 0.667285i
\(337\) −15.0717 −0.821007 −0.410504 0.911859i \(-0.634647\pi\)
−0.410504 + 0.911859i \(0.634647\pi\)
\(338\) 0 0
\(339\) 31.5773 1.71504
\(340\) − 5.97540i − 0.324062i
\(341\) 5.69803 0.308566
\(342\) −6.01616 −0.325317
\(343\) − 1.00000i − 0.0539949i
\(344\) − 1.12431i − 0.0606185i
\(345\) − 6.21594i − 0.334655i
\(346\) 3.97897i 0.213911i
\(347\) −16.4164 −0.881276 −0.440638 0.897685i \(-0.645248\pi\)
−0.440638 + 0.897685i \(0.645248\pi\)
\(348\) 3.92619 0.210466
\(349\) − 34.3722i − 1.83990i −0.392035 0.919950i \(-0.628229\pi\)
0.392035 0.919950i \(-0.371771\pi\)
\(350\) −0.00486525 −0.000260059 0
\(351\) 0 0
\(352\) 8.89672 0.474197
\(353\) − 23.9163i − 1.27293i −0.771304 0.636467i \(-0.780395\pi\)
0.771304 0.636467i \(-0.219605\pi\)
\(354\) −6.78120 −0.360416
\(355\) −11.6349 −0.617519
\(356\) − 17.7679i − 0.941697i
\(357\) − 4.57069i − 0.241907i
\(358\) − 3.35210i − 0.177164i
\(359\) 6.17875i 0.326102i 0.986618 + 0.163051i \(0.0521335\pi\)
−0.986618 + 0.163051i \(0.947867\pi\)
\(360\) 16.3750 0.863040
\(361\) 8.53276 0.449092
\(362\) − 1.58594i − 0.0833552i
\(363\) 0.124133 0.00651531
\(364\) 0 0
\(365\) 8.84411 0.462922
\(366\) − 4.20190i − 0.219637i
\(367\) 19.8560 1.03647 0.518236 0.855237i \(-0.326589\pi\)
0.518236 + 0.855237i \(0.326589\pi\)
\(368\) −3.08711 −0.160927
\(369\) − 73.7656i − 3.84008i
\(370\) − 0.802052i − 0.0416967i
\(371\) 8.39607i 0.435902i
\(372\) − 11.0911i − 0.575045i
\(373\) 30.1951 1.56344 0.781721 0.623628i \(-0.214342\pi\)
0.781721 + 0.623628i \(0.214342\pi\)
\(374\) 1.05748 0.0546809
\(375\) 37.2216i 1.92212i
\(376\) −1.48662 −0.0766664
\(377\) 0 0
\(378\) 3.87192 0.199150
\(379\) − 4.32242i − 0.222028i −0.993819 0.111014i \(-0.964590\pi\)
0.993819 0.111014i \(-0.0354098\pi\)
\(380\) 14.0519 0.720846
\(381\) 61.2405 3.13745
\(382\) − 0.659755i − 0.0337560i
\(383\) − 17.3481i − 0.886449i −0.896411 0.443224i \(-0.853834\pi\)
0.896411 0.443224i \(-0.146166\pi\)
\(384\) − 22.9770i − 1.17254i
\(385\) − 7.41314i − 0.377809i
\(386\) −2.32529 −0.118354
\(387\) −9.89690 −0.503087
\(388\) − 29.8956i − 1.51772i
\(389\) 25.3474 1.28516 0.642582 0.766217i \(-0.277863\pi\)
0.642582 + 0.766217i \(0.277863\pi\)
\(390\) 0 0
\(391\) −1.15360 −0.0583398
\(392\) − 0.913059i − 0.0461164i
\(393\) 5.80847 0.292999
\(394\) 5.88052 0.296256
\(395\) − 14.3934i − 0.724211i
\(396\) − 51.9750i − 2.61184i
\(397\) 27.0749i 1.35885i 0.733745 + 0.679425i \(0.237771\pi\)
−0.733745 + 0.679425i \(0.762229\pi\)
\(398\) 2.87965i 0.144344i
\(399\) 10.7485 0.538099
\(400\) 0.0774219 0.00387110
\(401\) 29.2858i 1.46246i 0.682129 + 0.731232i \(0.261054\pi\)
−0.682129 + 0.731232i \(0.738946\pi\)
\(402\) −7.82916 −0.390483
\(403\) 0 0
\(404\) 15.4753 0.769924
\(405\) − 70.2592i − 3.49121i
\(406\) 0.140469 0.00697135
\(407\) −5.16150 −0.255846
\(408\) − 4.17331i − 0.206610i
\(409\) 23.3713i 1.15563i 0.816166 + 0.577817i \(0.196095\pi\)
−0.816166 + 0.577817i \(0.803905\pi\)
\(410\) − 4.73805i − 0.233996i
\(411\) − 59.8112i − 2.95027i
\(412\) −1.35246 −0.0666308
\(413\) 8.82234 0.434119
\(414\) − 1.55922i − 0.0766315i
\(415\) −10.3548 −0.508295
\(416\) 0 0
\(417\) −46.2389 −2.26433
\(418\) 2.48679i 0.121633i
\(419\) 14.6064 0.713569 0.356785 0.934187i \(-0.383873\pi\)
0.356785 + 0.934187i \(0.383873\pi\)
\(420\) −14.4295 −0.704087
\(421\) 10.2728i 0.500668i 0.968160 + 0.250334i \(0.0805404\pi\)
−0.968160 + 0.250334i \(0.919460\pi\)
\(422\) − 5.64342i − 0.274717i
\(423\) 13.0862i 0.636273i
\(424\) 7.66611i 0.372299i
\(425\) 0.0289311 0.00140336
\(426\) −4.00790 −0.194183
\(427\) 5.46667i 0.264551i
\(428\) 17.4141 0.841740
\(429\) 0 0
\(430\) −0.635689 −0.0306557
\(431\) − 12.5017i − 0.602188i −0.953595 0.301094i \(-0.902648\pi\)
0.953595 0.301094i \(-0.0973518\pi\)
\(432\) −61.6147 −2.96444
\(433\) 10.9472 0.526090 0.263045 0.964784i \(-0.415273\pi\)
0.263045 + 0.964784i \(0.415273\pi\)
\(434\) − 0.396810i − 0.0190475i
\(435\) − 4.50083i − 0.215798i
\(436\) − 4.42370i − 0.211857i
\(437\) − 2.71282i − 0.129772i
\(438\) 3.04654 0.145569
\(439\) 17.9179 0.855176 0.427588 0.903974i \(-0.359363\pi\)
0.427588 + 0.903974i \(0.359363\pi\)
\(440\) − 6.76864i − 0.322682i
\(441\) −8.03736 −0.382732
\(442\) 0 0
\(443\) 27.7194 1.31699 0.658494 0.752586i \(-0.271194\pi\)
0.658494 + 0.752586i \(0.271194\pi\)
\(444\) 10.0467i 0.476796i
\(445\) −20.3684 −0.965556
\(446\) −5.24018 −0.248130
\(447\) 52.9245i 2.50324i
\(448\) 6.74383i 0.318616i
\(449\) 0.165980i 0.00783306i 0.999992 + 0.00391653i \(0.00124667\pi\)
−0.999992 + 0.00391653i \(0.998753\pi\)
\(450\) 0.0391038i 0.00184337i
\(451\) −30.4911 −1.43577
\(452\) −18.5008 −0.870204
\(453\) − 46.3444i − 2.17745i
\(454\) −0.297313 −0.0139536
\(455\) 0 0
\(456\) 9.81404 0.459584
\(457\) 31.7354i 1.48452i 0.670112 + 0.742260i \(0.266246\pi\)
−0.670112 + 0.742260i \(0.733754\pi\)
\(458\) −1.07456 −0.0502107
\(459\) −23.0242 −1.07468
\(460\) 3.64185i 0.169802i
\(461\) 29.0656i 1.35372i 0.736113 + 0.676859i \(0.236659\pi\)
−0.736113 + 0.676859i \(0.763341\pi\)
\(462\) − 2.55361i − 0.118805i
\(463\) − 6.31904i − 0.293671i −0.989161 0.146835i \(-0.953091\pi\)
0.989161 0.146835i \(-0.0469088\pi\)
\(464\) −2.23531 −0.103772
\(465\) −12.7144 −0.589615
\(466\) − 2.75035i − 0.127408i
\(467\) 34.6409 1.60299 0.801495 0.598002i \(-0.204038\pi\)
0.801495 + 0.598002i \(0.204038\pi\)
\(468\) 0 0
\(469\) 10.1857 0.470334
\(470\) 0.840542i 0.0387713i
\(471\) 43.1051 1.98618
\(472\) 8.05532 0.370776
\(473\) 4.09089i 0.188099i
\(474\) − 4.95811i − 0.227734i
\(475\) 0.0680349i 0.00312166i
\(476\) 2.67792i 0.122742i
\(477\) 67.4823 3.08980
\(478\) 0.966587 0.0442107
\(479\) 7.14230i 0.326340i 0.986598 + 0.163170i \(0.0521719\pi\)
−0.986598 + 0.163170i \(0.947828\pi\)
\(480\) −19.8518 −0.906107
\(481\) 0 0
\(482\) 0.933174 0.0425049
\(483\) 2.78572i 0.126755i
\(484\) −0.0727284 −0.00330584
\(485\) −34.2712 −1.55617
\(486\) − 12.5865i − 0.570935i
\(487\) − 18.5003i − 0.838327i −0.907911 0.419163i \(-0.862323\pi\)
0.907911 0.419163i \(-0.137677\pi\)
\(488\) 4.99140i 0.225950i
\(489\) 61.1575i 2.76564i
\(490\) −0.516249 −0.0233218
\(491\) 15.2781 0.689490 0.344745 0.938696i \(-0.387965\pi\)
0.344745 + 0.938696i \(0.387965\pi\)
\(492\) 59.3500i 2.67571i
\(493\) −0.835294 −0.0376197
\(494\) 0 0
\(495\) −59.5821 −2.67802
\(496\) 6.31452i 0.283530i
\(497\) 5.21428 0.233893
\(498\) −3.56691 −0.159837
\(499\) 12.4783i 0.558606i 0.960203 + 0.279303i \(0.0901034\pi\)
−0.960203 + 0.279303i \(0.909897\pi\)
\(500\) − 21.8077i − 0.975272i
\(501\) − 61.4595i − 2.74581i
\(502\) − 6.44768i − 0.287774i
\(503\) −2.58008 −0.115040 −0.0575200 0.998344i \(-0.518319\pi\)
−0.0575200 + 0.998344i \(0.518319\pi\)
\(504\) −7.33859 −0.326887
\(505\) − 17.7403i − 0.789431i
\(506\) −0.644506 −0.0286518
\(507\) 0 0
\(508\) −35.8802 −1.59192
\(509\) 35.6808i 1.58152i 0.612125 + 0.790761i \(0.290315\pi\)
−0.612125 + 0.790761i \(0.709685\pi\)
\(510\) −2.35962 −0.104486
\(511\) −3.96355 −0.175337
\(512\) 16.5825i 0.732850i
\(513\) − 54.1442i − 2.39053i
\(514\) 1.65206i 0.0728694i
\(515\) 1.55040i 0.0683190i
\(516\) 7.96281 0.350543
\(517\) 5.40919 0.237896
\(518\) 0.359445i 0.0157931i
\(519\) −57.1365 −2.50801
\(520\) 0 0
\(521\) −20.9637 −0.918437 −0.459219 0.888323i \(-0.651871\pi\)
−0.459219 + 0.888323i \(0.651871\pi\)
\(522\) − 1.12900i − 0.0494149i
\(523\) −22.8263 −0.998124 −0.499062 0.866566i \(-0.666322\pi\)
−0.499062 + 0.866566i \(0.666322\pi\)
\(524\) −3.40312 −0.148666
\(525\) − 0.0698632i − 0.00304908i
\(526\) 4.93243i 0.215064i
\(527\) 2.35962i 0.102787i
\(528\) 40.6362i 1.76847i
\(529\) −22.2969 −0.969431
\(530\) 4.33447 0.188277
\(531\) − 70.9083i − 3.07716i
\(532\) −6.29744 −0.273029
\(533\) 0 0
\(534\) −7.01634 −0.303627
\(535\) − 19.9628i − 0.863067i
\(536\) 9.30018 0.401707
\(537\) 48.1348 2.07717
\(538\) 0.645208i 0.0278169i
\(539\) 3.32225i 0.143100i
\(540\) 72.6865i 3.12793i
\(541\) − 30.2191i − 1.29922i −0.760266 0.649611i \(-0.774932\pi\)
0.760266 0.649611i \(-0.225068\pi\)
\(542\) −3.58025 −0.153785
\(543\) 22.7735 0.977305
\(544\) 3.68423i 0.157960i
\(545\) −5.07116 −0.217225
\(546\) 0 0
\(547\) −16.8223 −0.719271 −0.359636 0.933093i \(-0.617099\pi\)
−0.359636 + 0.933093i \(0.617099\pi\)
\(548\) 35.0427i 1.49695i
\(549\) 43.9376 1.87521
\(550\) 0.0161636 0.000689219 0
\(551\) − 1.96429i − 0.0836817i
\(552\) 2.54352i 0.108260i
\(553\) 6.45051i 0.274304i
\(554\) − 1.27932i − 0.0543531i
\(555\) 11.5172 0.488876
\(556\) 27.0909 1.14891
\(557\) 10.4918i 0.444553i 0.974984 + 0.222276i \(0.0713487\pi\)
−0.974984 + 0.222276i \(0.928651\pi\)
\(558\) −3.18930 −0.135014
\(559\) 0 0
\(560\) 8.21520 0.347155
\(561\) 15.1850i 0.641111i
\(562\) −7.28489 −0.307294
\(563\) −30.9474 −1.30428 −0.652138 0.758100i \(-0.726128\pi\)
−0.652138 + 0.758100i \(0.726128\pi\)
\(564\) − 10.5288i − 0.443344i
\(565\) 21.2086i 0.892252i
\(566\) 1.70078i 0.0714889i
\(567\) 31.4871i 1.32234i
\(568\) 4.76095 0.199765
\(569\) 36.9089 1.54730 0.773651 0.633612i \(-0.218428\pi\)
0.773651 + 0.633612i \(0.218428\pi\)
\(570\) − 5.54892i − 0.232419i
\(571\) 1.77093 0.0741113 0.0370556 0.999313i \(-0.488202\pi\)
0.0370556 + 0.999313i \(0.488202\pi\)
\(572\) 0 0
\(573\) 9.47383 0.395775
\(574\) 2.12339i 0.0886286i
\(575\) −0.0176328 −0.000735337 0
\(576\) 54.2026 2.25844
\(577\) − 9.83999i − 0.409644i −0.978799 0.204822i \(-0.934338\pi\)
0.978799 0.204822i \(-0.0656616\pi\)
\(578\) − 3.49522i − 0.145382i
\(579\) − 33.3903i − 1.38765i
\(580\) 2.63699i 0.109495i
\(581\) 4.64055 0.192523
\(582\) −11.8054 −0.489351
\(583\) − 27.8939i − 1.15525i
\(584\) −3.61896 −0.149753
\(585\) 0 0
\(586\) −3.13112 −0.129345
\(587\) − 15.1383i − 0.624826i −0.949946 0.312413i \(-0.898863\pi\)
0.949946 0.312413i \(-0.101137\pi\)
\(588\) 6.46667 0.266681
\(589\) −5.54892 −0.228639
\(590\) − 4.55453i − 0.187507i
\(591\) 84.4420i 3.47348i
\(592\) − 5.71994i − 0.235088i
\(593\) − 9.17148i − 0.376628i −0.982109 0.188314i \(-0.939698\pi\)
0.982109 0.188314i \(-0.0603022\pi\)
\(594\) −12.8635 −0.527795
\(595\) 3.06986 0.125852
\(596\) − 31.0079i − 1.27013i
\(597\) −41.3506 −1.69237
\(598\) 0 0
\(599\) −18.5811 −0.759202 −0.379601 0.925150i \(-0.623939\pi\)
−0.379601 + 0.925150i \(0.623939\pi\)
\(600\) − 0.0637892i − 0.00260418i
\(601\) −13.4036 −0.546744 −0.273372 0.961908i \(-0.588139\pi\)
−0.273372 + 0.961908i \(0.588139\pi\)
\(602\) 0.284889 0.0116112
\(603\) − 81.8665i − 3.33386i
\(604\) 27.1527i 1.10483i
\(605\) 0.0833731i 0.00338960i
\(606\) − 6.11101i − 0.248243i
\(607\) 12.6362 0.512889 0.256445 0.966559i \(-0.417449\pi\)
0.256445 + 0.966559i \(0.417449\pi\)
\(608\) −8.66390 −0.351368
\(609\) 2.01708i 0.0817361i
\(610\) 2.82217 0.114266
\(611\) 0 0
\(612\) 21.5234 0.870032
\(613\) 25.7079i 1.03833i 0.854673 + 0.519167i \(0.173758\pi\)
−0.854673 + 0.519167i \(0.826242\pi\)
\(614\) −0.764900 −0.0308688
\(615\) 68.0366 2.74350
\(616\) 3.03341i 0.122220i
\(617\) − 6.59620i − 0.265553i −0.991146 0.132777i \(-0.957611\pi\)
0.991146 0.132777i \(-0.0423893\pi\)
\(618\) 0.534070i 0.0214834i
\(619\) 21.0124i 0.844559i 0.906466 + 0.422280i \(0.138770\pi\)
−0.906466 + 0.422280i \(0.861230\pi\)
\(620\) 7.44921 0.299168
\(621\) 14.0327 0.563112
\(622\) 7.93712i 0.318250i
\(623\) 9.12826 0.365716
\(624\) 0 0
\(625\) −24.8944 −0.995777
\(626\) 1.66910i 0.0667108i
\(627\) −35.7093 −1.42609
\(628\) −25.2548 −1.00778
\(629\) − 2.13743i − 0.0852250i
\(630\) 4.14929i 0.165311i
\(631\) − 26.0210i − 1.03588i −0.855417 0.517940i \(-0.826699\pi\)
0.855417 0.517940i \(-0.173301\pi\)
\(632\) 5.88970i 0.234280i
\(633\) 81.0373 3.22095
\(634\) 1.86023 0.0738793
\(635\) 41.1316i 1.63226i
\(636\) −54.2946 −2.15292
\(637\) 0 0
\(638\) −0.466673 −0.0184758
\(639\) − 41.9091i − 1.65790i
\(640\) 15.4323 0.610015
\(641\) −18.5339 −0.732044 −0.366022 0.930606i \(-0.619281\pi\)
−0.366022 + 0.930606i \(0.619281\pi\)
\(642\) − 6.87661i − 0.271398i
\(643\) − 14.4466i − 0.569717i −0.958570 0.284858i \(-0.908053\pi\)
0.958570 0.284858i \(-0.0919466\pi\)
\(644\) − 1.63212i − 0.0643146i
\(645\) − 9.12826i − 0.359425i
\(646\) −1.02981 −0.0405172
\(647\) −29.2875 −1.15141 −0.575706 0.817657i \(-0.695273\pi\)
−0.575706 + 0.817657i \(0.695273\pi\)
\(648\) 28.7496i 1.12939i
\(649\) −29.3100 −1.15052
\(650\) 0 0
\(651\) 5.69803 0.223324
\(652\) − 35.8315i − 1.40327i
\(653\) −30.9615 −1.21162 −0.605808 0.795611i \(-0.707150\pi\)
−0.605808 + 0.795611i \(0.707150\pi\)
\(654\) −1.74687 −0.0683079
\(655\) 3.90121i 0.152433i
\(656\) − 33.7900i − 1.31928i
\(657\) 31.8565i 1.24284i
\(658\) − 0.376695i − 0.0146851i
\(659\) −37.0828 −1.44454 −0.722270 0.691611i \(-0.756901\pi\)
−0.722270 + 0.691611i \(0.756901\pi\)
\(660\) 47.9384 1.86600
\(661\) 20.4018i 0.793540i 0.917918 + 0.396770i \(0.129869\pi\)
−0.917918 + 0.396770i \(0.870131\pi\)
\(662\) −0.206764 −0.00803611
\(663\) 0 0
\(664\) 4.23710 0.164431
\(665\) 7.21915i 0.279947i
\(666\) 2.88899 0.111946
\(667\) 0.509090 0.0197120
\(668\) 36.0085i 1.39321i
\(669\) − 75.2469i − 2.90921i
\(670\) − 5.25838i − 0.203149i
\(671\) − 18.1617i − 0.701123i
\(672\) 8.89672 0.343199
\(673\) −14.5110 −0.559359 −0.279679 0.960093i \(-0.590228\pi\)
−0.279679 + 0.960093i \(0.590228\pi\)
\(674\) − 3.48700i − 0.134314i
\(675\) −0.351926 −0.0135457
\(676\) 0 0
\(677\) −3.51476 −0.135083 −0.0675417 0.997716i \(-0.521516\pi\)
−0.0675417 + 0.997716i \(0.521516\pi\)
\(678\) 7.30575i 0.280575i
\(679\) 15.3589 0.589420
\(680\) 2.80297 0.107489
\(681\) − 4.26930i − 0.163600i
\(682\) 1.31830i 0.0504804i
\(683\) 27.0753i 1.03601i 0.855379 + 0.518003i \(0.173324\pi\)
−0.855379 + 0.518003i \(0.826676\pi\)
\(684\) 50.6149i 1.93531i
\(685\) 40.1717 1.53488
\(686\) 0.231361 0.00883340
\(687\) − 15.4302i − 0.588700i
\(688\) −4.53350 −0.172838
\(689\) 0 0
\(690\) 1.43812 0.0547485
\(691\) 29.7404i 1.13138i 0.824618 + 0.565690i \(0.191390\pi\)
−0.824618 + 0.565690i \(0.808610\pi\)
\(692\) 33.4757 1.27255
\(693\) 26.7022 1.01433
\(694\) − 3.79810i − 0.144174i
\(695\) − 31.0559i − 1.17802i
\(696\) 1.84171i 0.0698099i
\(697\) − 12.6267i − 0.478270i
\(698\) 7.95237 0.301002
\(699\) 39.4940 1.49380
\(700\) 0.0409321i 0.00154709i
\(701\) −18.2888 −0.690760 −0.345380 0.938463i \(-0.612250\pi\)
−0.345380 + 0.938463i \(0.612250\pi\)
\(702\) 0 0
\(703\) 5.02642 0.189575
\(704\) − 22.4047i − 0.844409i
\(705\) −12.0699 −0.454577
\(706\) 5.53329 0.208248
\(707\) 7.95042i 0.299006i
\(708\) 57.0512i 2.14411i
\(709\) 28.6804i 1.07711i 0.842589 + 0.538557i \(0.181030\pi\)
−0.842589 + 0.538557i \(0.818970\pi\)
\(710\) − 2.69187i − 0.101024i
\(711\) 51.8451 1.94434
\(712\) 8.33464 0.312354
\(713\) − 1.43812i − 0.0538582i
\(714\) 1.05748 0.0395752
\(715\) 0 0
\(716\) −28.2017 −1.05395
\(717\) 13.8798i 0.518351i
\(718\) −1.42952 −0.0533492
\(719\) −25.4762 −0.950103 −0.475052 0.879958i \(-0.657571\pi\)
−0.475052 + 0.879958i \(0.657571\pi\)
\(720\) − 66.0285i − 2.46074i
\(721\) − 0.694825i − 0.0258766i
\(722\) 1.97415i 0.0734701i
\(723\) 13.4000i 0.498352i
\(724\) −13.3428 −0.495879
\(725\) −0.0127675 −0.000474173 0
\(726\) 0.0287196i 0.00106588i
\(727\) 9.02572 0.334746 0.167373 0.985894i \(-0.446472\pi\)
0.167373 + 0.985894i \(0.446472\pi\)
\(728\) 0 0
\(729\) 86.2759 3.19540
\(730\) 2.04618i 0.0757325i
\(731\) −1.69408 −0.0626579
\(732\) −35.3512 −1.30662
\(733\) − 7.57069i − 0.279630i −0.990178 0.139815i \(-0.955349\pi\)
0.990178 0.139815i \(-0.0446508\pi\)
\(734\) 4.59389i 0.169564i
\(735\) − 7.41314i − 0.273438i
\(736\) − 2.24544i − 0.0827681i
\(737\) −33.8396 −1.24650
\(738\) 17.0665 0.628225
\(739\) 6.37296i 0.234433i 0.993106 + 0.117216i \(0.0373971\pi\)
−0.993106 + 0.117216i \(0.962603\pi\)
\(740\) −6.74778 −0.248053
\(741\) 0 0
\(742\) −1.94252 −0.0713122
\(743\) − 22.8297i − 0.837539i −0.908092 0.418770i \(-0.862461\pi\)
0.908092 0.418770i \(-0.137539\pi\)
\(744\) 5.20264 0.190738
\(745\) −35.5463 −1.30231
\(746\) 6.98596i 0.255774i
\(747\) − 37.2978i − 1.36466i
\(748\) − 8.89672i − 0.325296i
\(749\) 8.94647i 0.326897i
\(750\) −8.61162 −0.314452
\(751\) −39.3695 −1.43661 −0.718307 0.695726i \(-0.755083\pi\)
−0.718307 + 0.695726i \(0.755083\pi\)
\(752\) 5.99443i 0.218594i
\(753\) 92.5863 3.37403
\(754\) 0 0
\(755\) 31.1268 1.13282
\(756\) − 32.5750i − 1.18474i
\(757\) 8.72714 0.317193 0.158597 0.987343i \(-0.449303\pi\)
0.158597 + 0.987343i \(0.449303\pi\)
\(758\) 1.00004 0.0363231
\(759\) − 9.25486i − 0.335930i
\(760\) 6.59151i 0.239099i
\(761\) 22.8391i 0.827915i 0.910296 + 0.413958i \(0.135854\pi\)
−0.910296 + 0.413958i \(0.864146\pi\)
\(762\) 14.1687i 0.513276i
\(763\) 2.27268 0.0822764
\(764\) −5.55062 −0.200814
\(765\) − 24.6736i − 0.892076i
\(766\) 4.01368 0.145020
\(767\) 0 0
\(768\) −39.4934 −1.42510
\(769\) 34.8349i 1.25618i 0.778141 + 0.628089i \(0.216163\pi\)
−0.778141 + 0.628089i \(0.783837\pi\)
\(770\) 1.71511 0.0618083
\(771\) −23.7230 −0.854363
\(772\) 19.5630i 0.704088i
\(773\) − 33.2743i − 1.19679i −0.801200 0.598397i \(-0.795804\pi\)
0.801200 0.598397i \(-0.204196\pi\)
\(774\) − 2.28975i − 0.0823035i
\(775\) 0.0360668i 0.00129556i
\(776\) 14.0236 0.503416
\(777\) −5.16150 −0.185168
\(778\) 5.86440i 0.210249i
\(779\) 29.6932 1.06387
\(780\) 0 0
\(781\) −17.3232 −0.619872
\(782\) − 0.266897i − 0.00954421i
\(783\) 10.1608 0.363116
\(784\) −3.68170 −0.131489
\(785\) 28.9512i 1.03331i
\(786\) 1.34385i 0.0479336i
\(787\) 27.8157i 0.991524i 0.868458 + 0.495762i \(0.165111\pi\)
−0.868458 + 0.495762i \(0.834889\pi\)
\(788\) − 49.4737i − 1.76243i
\(789\) −70.8278 −2.52154
\(790\) 3.33007 0.118479
\(791\) − 9.50478i − 0.337951i
\(792\) 24.3806 0.866329
\(793\) 0 0
\(794\) −6.26407 −0.222304
\(795\) 62.2413i 2.20747i
\(796\) 24.2269 0.858699
\(797\) −35.8685 −1.27053 −0.635264 0.772295i \(-0.719109\pi\)
−0.635264 + 0.772295i \(0.719109\pi\)
\(798\) 2.48679i 0.0880313i
\(799\) 2.24001i 0.0792457i
\(800\) 0.0563136i 0.00199099i
\(801\) − 73.3671i − 2.59230i
\(802\) −6.77559 −0.239254
\(803\) 13.1679 0.464686
\(804\) 65.8678i 2.32298i
\(805\) −1.87100 −0.0659441
\(806\) 0 0
\(807\) −9.26495 −0.326142
\(808\) 7.25921i 0.255378i
\(809\) −17.8245 −0.626675 −0.313337 0.949642i \(-0.601447\pi\)
−0.313337 + 0.949642i \(0.601447\pi\)
\(810\) 16.2552 0.571150
\(811\) 25.2152i 0.885425i 0.896664 + 0.442713i \(0.145984\pi\)
−0.896664 + 0.442713i \(0.854016\pi\)
\(812\) − 1.18178i − 0.0414725i
\(813\) − 51.4110i − 1.80306i
\(814\) − 1.19417i − 0.0418556i
\(815\) −41.0759 −1.43883
\(816\) −16.8279 −0.589095
\(817\) − 3.98384i − 0.139377i
\(818\) −5.40719 −0.189058
\(819\) 0 0
\(820\) −39.8619 −1.39204
\(821\) − 10.4559i − 0.364915i −0.983214 0.182457i \(-0.941595\pi\)
0.983214 0.182457i \(-0.0584052\pi\)
\(822\) 13.8380 0.482655
\(823\) 33.2405 1.15869 0.579346 0.815082i \(-0.303308\pi\)
0.579346 + 0.815082i \(0.303308\pi\)
\(824\) − 0.634416i − 0.0221009i
\(825\) 0.232103i 0.00808080i
\(826\) 2.04114i 0.0710205i
\(827\) 37.9927i 1.32113i 0.750767 + 0.660567i \(0.229684\pi\)
−0.750767 + 0.660567i \(0.770316\pi\)
\(828\) −13.1180 −0.455880
\(829\) −16.6944 −0.579821 −0.289911 0.957054i \(-0.593626\pi\)
−0.289911 + 0.957054i \(0.593626\pi\)
\(830\) − 2.39568i − 0.0831554i
\(831\) 18.3706 0.637268
\(832\) 0 0
\(833\) −1.37578 −0.0476680
\(834\) − 10.6979i − 0.370437i
\(835\) 41.2787 1.42851
\(836\) 20.9217 0.723592
\(837\) − 28.7031i − 0.992123i
\(838\) 3.37935i 0.116738i
\(839\) 46.6411i 1.61023i 0.593119 + 0.805115i \(0.297896\pi\)
−0.593119 + 0.805115i \(0.702104\pi\)
\(840\) − 6.76864i − 0.233540i
\(841\) −28.6314 −0.987289
\(842\) −2.37673 −0.0819077
\(843\) − 104.608i − 3.60290i
\(844\) −47.4789 −1.63429
\(845\) 0 0
\(846\) −3.02763 −0.104092
\(847\) − 0.0373642i − 0.00128385i
\(848\) 30.9118 1.06152
\(849\) −24.4225 −0.838178
\(850\) 0.00669352i 0 0.000229586i
\(851\) 1.30271i 0.0446563i
\(852\) 33.7191i 1.15520i
\(853\) − 39.5640i − 1.35464i −0.735686 0.677322i \(-0.763140\pi\)
0.735686 0.677322i \(-0.236860\pi\)
\(854\) −1.26477 −0.0432797
\(855\) 58.0229 1.98434
\(856\) 8.16866i 0.279199i
\(857\) 19.5613 0.668201 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(858\) 0 0
\(859\) −10.1632 −0.346762 −0.173381 0.984855i \(-0.555469\pi\)
−0.173381 + 0.984855i \(0.555469\pi\)
\(860\) 5.34815i 0.182370i
\(861\) −30.4911 −1.03913
\(862\) 2.89241 0.0985160
\(863\) 26.8903i 0.915356i 0.889118 + 0.457678i \(0.151319\pi\)
−0.889118 + 0.457678i \(0.848681\pi\)
\(864\) − 44.8160i − 1.52467i
\(865\) − 38.3752i − 1.30480i
\(866\) 2.53276i 0.0860666i
\(867\) 50.1900 1.70454
\(868\) −3.33842 −0.113313
\(869\) − 21.4302i − 0.726971i
\(870\) 1.04132 0.0353039
\(871\) 0 0
\(872\) 2.07509 0.0702713
\(873\) − 123.445i − 4.17798i
\(874\) 0.627640 0.0212302
\(875\) 11.2037 0.378755
\(876\) − 25.6310i − 0.865991i
\(877\) − 1.70160i − 0.0574590i −0.999587 0.0287295i \(-0.990854\pi\)
0.999587 0.0287295i \(-0.00914614\pi\)
\(878\) 4.14551i 0.139904i
\(879\) − 44.9617i − 1.51652i
\(880\) −27.2930 −0.920046
\(881\) 11.3090 0.381008 0.190504 0.981686i \(-0.438988\pi\)
0.190504 + 0.981686i \(0.438988\pi\)
\(882\) − 1.85953i − 0.0626137i
\(883\) 46.9068 1.57854 0.789270 0.614047i \(-0.210459\pi\)
0.789270 + 0.614047i \(0.210459\pi\)
\(884\) 0 0
\(885\) 65.4013 2.19844
\(886\) 6.41318i 0.215455i
\(887\) −2.44692 −0.0821594 −0.0410797 0.999156i \(-0.513080\pi\)
−0.0410797 + 0.999156i \(0.513080\pi\)
\(888\) −4.71275 −0.158150
\(889\) − 18.4334i − 0.618237i
\(890\) − 4.71246i − 0.157962i
\(891\) − 104.608i − 3.50451i
\(892\) 44.0864i 1.47612i
\(893\) −5.26764 −0.176275
\(894\) −12.2447 −0.409523
\(895\) 32.3293i 1.08065i
\(896\) −6.91610 −0.231051
\(897\) 0 0
\(898\) −0.0384012 −0.00128146
\(899\) − 1.04132i − 0.0347298i
\(900\) 0.328986 0.0109662
\(901\) 11.5511 0.384825
\(902\) − 7.05444i − 0.234887i
\(903\) 4.09089i 0.136136i
\(904\) − 8.67842i − 0.288640i
\(905\) 15.2956i 0.508443i
\(906\) 10.7223 0.356224
\(907\) 41.4165 1.37521 0.687607 0.726083i \(-0.258661\pi\)
0.687607 + 0.726083i \(0.258661\pi\)
\(908\) 2.50133i 0.0830097i
\(909\) 63.9004 2.11944
\(910\) 0 0
\(911\) −11.9951 −0.397416 −0.198708 0.980059i \(-0.563675\pi\)
−0.198708 + 0.980059i \(0.563675\pi\)
\(912\) − 39.5728i − 1.31039i
\(913\) −15.4171 −0.510231
\(914\) −7.34233 −0.242863
\(915\) 40.5252i 1.33972i
\(916\) 9.04040i 0.298703i
\(917\) − 1.74835i − 0.0577357i
\(918\) − 5.32691i − 0.175814i
\(919\) 44.9416 1.48249 0.741243 0.671237i \(-0.234237\pi\)
0.741243 + 0.671237i \(0.234237\pi\)
\(920\) −1.70833 −0.0563221
\(921\) − 10.9837i − 0.361924i
\(922\) −6.72463 −0.221464
\(923\) 0 0
\(924\) −21.4839 −0.706769
\(925\) − 0.0326707i − 0.00107421i
\(926\) 1.46198 0.0480436
\(927\) −5.58456 −0.183421
\(928\) − 1.62588i − 0.0533720i
\(929\) 28.2595i 0.927164i 0.886054 + 0.463582i \(0.153436\pi\)
−0.886054 + 0.463582i \(0.846564\pi\)
\(930\) − 2.94161i − 0.0964591i
\(931\) − 3.23531i − 0.106033i
\(932\) −23.1391 −0.757947
\(933\) −113.974 −3.73134
\(934\) 8.01455i 0.262244i
\(935\) −10.1989 −0.333538
\(936\) 0 0
\(937\) 32.4601 1.06042 0.530212 0.847865i \(-0.322112\pi\)
0.530212 + 0.847865i \(0.322112\pi\)
\(938\) 2.35658i 0.0769451i
\(939\) −23.9677 −0.782155
\(940\) 7.07160 0.230650
\(941\) − 12.6051i − 0.410913i −0.978666 0.205457i \(-0.934132\pi\)
0.978666 0.205457i \(-0.0658679\pi\)
\(942\) 9.97283i 0.324932i
\(943\) 7.69563i 0.250604i
\(944\) − 32.4812i − 1.05717i
\(945\) −37.3427 −1.21476
\(946\) −0.946472 −0.0307725
\(947\) 13.2802i 0.431548i 0.976443 + 0.215774i \(0.0692275\pi\)
−0.976443 + 0.215774i \(0.930772\pi\)
\(948\) −41.7133 −1.35479
\(949\) 0 0
\(950\) −0.0157406 −0.000510693 0
\(951\) 26.7122i 0.866204i
\(952\) −1.25617 −0.0407127
\(953\) 58.4319 1.89279 0.946397 0.323006i \(-0.104693\pi\)
0.946397 + 0.323006i \(0.104693\pi\)
\(954\) 15.6127i 0.505481i
\(955\) 6.36301i 0.205902i
\(956\) − 8.13204i − 0.263009i
\(957\) − 6.70124i − 0.216620i
\(958\) −1.65245 −0.0533882
\(959\) −18.0032 −0.581354
\(960\) 49.9930i 1.61352i
\(961\) 28.0584 0.905109
\(962\) 0 0
\(963\) 71.9061 2.31714
\(964\) − 7.85093i − 0.252861i
\(965\) 22.4263 0.721927
\(966\) −0.644506 −0.0207366
\(967\) 33.2182i 1.06823i 0.845413 + 0.534113i \(0.179354\pi\)
−0.845413 + 0.534113i \(0.820646\pi\)
\(968\) − 0.0341157i − 0.00109652i
\(969\) − 14.7876i − 0.475047i
\(970\) − 7.92901i − 0.254585i
\(971\) 16.7778 0.538425 0.269213 0.963081i \(-0.413237\pi\)
0.269213 + 0.963081i \(0.413237\pi\)
\(972\) −105.892 −3.39649
\(973\) 13.9179i 0.446188i
\(974\) 4.28023 0.137148
\(975\) 0 0
\(976\) 20.1266 0.644238
\(977\) − 50.0422i − 1.60099i −0.599338 0.800496i \(-0.704569\pi\)
0.599338 0.800496i \(-0.295431\pi\)
\(978\) −14.1495 −0.452450
\(979\) −30.3264 −0.969235
\(980\) 4.34328i 0.138741i
\(981\) − 18.2663i − 0.583199i
\(982\) 3.53475i 0.112798i
\(983\) − 16.6741i − 0.531822i −0.963998 0.265911i \(-0.914327\pi\)
0.963998 0.265911i \(-0.0856728\pi\)
\(984\) −27.8402 −0.887512
\(985\) −56.7147 −1.80708
\(986\) − 0.193254i − 0.00615447i
\(987\) 5.40919 0.172177
\(988\) 0 0
\(989\) 1.03250 0.0328316
\(990\) − 13.7850i − 0.438115i
\(991\) 20.3285 0.645756 0.322878 0.946441i \(-0.395350\pi\)
0.322878 + 0.946441i \(0.395350\pi\)
\(992\) −4.59293 −0.145826
\(993\) − 2.96905i − 0.0942201i
\(994\) 1.20638i 0.0382641i
\(995\) − 27.7728i − 0.880456i
\(996\) 30.0089i 0.950870i
\(997\) 6.27646 0.198777 0.0993887 0.995049i \(-0.468311\pi\)
0.0993887 + 0.995049i \(0.468311\pi\)
\(998\) −2.88699 −0.0913862
\(999\) 26.0003i 0.822614i
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 1183.2.c.g.337.5 8
13.2 odd 12 91.2.f.c.22.2 8
13.5 odd 4 1183.2.a.k.1.3 4
13.6 odd 12 91.2.f.c.29.2 yes 8
13.8 odd 4 1183.2.a.l.1.2 4
13.12 even 2 inner 1183.2.c.g.337.4 8
39.2 even 12 819.2.o.h.568.3 8
39.32 even 12 819.2.o.h.757.3 8
52.15 even 12 1456.2.s.q.113.1 8
52.19 even 12 1456.2.s.q.1121.1 8
91.2 odd 12 637.2.h.h.165.3 8
91.6 even 12 637.2.f.i.393.2 8
91.19 even 12 637.2.g.j.263.2 8
91.32 odd 12 637.2.h.h.471.3 8
91.34 even 4 8281.2.a.bt.1.2 4
91.41 even 12 637.2.f.i.295.2 8
91.45 even 12 637.2.h.i.471.3 8
91.54 even 12 637.2.h.i.165.3 8
91.58 odd 12 637.2.g.k.263.2 8
91.67 odd 12 637.2.g.k.373.2 8
91.80 even 12 637.2.g.j.373.2 8
91.83 even 4 8281.2.a.bp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 8 13.2 odd 12
91.2.f.c.29.2 yes 8 13.6 odd 12
637.2.f.i.295.2 8 91.41 even 12
637.2.f.i.393.2 8 91.6 even 12
637.2.g.j.263.2 8 91.19 even 12
637.2.g.j.373.2 8 91.80 even 12
637.2.g.k.263.2 8 91.58 odd 12
637.2.g.k.373.2 8 91.67 odd 12
637.2.h.h.165.3 8 91.2 odd 12
637.2.h.h.471.3 8 91.32 odd 12
637.2.h.i.165.3 8 91.54 even 12
637.2.h.i.471.3 8 91.45 even 12
819.2.o.h.568.3 8 39.2 even 12
819.2.o.h.757.3 8 39.32 even 12
1183.2.a.k.1.3 4 13.5 odd 4
1183.2.a.l.1.2 4 13.8 odd 4
1183.2.c.g.337.4 8 13.12 even 2 inner
1183.2.c.g.337.5 8 1.1 even 1 trivial
1456.2.s.q.113.1 8 52.15 even 12
1456.2.s.q.1121.1 8 52.19 even 12
8281.2.a.bp.1.3 4 91.83 even 4
8281.2.a.bt.1.2 4 91.34 even 4