Properties

Label 363.2.a.e.1.1
Level $363$
Weight $2$
Character 363.1
Self dual yes
Analytic conductor $2.899$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.89856959337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -0.381966 q^{5} -1.61803 q^{6} -3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -0.381966 q^{5} -1.61803 q^{6} -3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{10} +0.618034 q^{12} -6.23607 q^{13} +4.85410 q^{14} -0.381966 q^{15} -4.85410 q^{16} +0.618034 q^{17} -1.61803 q^{18} +0.854102 q^{19} -0.236068 q^{20} -3.00000 q^{21} -5.47214 q^{23} +2.23607 q^{24} -4.85410 q^{25} +10.0902 q^{26} +1.00000 q^{27} -1.85410 q^{28} +4.47214 q^{29} +0.618034 q^{30} -3.85410 q^{31} +3.38197 q^{32} -1.00000 q^{34} +1.14590 q^{35} +0.618034 q^{36} -4.23607 q^{37} -1.38197 q^{38} -6.23607 q^{39} -0.854102 q^{40} -5.94427 q^{41} +4.85410 q^{42} -1.76393 q^{43} -0.381966 q^{45} +8.85410 q^{46} -0.618034 q^{47} -4.85410 q^{48} +2.00000 q^{49} +7.85410 q^{50} +0.618034 q^{51} -3.85410 q^{52} -7.38197 q^{53} -1.61803 q^{54} -6.70820 q^{56} +0.854102 q^{57} -7.23607 q^{58} -5.32624 q^{59} -0.236068 q^{60} -1.14590 q^{61} +6.23607 q^{62} -3.00000 q^{63} +4.23607 q^{64} +2.38197 q^{65} +10.5623 q^{67} +0.381966 q^{68} -5.47214 q^{69} -1.85410 q^{70} +14.5623 q^{71} +2.23607 q^{72} -1.23607 q^{73} +6.85410 q^{74} -4.85410 q^{75} +0.527864 q^{76} +10.0902 q^{78} -0.527864 q^{79} +1.85410 q^{80} +1.00000 q^{81} +9.61803 q^{82} +12.7082 q^{83} -1.85410 q^{84} -0.236068 q^{85} +2.85410 q^{86} +4.47214 q^{87} +9.47214 q^{89} +0.618034 q^{90} +18.7082 q^{91} -3.38197 q^{92} -3.85410 q^{93} +1.00000 q^{94} -0.326238 q^{95} +3.38197 q^{96} +15.0344 q^{97} -3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} - q^{10} - q^{12} - 8 q^{13} + 3 q^{14} - 3 q^{15} - 3 q^{16} - q^{17} - q^{18} - 5 q^{19} + 4 q^{20} - 6 q^{21} - 2 q^{23} - 3 q^{25} + 9 q^{26} + 2 q^{27} + 3 q^{28} - q^{30} - q^{31} + 9 q^{32} - 2 q^{34} + 9 q^{35} - q^{36} - 4 q^{37} - 5 q^{38} - 8 q^{39} + 5 q^{40} + 6 q^{41} + 3 q^{42} - 8 q^{43} - 3 q^{45} + 11 q^{46} + q^{47} - 3 q^{48} + 4 q^{49} + 9 q^{50} - q^{51} - q^{52} - 17 q^{53} - q^{54} - 5 q^{57} - 10 q^{58} + 5 q^{59} + 4 q^{60} - 9 q^{61} + 8 q^{62} - 6 q^{63} + 4 q^{64} + 7 q^{65} + q^{67} + 3 q^{68} - 2 q^{69} + 3 q^{70} + 9 q^{71} + 2 q^{73} + 7 q^{74} - 3 q^{75} + 10 q^{76} + 9 q^{78} - 10 q^{79} - 3 q^{80} + 2 q^{81} + 17 q^{82} + 12 q^{83} + 3 q^{84} + 4 q^{85} - q^{86} + 10 q^{89} - q^{90} + 24 q^{91} - 9 q^{92} - q^{93} + 2 q^{94} + 15 q^{95} + 9 q^{96} + q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) −1.61803 −0.660560
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0.618034 0.195440
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) −6.23607 −1.72957 −0.864787 0.502139i \(-0.832547\pi\)
−0.864787 + 0.502139i \(0.832547\pi\)
\(14\) 4.85410 1.29731
\(15\) −0.381966 −0.0986232
\(16\) −4.85410 −1.21353
\(17\) 0.618034 0.149895 0.0749476 0.997187i \(-0.476121\pi\)
0.0749476 + 0.997187i \(0.476121\pi\)
\(18\) −1.61803 −0.381374
\(19\) 0.854102 0.195944 0.0979722 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) −0.236068 −0.0527864
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −5.47214 −1.14102 −0.570510 0.821291i \(-0.693254\pi\)
−0.570510 + 0.821291i \(0.693254\pi\)
\(24\) 2.23607 0.456435
\(25\) −4.85410 −0.970820
\(26\) 10.0902 1.97885
\(27\) 1.00000 0.192450
\(28\) −1.85410 −0.350392
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0.618034 0.112837
\(31\) −3.85410 −0.692217 −0.346109 0.938194i \(-0.612497\pi\)
−0.346109 + 0.938194i \(0.612497\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 1.14590 0.193692
\(36\) 0.618034 0.103006
\(37\) −4.23607 −0.696405 −0.348203 0.937419i \(-0.613208\pi\)
−0.348203 + 0.937419i \(0.613208\pi\)
\(38\) −1.38197 −0.224184
\(39\) −6.23607 −0.998570
\(40\) −0.854102 −0.135045
\(41\) −5.94427 −0.928339 −0.464170 0.885746i \(-0.653647\pi\)
−0.464170 + 0.885746i \(0.653647\pi\)
\(42\) 4.85410 0.749004
\(43\) −1.76393 −0.268997 −0.134499 0.990914i \(-0.542942\pi\)
−0.134499 + 0.990914i \(0.542942\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) 8.85410 1.30547
\(47\) −0.618034 −0.0901495 −0.0450748 0.998984i \(-0.514353\pi\)
−0.0450748 + 0.998984i \(0.514353\pi\)
\(48\) −4.85410 −0.700629
\(49\) 2.00000 0.285714
\(50\) 7.85410 1.11074
\(51\) 0.618034 0.0865421
\(52\) −3.85410 −0.534468
\(53\) −7.38197 −1.01399 −0.506996 0.861949i \(-0.669244\pi\)
−0.506996 + 0.861949i \(0.669244\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) −6.70820 −0.896421
\(57\) 0.854102 0.113129
\(58\) −7.23607 −0.950142
\(59\) −5.32624 −0.693417 −0.346709 0.937973i \(-0.612701\pi\)
−0.346709 + 0.937973i \(0.612701\pi\)
\(60\) −0.236068 −0.0304762
\(61\) −1.14590 −0.146717 −0.0733586 0.997306i \(-0.523372\pi\)
−0.0733586 + 0.997306i \(0.523372\pi\)
\(62\) 6.23607 0.791981
\(63\) −3.00000 −0.377964
\(64\) 4.23607 0.529508
\(65\) 2.38197 0.295447
\(66\) 0 0
\(67\) 10.5623 1.29039 0.645196 0.764017i \(-0.276776\pi\)
0.645196 + 0.764017i \(0.276776\pi\)
\(68\) 0.381966 0.0463202
\(69\) −5.47214 −0.658768
\(70\) −1.85410 −0.221608
\(71\) 14.5623 1.72823 0.864114 0.503296i \(-0.167880\pi\)
0.864114 + 0.503296i \(0.167880\pi\)
\(72\) 2.23607 0.263523
\(73\) −1.23607 −0.144671 −0.0723354 0.997380i \(-0.523045\pi\)
−0.0723354 + 0.997380i \(0.523045\pi\)
\(74\) 6.85410 0.796773
\(75\) −4.85410 −0.560503
\(76\) 0.527864 0.0605502
\(77\) 0 0
\(78\) 10.0902 1.14249
\(79\) −0.527864 −0.0593893 −0.0296947 0.999559i \(-0.509453\pi\)
−0.0296947 + 0.999559i \(0.509453\pi\)
\(80\) 1.85410 0.207295
\(81\) 1.00000 0.111111
\(82\) 9.61803 1.06213
\(83\) 12.7082 1.39491 0.697453 0.716630i \(-0.254316\pi\)
0.697453 + 0.716630i \(0.254316\pi\)
\(84\) −1.85410 −0.202299
\(85\) −0.236068 −0.0256052
\(86\) 2.85410 0.307766
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) 9.47214 1.00404 0.502022 0.864855i \(-0.332590\pi\)
0.502022 + 0.864855i \(0.332590\pi\)
\(90\) 0.618034 0.0651465
\(91\) 18.7082 1.96115
\(92\) −3.38197 −0.352594
\(93\) −3.85410 −0.399652
\(94\) 1.00000 0.103142
\(95\) −0.326238 −0.0334713
\(96\) 3.38197 0.345170
\(97\) 15.0344 1.52652 0.763258 0.646094i \(-0.223598\pi\)
0.763258 + 0.646094i \(0.223598\pi\)
\(98\) −3.23607 −0.326892
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −13.9443 −1.36735
\(105\) 1.14590 0.111828
\(106\) 11.9443 1.16013
\(107\) −0.236068 −0.0228216 −0.0114108 0.999935i \(-0.503632\pi\)
−0.0114108 + 0.999935i \(0.503632\pi\)
\(108\) 0.618034 0.0594703
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −4.23607 −0.402070
\(112\) 14.5623 1.37601
\(113\) −12.7082 −1.19549 −0.597744 0.801687i \(-0.703936\pi\)
−0.597744 + 0.801687i \(0.703936\pi\)
\(114\) −1.38197 −0.129433
\(115\) 2.09017 0.194909
\(116\) 2.76393 0.256625
\(117\) −6.23607 −0.576525
\(118\) 8.61803 0.793354
\(119\) −1.85410 −0.169965
\(120\) −0.854102 −0.0779685
\(121\) 0 0
\(122\) 1.85410 0.167863
\(123\) −5.94427 −0.535977
\(124\) −2.38197 −0.213907
\(125\) 3.76393 0.336656
\(126\) 4.85410 0.432438
\(127\) −9.70820 −0.861464 −0.430732 0.902480i \(-0.641745\pi\)
−0.430732 + 0.902480i \(0.641745\pi\)
\(128\) −13.6180 −1.20368
\(129\) −1.76393 −0.155306
\(130\) −3.85410 −0.338027
\(131\) 13.8541 1.21044 0.605219 0.796059i \(-0.293085\pi\)
0.605219 + 0.796059i \(0.293085\pi\)
\(132\) 0 0
\(133\) −2.56231 −0.222180
\(134\) −17.0902 −1.47637
\(135\) −0.381966 −0.0328744
\(136\) 1.38197 0.118503
\(137\) −1.47214 −0.125773 −0.0628865 0.998021i \(-0.520031\pi\)
−0.0628865 + 0.998021i \(0.520031\pi\)
\(138\) 8.85410 0.753711
\(139\) −5.85410 −0.496538 −0.248269 0.968691i \(-0.579862\pi\)
−0.248269 + 0.968691i \(0.579862\pi\)
\(140\) 0.708204 0.0598542
\(141\) −0.618034 −0.0520479
\(142\) −23.5623 −1.97730
\(143\) 0 0
\(144\) −4.85410 −0.404508
\(145\) −1.70820 −0.141859
\(146\) 2.00000 0.165521
\(147\) 2.00000 0.164957
\(148\) −2.61803 −0.215201
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 7.85410 0.641285
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 1.90983 0.154908
\(153\) 0.618034 0.0499651
\(154\) 0 0
\(155\) 1.47214 0.118245
\(156\) −3.85410 −0.308575
\(157\) 9.70820 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(158\) 0.854102 0.0679487
\(159\) −7.38197 −0.585428
\(160\) −1.29180 −0.102125
\(161\) 16.4164 1.29379
\(162\) −1.61803 −0.127125
\(163\) −15.2705 −1.19608 −0.598039 0.801467i \(-0.704053\pi\)
−0.598039 + 0.801467i \(0.704053\pi\)
\(164\) −3.67376 −0.286873
\(165\) 0 0
\(166\) −20.5623 −1.59594
\(167\) 19.0344 1.47293 0.736465 0.676476i \(-0.236494\pi\)
0.736465 + 0.676476i \(0.236494\pi\)
\(168\) −6.70820 −0.517549
\(169\) 25.8885 1.99143
\(170\) 0.381966 0.0292955
\(171\) 0.854102 0.0653148
\(172\) −1.09017 −0.0831247
\(173\) −17.6180 −1.33947 −0.669737 0.742598i \(-0.733593\pi\)
−0.669737 + 0.742598i \(0.733593\pi\)
\(174\) −7.23607 −0.548565
\(175\) 14.5623 1.10081
\(176\) 0 0
\(177\) −5.32624 −0.400345
\(178\) −15.3262 −1.14875
\(179\) 2.23607 0.167132 0.0835658 0.996502i \(-0.473369\pi\)
0.0835658 + 0.996502i \(0.473369\pi\)
\(180\) −0.236068 −0.0175955
\(181\) −8.52786 −0.633871 −0.316936 0.948447i \(-0.602654\pi\)
−0.316936 + 0.948447i \(0.602654\pi\)
\(182\) −30.2705 −2.24380
\(183\) −1.14590 −0.0847072
\(184\) −12.2361 −0.902055
\(185\) 1.61803 0.118960
\(186\) 6.23607 0.457251
\(187\) 0 0
\(188\) −0.381966 −0.0278577
\(189\) −3.00000 −0.218218
\(190\) 0.527864 0.0382953
\(191\) 1.47214 0.106520 0.0532600 0.998581i \(-0.483039\pi\)
0.0532600 + 0.998581i \(0.483039\pi\)
\(192\) 4.23607 0.305712
\(193\) −1.56231 −0.112457 −0.0562286 0.998418i \(-0.517908\pi\)
−0.0562286 + 0.998418i \(0.517908\pi\)
\(194\) −24.3262 −1.74652
\(195\) 2.38197 0.170576
\(196\) 1.23607 0.0882906
\(197\) −26.6180 −1.89646 −0.948228 0.317590i \(-0.897127\pi\)
−0.948228 + 0.317590i \(0.897127\pi\)
\(198\) 0 0
\(199\) −3.29180 −0.233349 −0.116675 0.993170i \(-0.537223\pi\)
−0.116675 + 0.993170i \(0.537223\pi\)
\(200\) −10.8541 −0.767501
\(201\) 10.5623 0.745008
\(202\) −4.85410 −0.341533
\(203\) −13.4164 −0.941647
\(204\) 0.381966 0.0267430
\(205\) 2.27051 0.158579
\(206\) 9.70820 0.676403
\(207\) −5.47214 −0.380340
\(208\) 30.2705 2.09888
\(209\) 0 0
\(210\) −1.85410 −0.127945
\(211\) −11.2705 −0.775894 −0.387947 0.921682i \(-0.626816\pi\)
−0.387947 + 0.921682i \(0.626816\pi\)
\(212\) −4.56231 −0.313340
\(213\) 14.5623 0.997793
\(214\) 0.381966 0.0261107
\(215\) 0.673762 0.0459502
\(216\) 2.23607 0.152145
\(217\) 11.5623 0.784900
\(218\) 0 0
\(219\) −1.23607 −0.0835257
\(220\) 0 0
\(221\) −3.85410 −0.259255
\(222\) 6.85410 0.460017
\(223\) −12.7082 −0.851004 −0.425502 0.904957i \(-0.639903\pi\)
−0.425502 + 0.904957i \(0.639903\pi\)
\(224\) −10.1459 −0.677901
\(225\) −4.85410 −0.323607
\(226\) 20.5623 1.36778
\(227\) −10.8885 −0.722698 −0.361349 0.932431i \(-0.617684\pi\)
−0.361349 + 0.932431i \(0.617684\pi\)
\(228\) 0.527864 0.0349587
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −3.38197 −0.223000
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −8.67376 −0.568237 −0.284119 0.958789i \(-0.591701\pi\)
−0.284119 + 0.958789i \(0.591701\pi\)
\(234\) 10.0902 0.659615
\(235\) 0.236068 0.0153994
\(236\) −3.29180 −0.214278
\(237\) −0.527864 −0.0342885
\(238\) 3.00000 0.194461
\(239\) 17.5623 1.13601 0.568006 0.823025i \(-0.307715\pi\)
0.568006 + 0.823025i \(0.307715\pi\)
\(240\) 1.85410 0.119682
\(241\) −17.1246 −1.10309 −0.551547 0.834144i \(-0.685962\pi\)
−0.551547 + 0.834144i \(0.685962\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −0.708204 −0.0453381
\(245\) −0.763932 −0.0488058
\(246\) 9.61803 0.613223
\(247\) −5.32624 −0.338900
\(248\) −8.61803 −0.547246
\(249\) 12.7082 0.805350
\(250\) −6.09017 −0.385176
\(251\) 16.7984 1.06030 0.530152 0.847903i \(-0.322135\pi\)
0.530152 + 0.847903i \(0.322135\pi\)
\(252\) −1.85410 −0.116797
\(253\) 0 0
\(254\) 15.7082 0.985620
\(255\) −0.236068 −0.0147832
\(256\) 13.5623 0.847644
\(257\) −27.3262 −1.70456 −0.852282 0.523083i \(-0.824782\pi\)
−0.852282 + 0.523083i \(0.824782\pi\)
\(258\) 2.85410 0.177689
\(259\) 12.7082 0.789649
\(260\) 1.47214 0.0912980
\(261\) 4.47214 0.276818
\(262\) −22.4164 −1.38489
\(263\) 0.673762 0.0415459 0.0207730 0.999784i \(-0.493387\pi\)
0.0207730 + 0.999784i \(0.493387\pi\)
\(264\) 0 0
\(265\) 2.81966 0.173210
\(266\) 4.14590 0.254201
\(267\) 9.47214 0.579685
\(268\) 6.52786 0.398753
\(269\) 24.4721 1.49209 0.746046 0.665894i \(-0.231950\pi\)
0.746046 + 0.665894i \(0.231950\pi\)
\(270\) 0.618034 0.0376124
\(271\) −6.27051 −0.380906 −0.190453 0.981696i \(-0.560996\pi\)
−0.190453 + 0.981696i \(0.560996\pi\)
\(272\) −3.00000 −0.181902
\(273\) 18.7082 1.13227
\(274\) 2.38197 0.143900
\(275\) 0 0
\(276\) −3.38197 −0.203570
\(277\) −10.4377 −0.627140 −0.313570 0.949565i \(-0.601525\pi\)
−0.313570 + 0.949565i \(0.601525\pi\)
\(278\) 9.47214 0.568101
\(279\) −3.85410 −0.230739
\(280\) 2.56231 0.153127
\(281\) 5.23607 0.312358 0.156179 0.987729i \(-0.450082\pi\)
0.156179 + 0.987729i \(0.450082\pi\)
\(282\) 1.00000 0.0595491
\(283\) 22.1803 1.31848 0.659242 0.751931i \(-0.270877\pi\)
0.659242 + 0.751931i \(0.270877\pi\)
\(284\) 9.00000 0.534052
\(285\) −0.326238 −0.0193247
\(286\) 0 0
\(287\) 17.8328 1.05264
\(288\) 3.38197 0.199284
\(289\) −16.6180 −0.977531
\(290\) 2.76393 0.162304
\(291\) 15.0344 0.881335
\(292\) −0.763932 −0.0447057
\(293\) −17.9443 −1.04832 −0.524158 0.851621i \(-0.675620\pi\)
−0.524158 + 0.851621i \(0.675620\pi\)
\(294\) −3.23607 −0.188731
\(295\) 2.03444 0.118450
\(296\) −9.47214 −0.550557
\(297\) 0 0
\(298\) 24.2705 1.40595
\(299\) 34.1246 1.97348
\(300\) −3.00000 −0.173205
\(301\) 5.29180 0.305014
\(302\) 3.23607 0.186215
\(303\) 3.00000 0.172345
\(304\) −4.14590 −0.237784
\(305\) 0.437694 0.0250623
\(306\) −1.00000 −0.0571662
\(307\) 19.5623 1.11648 0.558240 0.829680i \(-0.311477\pi\)
0.558240 + 0.829680i \(0.311477\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) −2.38197 −0.135287
\(311\) 11.4721 0.650525 0.325263 0.945624i \(-0.394547\pi\)
0.325263 + 0.945624i \(0.394547\pi\)
\(312\) −13.9443 −0.789439
\(313\) −27.8328 −1.57320 −0.786602 0.617461i \(-0.788162\pi\)
−0.786602 + 0.617461i \(0.788162\pi\)
\(314\) −15.7082 −0.886465
\(315\) 1.14590 0.0645640
\(316\) −0.326238 −0.0183523
\(317\) 25.3607 1.42440 0.712199 0.701978i \(-0.247699\pi\)
0.712199 + 0.701978i \(0.247699\pi\)
\(318\) 11.9443 0.669802
\(319\) 0 0
\(320\) −1.61803 −0.0904508
\(321\) −0.236068 −0.0131760
\(322\) −26.5623 −1.48026
\(323\) 0.527864 0.0293711
\(324\) 0.618034 0.0343352
\(325\) 30.2705 1.67911
\(326\) 24.7082 1.36846
\(327\) 0 0
\(328\) −13.2918 −0.733917
\(329\) 1.85410 0.102220
\(330\) 0 0
\(331\) −22.5967 −1.24203 −0.621015 0.783799i \(-0.713279\pi\)
−0.621015 + 0.783799i \(0.713279\pi\)
\(332\) 7.85410 0.431050
\(333\) −4.23607 −0.232135
\(334\) −30.7984 −1.68521
\(335\) −4.03444 −0.220425
\(336\) 14.5623 0.794439
\(337\) 13.7082 0.746733 0.373367 0.927684i \(-0.378203\pi\)
0.373367 + 0.927684i \(0.378203\pi\)
\(338\) −41.8885 −2.27844
\(339\) −12.7082 −0.690215
\(340\) −0.145898 −0.00791243
\(341\) 0 0
\(342\) −1.38197 −0.0747282
\(343\) 15.0000 0.809924
\(344\) −3.94427 −0.212661
\(345\) 2.09017 0.112531
\(346\) 28.5066 1.53252
\(347\) 3.05573 0.164040 0.0820200 0.996631i \(-0.473863\pi\)
0.0820200 + 0.996631i \(0.473863\pi\)
\(348\) 2.76393 0.148162
\(349\) −30.1246 −1.61253 −0.806267 0.591552i \(-0.798515\pi\)
−0.806267 + 0.591552i \(0.798515\pi\)
\(350\) −23.5623 −1.25946
\(351\) −6.23607 −0.332857
\(352\) 0 0
\(353\) −1.52786 −0.0813200 −0.0406600 0.999173i \(-0.512946\pi\)
−0.0406600 + 0.999173i \(0.512946\pi\)
\(354\) 8.61803 0.458043
\(355\) −5.56231 −0.295217
\(356\) 5.85410 0.310267
\(357\) −1.85410 −0.0981295
\(358\) −3.61803 −0.191219
\(359\) 17.2361 0.909685 0.454842 0.890572i \(-0.349696\pi\)
0.454842 + 0.890572i \(0.349696\pi\)
\(360\) −0.854102 −0.0450151
\(361\) −18.2705 −0.961606
\(362\) 13.7984 0.725226
\(363\) 0 0
\(364\) 11.5623 0.606029
\(365\) 0.472136 0.0247127
\(366\) 1.85410 0.0969155
\(367\) −14.5623 −0.760146 −0.380073 0.924956i \(-0.624101\pi\)
−0.380073 + 0.924956i \(0.624101\pi\)
\(368\) 26.5623 1.38466
\(369\) −5.94427 −0.309446
\(370\) −2.61803 −0.136105
\(371\) 22.1459 1.14976
\(372\) −2.38197 −0.123499
\(373\) −22.4164 −1.16068 −0.580339 0.814375i \(-0.697080\pi\)
−0.580339 + 0.814375i \(0.697080\pi\)
\(374\) 0 0
\(375\) 3.76393 0.194369
\(376\) −1.38197 −0.0712695
\(377\) −27.8885 −1.43633
\(378\) 4.85410 0.249668
\(379\) 28.4164 1.45965 0.729826 0.683633i \(-0.239601\pi\)
0.729826 + 0.683633i \(0.239601\pi\)
\(380\) −0.201626 −0.0103432
\(381\) −9.70820 −0.497366
\(382\) −2.38197 −0.121872
\(383\) −8.88854 −0.454183 −0.227092 0.973873i \(-0.572922\pi\)
−0.227092 + 0.973873i \(0.572922\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) 2.52786 0.128665
\(387\) −1.76393 −0.0896657
\(388\) 9.29180 0.471719
\(389\) −9.27051 −0.470034 −0.235017 0.971991i \(-0.575515\pi\)
−0.235017 + 0.971991i \(0.575515\pi\)
\(390\) −3.85410 −0.195160
\(391\) −3.38197 −0.171033
\(392\) 4.47214 0.225877
\(393\) 13.8541 0.698847
\(394\) 43.0689 2.16978
\(395\) 0.201626 0.0101449
\(396\) 0 0
\(397\) −25.2918 −1.26936 −0.634679 0.772776i \(-0.718868\pi\)
−0.634679 + 0.772776i \(0.718868\pi\)
\(398\) 5.32624 0.266980
\(399\) −2.56231 −0.128276
\(400\) 23.5623 1.17812
\(401\) −14.9098 −0.744561 −0.372281 0.928120i \(-0.621424\pi\)
−0.372281 + 0.928120i \(0.621424\pi\)
\(402\) −17.0902 −0.852380
\(403\) 24.0344 1.19724
\(404\) 1.85410 0.0922450
\(405\) −0.381966 −0.0189800
\(406\) 21.7082 1.07736
\(407\) 0 0
\(408\) 1.38197 0.0684175
\(409\) 28.9443 1.43120 0.715601 0.698509i \(-0.246153\pi\)
0.715601 + 0.698509i \(0.246153\pi\)
\(410\) −3.67376 −0.181434
\(411\) −1.47214 −0.0726151
\(412\) −3.70820 −0.182690
\(413\) 15.9787 0.786261
\(414\) 8.85410 0.435155
\(415\) −4.85410 −0.238278
\(416\) −21.0902 −1.03403
\(417\) −5.85410 −0.286677
\(418\) 0 0
\(419\) −21.5066 −1.05067 −0.525333 0.850897i \(-0.676059\pi\)
−0.525333 + 0.850897i \(0.676059\pi\)
\(420\) 0.708204 0.0345568
\(421\) −3.72949 −0.181764 −0.0908821 0.995862i \(-0.528969\pi\)
−0.0908821 + 0.995862i \(0.528969\pi\)
\(422\) 18.2361 0.887718
\(423\) −0.618034 −0.0300498
\(424\) −16.5066 −0.801630
\(425\) −3.00000 −0.145521
\(426\) −23.5623 −1.14160
\(427\) 3.43769 0.166362
\(428\) −0.145898 −0.00705225
\(429\) 0 0
\(430\) −1.09017 −0.0525727
\(431\) 1.49342 0.0719356 0.0359678 0.999353i \(-0.488549\pi\)
0.0359678 + 0.999353i \(0.488549\pi\)
\(432\) −4.85410 −0.233543
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) −18.7082 −0.898023
\(435\) −1.70820 −0.0819021
\(436\) 0 0
\(437\) −4.67376 −0.223576
\(438\) 2.00000 0.0955637
\(439\) −16.7082 −0.797439 −0.398720 0.917073i \(-0.630545\pi\)
−0.398720 + 0.917073i \(0.630545\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 6.23607 0.296620
\(443\) −0.875388 −0.0415909 −0.0207955 0.999784i \(-0.506620\pi\)
−0.0207955 + 0.999784i \(0.506620\pi\)
\(444\) −2.61803 −0.124246
\(445\) −3.61803 −0.171511
\(446\) 20.5623 0.973653
\(447\) −15.0000 −0.709476
\(448\) −12.7082 −0.600406
\(449\) −15.5279 −0.732805 −0.366403 0.930456i \(-0.619411\pi\)
−0.366403 + 0.930456i \(0.619411\pi\)
\(450\) 7.85410 0.370246
\(451\) 0 0
\(452\) −7.85410 −0.369426
\(453\) −2.00000 −0.0939682
\(454\) 17.6180 0.826855
\(455\) −7.14590 −0.335005
\(456\) 1.90983 0.0894360
\(457\) −32.7984 −1.53424 −0.767122 0.641502i \(-0.778312\pi\)
−0.767122 + 0.641502i \(0.778312\pi\)
\(458\) −16.1803 −0.756058
\(459\) 0.618034 0.0288474
\(460\) 1.29180 0.0602303
\(461\) 9.90983 0.461547 0.230773 0.973008i \(-0.425874\pi\)
0.230773 + 0.973008i \(0.425874\pi\)
\(462\) 0 0
\(463\) 8.79837 0.408895 0.204448 0.978878i \(-0.434460\pi\)
0.204448 + 0.978878i \(0.434460\pi\)
\(464\) −21.7082 −1.00778
\(465\) 1.47214 0.0682687
\(466\) 14.0344 0.650133
\(467\) −14.2361 −0.658767 −0.329383 0.944196i \(-0.606841\pi\)
−0.329383 + 0.944196i \(0.606841\pi\)
\(468\) −3.85410 −0.178156
\(469\) −31.6869 −1.46317
\(470\) −0.381966 −0.0176188
\(471\) 9.70820 0.447330
\(472\) −11.9098 −0.548194
\(473\) 0 0
\(474\) 0.854102 0.0392302
\(475\) −4.14590 −0.190227
\(476\) −1.14590 −0.0525222
\(477\) −7.38197 −0.337997
\(478\) −28.4164 −1.29974
\(479\) 16.9098 0.772630 0.386315 0.922367i \(-0.373748\pi\)
0.386315 + 0.922367i \(0.373748\pi\)
\(480\) −1.29180 −0.0589622
\(481\) 26.4164 1.20448
\(482\) 27.7082 1.26207
\(483\) 16.4164 0.746972
\(484\) 0 0
\(485\) −5.74265 −0.260760
\(486\) −1.61803 −0.0733955
\(487\) 39.1803 1.77543 0.887715 0.460393i \(-0.152291\pi\)
0.887715 + 0.460393i \(0.152291\pi\)
\(488\) −2.56231 −0.115990
\(489\) −15.2705 −0.690556
\(490\) 1.23607 0.0558399
\(491\) 26.2148 1.18306 0.591528 0.806284i \(-0.298525\pi\)
0.591528 + 0.806284i \(0.298525\pi\)
\(492\) −3.67376 −0.165626
\(493\) 2.76393 0.124481
\(494\) 8.61803 0.387744
\(495\) 0 0
\(496\) 18.7082 0.840023
\(497\) −43.6869 −1.95963
\(498\) −20.5623 −0.921419
\(499\) 2.56231 0.114705 0.0573523 0.998354i \(-0.481734\pi\)
0.0573523 + 0.998354i \(0.481734\pi\)
\(500\) 2.32624 0.104033
\(501\) 19.0344 0.850396
\(502\) −27.1803 −1.21312
\(503\) 30.0689 1.34071 0.670353 0.742043i \(-0.266143\pi\)
0.670353 + 0.742043i \(0.266143\pi\)
\(504\) −6.70820 −0.298807
\(505\) −1.14590 −0.0509918
\(506\) 0 0
\(507\) 25.8885 1.14975
\(508\) −6.00000 −0.266207
\(509\) 21.3820 0.947739 0.473869 0.880595i \(-0.342857\pi\)
0.473869 + 0.880595i \(0.342857\pi\)
\(510\) 0.381966 0.0169137
\(511\) 3.70820 0.164041
\(512\) 5.29180 0.233867
\(513\) 0.854102 0.0377095
\(514\) 44.2148 1.95023
\(515\) 2.29180 0.100989
\(516\) −1.09017 −0.0479921
\(517\) 0 0
\(518\) −20.5623 −0.903456
\(519\) −17.6180 −0.773346
\(520\) 5.32624 0.233571
\(521\) 38.8328 1.70130 0.850648 0.525735i \(-0.176210\pi\)
0.850648 + 0.525735i \(0.176210\pi\)
\(522\) −7.23607 −0.316714
\(523\) −34.9787 −1.52951 −0.764756 0.644320i \(-0.777141\pi\)
−0.764756 + 0.644320i \(0.777141\pi\)
\(524\) 8.56231 0.374046
\(525\) 14.5623 0.635551
\(526\) −1.09017 −0.0475337
\(527\) −2.38197 −0.103760
\(528\) 0 0
\(529\) 6.94427 0.301925
\(530\) −4.56231 −0.198174
\(531\) −5.32624 −0.231139
\(532\) −1.58359 −0.0686574
\(533\) 37.0689 1.60563
\(534\) −15.3262 −0.663231
\(535\) 0.0901699 0.00389839
\(536\) 23.6180 1.02014
\(537\) 2.23607 0.0964935
\(538\) −39.5967 −1.70714
\(539\) 0 0
\(540\) −0.236068 −0.0101587
\(541\) 0.562306 0.0241754 0.0120877 0.999927i \(-0.496152\pi\)
0.0120877 + 0.999927i \(0.496152\pi\)
\(542\) 10.1459 0.435804
\(543\) −8.52786 −0.365966
\(544\) 2.09017 0.0896153
\(545\) 0 0
\(546\) −30.2705 −1.29546
\(547\) −19.3820 −0.828713 −0.414357 0.910115i \(-0.635993\pi\)
−0.414357 + 0.910115i \(0.635993\pi\)
\(548\) −0.909830 −0.0388660
\(549\) −1.14590 −0.0489057
\(550\) 0 0
\(551\) 3.81966 0.162723
\(552\) −12.2361 −0.520802
\(553\) 1.58359 0.0673412
\(554\) 16.8885 0.717525
\(555\) 1.61803 0.0686817
\(556\) −3.61803 −0.153439
\(557\) −26.0902 −1.10548 −0.552738 0.833355i \(-0.686417\pi\)
−0.552738 + 0.833355i \(0.686417\pi\)
\(558\) 6.23607 0.263994
\(559\) 11.0000 0.465250
\(560\) −5.56231 −0.235050
\(561\) 0 0
\(562\) −8.47214 −0.357375
\(563\) −26.8885 −1.13322 −0.566609 0.823987i \(-0.691745\pi\)
−0.566609 + 0.823987i \(0.691745\pi\)
\(564\) −0.381966 −0.0160837
\(565\) 4.85410 0.204214
\(566\) −35.8885 −1.50851
\(567\) −3.00000 −0.125988
\(568\) 32.5623 1.36628
\(569\) −34.0689 −1.42824 −0.714121 0.700022i \(-0.753173\pi\)
−0.714121 + 0.700022i \(0.753173\pi\)
\(570\) 0.527864 0.0221098
\(571\) 25.6869 1.07496 0.537482 0.843275i \(-0.319376\pi\)
0.537482 + 0.843275i \(0.319376\pi\)
\(572\) 0 0
\(573\) 1.47214 0.0614994
\(574\) −28.8541 −1.20435
\(575\) 26.5623 1.10772
\(576\) 4.23607 0.176503
\(577\) 15.2361 0.634286 0.317143 0.948378i \(-0.397277\pi\)
0.317143 + 0.948378i \(0.397277\pi\)
\(578\) 26.8885 1.11842
\(579\) −1.56231 −0.0649272
\(580\) −1.05573 −0.0438367
\(581\) −38.1246 −1.58168
\(582\) −24.3262 −1.00836
\(583\) 0 0
\(584\) −2.76393 −0.114372
\(585\) 2.38197 0.0984822
\(586\) 29.0344 1.19940
\(587\) 24.3050 1.00317 0.501586 0.865108i \(-0.332750\pi\)
0.501586 + 0.865108i \(0.332750\pi\)
\(588\) 1.23607 0.0509746
\(589\) −3.29180 −0.135636
\(590\) −3.29180 −0.135521
\(591\) −26.6180 −1.09492
\(592\) 20.5623 0.845106
\(593\) 29.2148 1.19971 0.599854 0.800110i \(-0.295225\pi\)
0.599854 + 0.800110i \(0.295225\pi\)
\(594\) 0 0
\(595\) 0.708204 0.0290335
\(596\) −9.27051 −0.379735
\(597\) −3.29180 −0.134724
\(598\) −55.2148 −2.25790
\(599\) −21.7082 −0.886973 −0.443487 0.896281i \(-0.646259\pi\)
−0.443487 + 0.896281i \(0.646259\pi\)
\(600\) −10.8541 −0.443117
\(601\) 19.8328 0.808997 0.404499 0.914539i \(-0.367446\pi\)
0.404499 + 0.914539i \(0.367446\pi\)
\(602\) −8.56231 −0.348974
\(603\) 10.5623 0.430130
\(604\) −1.23607 −0.0502949
\(605\) 0 0
\(606\) −4.85410 −0.197184
\(607\) 2.32624 0.0944191 0.0472095 0.998885i \(-0.484967\pi\)
0.0472095 + 0.998885i \(0.484967\pi\)
\(608\) 2.88854 0.117146
\(609\) −13.4164 −0.543660
\(610\) −0.708204 −0.0286743
\(611\) 3.85410 0.155920
\(612\) 0.381966 0.0154401
\(613\) −3.34752 −0.135205 −0.0676026 0.997712i \(-0.521535\pi\)
−0.0676026 + 0.997712i \(0.521535\pi\)
\(614\) −31.6525 −1.27739
\(615\) 2.27051 0.0915558
\(616\) 0 0
\(617\) 46.4164 1.86865 0.934327 0.356417i \(-0.116002\pi\)
0.934327 + 0.356417i \(0.116002\pi\)
\(618\) 9.70820 0.390521
\(619\) −31.1803 −1.25324 −0.626622 0.779323i \(-0.715563\pi\)
−0.626622 + 0.779323i \(0.715563\pi\)
\(620\) 0.909830 0.0365397
\(621\) −5.47214 −0.219589
\(622\) −18.5623 −0.744281
\(623\) −28.4164 −1.13848
\(624\) 30.2705 1.21179
\(625\) 22.8328 0.913313
\(626\) 45.0344 1.79994
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) −2.61803 −0.104388
\(630\) −1.85410 −0.0738692
\(631\) 12.7295 0.506753 0.253377 0.967368i \(-0.418459\pi\)
0.253377 + 0.967368i \(0.418459\pi\)
\(632\) −1.18034 −0.0469514
\(633\) −11.2705 −0.447963
\(634\) −41.0344 −1.62969
\(635\) 3.70820 0.147156
\(636\) −4.56231 −0.180907
\(637\) −12.4721 −0.494164
\(638\) 0 0
\(639\) 14.5623 0.576076
\(640\) 5.20163 0.205612
\(641\) −6.74265 −0.266318 −0.133159 0.991095i \(-0.542512\pi\)
−0.133159 + 0.991095i \(0.542512\pi\)
\(642\) 0.381966 0.0150750
\(643\) −18.4377 −0.727112 −0.363556 0.931572i \(-0.618437\pi\)
−0.363556 + 0.931572i \(0.618437\pi\)
\(644\) 10.1459 0.399804
\(645\) 0.673762 0.0265294
\(646\) −0.854102 −0.0336042
\(647\) 3.20163 0.125869 0.0629345 0.998018i \(-0.479954\pi\)
0.0629345 + 0.998018i \(0.479954\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) −48.9787 −1.92110
\(651\) 11.5623 0.453162
\(652\) −9.43769 −0.369609
\(653\) 21.9656 0.859579 0.429789 0.902929i \(-0.358588\pi\)
0.429789 + 0.902929i \(0.358588\pi\)
\(654\) 0 0
\(655\) −5.29180 −0.206768
\(656\) 28.8541 1.12656
\(657\) −1.23607 −0.0482236
\(658\) −3.00000 −0.116952
\(659\) −20.6525 −0.804506 −0.402253 0.915528i \(-0.631773\pi\)
−0.402253 + 0.915528i \(0.631773\pi\)
\(660\) 0 0
\(661\) −21.0902 −0.820313 −0.410156 0.912015i \(-0.634526\pi\)
−0.410156 + 0.912015i \(0.634526\pi\)
\(662\) 36.5623 1.42103
\(663\) −3.85410 −0.149681
\(664\) 28.4164 1.10277
\(665\) 0.978714 0.0379529
\(666\) 6.85410 0.265591
\(667\) −24.4721 −0.947565
\(668\) 11.7639 0.455160
\(669\) −12.7082 −0.491328
\(670\) 6.52786 0.252193
\(671\) 0 0
\(672\) −10.1459 −0.391387
\(673\) 14.4164 0.555712 0.277856 0.960623i \(-0.410376\pi\)
0.277856 + 0.960623i \(0.410376\pi\)
\(674\) −22.1803 −0.854355
\(675\) −4.85410 −0.186834
\(676\) 16.0000 0.615385
\(677\) −22.4721 −0.863674 −0.431837 0.901952i \(-0.642134\pi\)
−0.431837 + 0.901952i \(0.642134\pi\)
\(678\) 20.5623 0.789691
\(679\) −45.1033 −1.73091
\(680\) −0.527864 −0.0202427
\(681\) −10.8885 −0.417250
\(682\) 0 0
\(683\) −38.8885 −1.48803 −0.744014 0.668164i \(-0.767081\pi\)
−0.744014 + 0.668164i \(0.767081\pi\)
\(684\) 0.527864 0.0201834
\(685\) 0.562306 0.0214846
\(686\) −24.2705 −0.926652
\(687\) 10.0000 0.381524
\(688\) 8.56231 0.326435
\(689\) 46.0344 1.75377
\(690\) −3.38197 −0.128749
\(691\) −39.7082 −1.51057 −0.755286 0.655396i \(-0.772502\pi\)
−0.755286 + 0.655396i \(0.772502\pi\)
\(692\) −10.8885 −0.413920
\(693\) 0 0
\(694\) −4.94427 −0.187682
\(695\) 2.23607 0.0848189
\(696\) 10.0000 0.379049
\(697\) −3.67376 −0.139154
\(698\) 48.7426 1.84494
\(699\) −8.67376 −0.328072
\(700\) 9.00000 0.340168
\(701\) −49.6869 −1.87665 −0.938324 0.345756i \(-0.887623\pi\)
−0.938324 + 0.345756i \(0.887623\pi\)
\(702\) 10.0902 0.380829
\(703\) −3.61803 −0.136457
\(704\) 0 0
\(705\) 0.236068 0.00889083
\(706\) 2.47214 0.0930401
\(707\) −9.00000 −0.338480
\(708\) −3.29180 −0.123713
\(709\) 6.25735 0.235000 0.117500 0.993073i \(-0.462512\pi\)
0.117500 + 0.993073i \(0.462512\pi\)
\(710\) 9.00000 0.337764
\(711\) −0.527864 −0.0197964
\(712\) 21.1803 0.793767
\(713\) 21.0902 0.789833
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 1.38197 0.0516465
\(717\) 17.5623 0.655876
\(718\) −27.8885 −1.04079
\(719\) −28.4164 −1.05975 −0.529877 0.848075i \(-0.677762\pi\)
−0.529877 + 0.848075i \(0.677762\pi\)
\(720\) 1.85410 0.0690983
\(721\) 18.0000 0.670355
\(722\) 29.5623 1.10020
\(723\) −17.1246 −0.636871
\(724\) −5.27051 −0.195877
\(725\) −21.7082 −0.806222
\(726\) 0 0
\(727\) 32.1459 1.19223 0.596113 0.802901i \(-0.296711\pi\)
0.596113 + 0.802901i \(0.296711\pi\)
\(728\) 41.8328 1.55043
\(729\) 1.00000 0.0370370
\(730\) −0.763932 −0.0282744
\(731\) −1.09017 −0.0403214
\(732\) −0.708204 −0.0261760
\(733\) 24.2918 0.897238 0.448619 0.893723i \(-0.351916\pi\)
0.448619 + 0.893723i \(0.351916\pi\)
\(734\) 23.5623 0.869701
\(735\) −0.763932 −0.0281781
\(736\) −18.5066 −0.682162
\(737\) 0 0
\(738\) 9.61803 0.354045
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 1.00000 0.0367607
\(741\) −5.32624 −0.195664
\(742\) −35.8328 −1.31546
\(743\) −12.8197 −0.470308 −0.235154 0.971958i \(-0.575559\pi\)
−0.235154 + 0.971958i \(0.575559\pi\)
\(744\) −8.61803 −0.315952
\(745\) 5.72949 0.209912
\(746\) 36.2705 1.32796
\(747\) 12.7082 0.464969
\(748\) 0 0
\(749\) 0.708204 0.0258772
\(750\) −6.09017 −0.222382
\(751\) −52.9230 −1.93119 −0.965594 0.260056i \(-0.916259\pi\)
−0.965594 + 0.260056i \(0.916259\pi\)
\(752\) 3.00000 0.109399
\(753\) 16.7984 0.612167
\(754\) 45.1246 1.64334
\(755\) 0.763932 0.0278023
\(756\) −1.85410 −0.0674330
\(757\) −15.9443 −0.579504 −0.289752 0.957102i \(-0.593573\pi\)
−0.289752 + 0.957102i \(0.593573\pi\)
\(758\) −45.9787 −1.67002
\(759\) 0 0
\(760\) −0.729490 −0.0264614
\(761\) −4.88854 −0.177210 −0.0886048 0.996067i \(-0.528241\pi\)
−0.0886048 + 0.996067i \(0.528241\pi\)
\(762\) 15.7082 0.569048
\(763\) 0 0
\(764\) 0.909830 0.0329165
\(765\) −0.236068 −0.00853506
\(766\) 14.3820 0.519642
\(767\) 33.2148 1.19932
\(768\) 13.5623 0.489388
\(769\) 47.6869 1.71963 0.859817 0.510602i \(-0.170577\pi\)
0.859817 + 0.510602i \(0.170577\pi\)
\(770\) 0 0
\(771\) −27.3262 −0.984130
\(772\) −0.965558 −0.0347512
\(773\) 49.2492 1.77137 0.885686 0.464285i \(-0.153689\pi\)
0.885686 + 0.464285i \(0.153689\pi\)
\(774\) 2.85410 0.102589
\(775\) 18.7082 0.672019
\(776\) 33.6180 1.20682
\(777\) 12.7082 0.455904
\(778\) 15.0000 0.537776
\(779\) −5.07701 −0.181903
\(780\) 1.47214 0.0527109
\(781\) 0 0
\(782\) 5.47214 0.195683
\(783\) 4.47214 0.159821
\(784\) −9.70820 −0.346722
\(785\) −3.70820 −0.132351
\(786\) −22.4164 −0.799567
\(787\) 23.7082 0.845106 0.422553 0.906338i \(-0.361134\pi\)
0.422553 + 0.906338i \(0.361134\pi\)
\(788\) −16.4508 −0.586037
\(789\) 0.673762 0.0239866
\(790\) −0.326238 −0.0116070
\(791\) 38.1246 1.35556
\(792\) 0 0
\(793\) 7.14590 0.253758
\(794\) 40.9230 1.45230
\(795\) 2.81966 0.100003
\(796\) −2.03444 −0.0721089
\(797\) 15.2361 0.539689 0.269845 0.962904i \(-0.413028\pi\)
0.269845 + 0.962904i \(0.413028\pi\)
\(798\) 4.14590 0.146763
\(799\) −0.381966 −0.0135130
\(800\) −16.4164 −0.580408
\(801\) 9.47214 0.334681
\(802\) 24.1246 0.851870
\(803\) 0 0
\(804\) 6.52786 0.230220
\(805\) −6.27051 −0.221006
\(806\) −38.8885 −1.36979
\(807\) 24.4721 0.861460
\(808\) 6.70820 0.235994
\(809\) −25.3262 −0.890423 −0.445212 0.895425i \(-0.646872\pi\)
−0.445212 + 0.895425i \(0.646872\pi\)
\(810\) 0.618034 0.0217155
\(811\) −45.7426 −1.60624 −0.803121 0.595816i \(-0.796829\pi\)
−0.803121 + 0.595816i \(0.796829\pi\)
\(812\) −8.29180 −0.290985
\(813\) −6.27051 −0.219916
\(814\) 0 0
\(815\) 5.83282 0.204315
\(816\) −3.00000 −0.105021
\(817\) −1.50658 −0.0527085
\(818\) −46.8328 −1.63747
\(819\) 18.7082 0.653718
\(820\) 1.40325 0.0490037
\(821\) −13.5836 −0.474071 −0.237035 0.971501i \(-0.576176\pi\)
−0.237035 + 0.971501i \(0.576176\pi\)
\(822\) 2.38197 0.0830806
\(823\) −26.5279 −0.924703 −0.462352 0.886697i \(-0.652994\pi\)
−0.462352 + 0.886697i \(0.652994\pi\)
\(824\) −13.4164 −0.467383
\(825\) 0 0
\(826\) −25.8541 −0.899579
\(827\) −12.8754 −0.447721 −0.223861 0.974621i \(-0.571866\pi\)
−0.223861 + 0.974621i \(0.571866\pi\)
\(828\) −3.38197 −0.117531
\(829\) −42.6869 −1.48258 −0.741289 0.671186i \(-0.765785\pi\)
−0.741289 + 0.671186i \(0.765785\pi\)
\(830\) 7.85410 0.272620
\(831\) −10.4377 −0.362080
\(832\) −26.4164 −0.915824
\(833\) 1.23607 0.0428272
\(834\) 9.47214 0.327993
\(835\) −7.27051 −0.251606
\(836\) 0 0
\(837\) −3.85410 −0.133217
\(838\) 34.7984 1.20209
\(839\) −23.2918 −0.804122 −0.402061 0.915613i \(-0.631706\pi\)
−0.402061 + 0.915613i \(0.631706\pi\)
\(840\) 2.56231 0.0884080
\(841\) −9.00000 −0.310345
\(842\) 6.03444 0.207961
\(843\) 5.23607 0.180340
\(844\) −6.96556 −0.239764
\(845\) −9.88854 −0.340176
\(846\) 1.00000 0.0343807
\(847\) 0 0
\(848\) 35.8328 1.23050
\(849\) 22.1803 0.761227
\(850\) 4.85410 0.166494
\(851\) 23.1803 0.794612
\(852\) 9.00000 0.308335
\(853\) −7.94427 −0.272007 −0.136003 0.990708i \(-0.543426\pi\)
−0.136003 + 0.990708i \(0.543426\pi\)
\(854\) −5.56231 −0.190338
\(855\) −0.326238 −0.0111571
\(856\) −0.527864 −0.0180420
\(857\) 41.7214 1.42517 0.712587 0.701584i \(-0.247523\pi\)
0.712587 + 0.701584i \(0.247523\pi\)
\(858\) 0 0
\(859\) 42.8885 1.46334 0.731669 0.681660i \(-0.238742\pi\)
0.731669 + 0.681660i \(0.238742\pi\)
\(860\) 0.416408 0.0141994
\(861\) 17.8328 0.607741
\(862\) −2.41641 −0.0823032
\(863\) −23.8885 −0.813175 −0.406588 0.913612i \(-0.633281\pi\)
−0.406588 + 0.913612i \(0.633281\pi\)
\(864\) 3.38197 0.115057
\(865\) 6.72949 0.228810
\(866\) 9.70820 0.329898
\(867\) −16.6180 −0.564378
\(868\) 7.14590 0.242548
\(869\) 0 0
\(870\) 2.76393 0.0937061
\(871\) −65.8673 −2.23183
\(872\) 0 0
\(873\) 15.0344 0.508839
\(874\) 7.56231 0.255799
\(875\) −11.2918 −0.381732
\(876\) −0.763932 −0.0258109
\(877\) 20.4164 0.689413 0.344707 0.938710i \(-0.387978\pi\)
0.344707 + 0.938710i \(0.387978\pi\)
\(878\) 27.0344 0.912368
\(879\) −17.9443 −0.605245
\(880\) 0 0
\(881\) 25.0902 0.845309 0.422655 0.906291i \(-0.361098\pi\)
0.422655 + 0.906291i \(0.361098\pi\)
\(882\) −3.23607 −0.108964
\(883\) 37.4164 1.25916 0.629581 0.776935i \(-0.283226\pi\)
0.629581 + 0.776935i \(0.283226\pi\)
\(884\) −2.38197 −0.0801142
\(885\) 2.03444 0.0683870
\(886\) 1.41641 0.0475852
\(887\) −3.00000 −0.100730 −0.0503651 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) −9.47214 −0.317864
\(889\) 29.1246 0.976808
\(890\) 5.85410 0.196230
\(891\) 0 0
\(892\) −7.85410 −0.262975
\(893\) −0.527864 −0.0176643
\(894\) 24.2705 0.811727
\(895\) −0.854102 −0.0285495
\(896\) 40.8541 1.36484
\(897\) 34.1246 1.13939
\(898\) 25.1246 0.838419
\(899\) −17.2361 −0.574855
\(900\) −3.00000 −0.100000
\(901\) −4.56231 −0.151992
\(902\) 0 0
\(903\) 5.29180 0.176100
\(904\) −28.4164 −0.945116
\(905\) 3.25735 0.108278
\(906\) 3.23607 0.107511
\(907\) 3.97871 0.132111 0.0660555 0.997816i \(-0.478959\pi\)
0.0660555 + 0.997816i \(0.478959\pi\)
\(908\) −6.72949 −0.223326
\(909\) 3.00000 0.0995037
\(910\) 11.5623 0.383287
\(911\) 35.9443 1.19089 0.595443 0.803397i \(-0.296976\pi\)
0.595443 + 0.803397i \(0.296976\pi\)
\(912\) −4.14590 −0.137284
\(913\) 0 0
\(914\) 53.0689 1.75536
\(915\) 0.437694 0.0144697
\(916\) 6.18034 0.204204
\(917\) −41.5623 −1.37251
\(918\) −1.00000 −0.0330049
\(919\) 46.9574 1.54898 0.774491 0.632585i \(-0.218006\pi\)
0.774491 + 0.632585i \(0.218006\pi\)
\(920\) 4.67376 0.154089
\(921\) 19.5623 0.644600
\(922\) −16.0344 −0.528066
\(923\) −90.8115 −2.98910
\(924\) 0 0
\(925\) 20.5623 0.676084
\(926\) −14.2361 −0.467826
\(927\) −6.00000 −0.197066
\(928\) 15.1246 0.496490
\(929\) 2.88854 0.0947700 0.0473850 0.998877i \(-0.484911\pi\)
0.0473850 + 0.998877i \(0.484911\pi\)
\(930\) −2.38197 −0.0781077
\(931\) 1.70820 0.0559841
\(932\) −5.36068 −0.175595
\(933\) 11.4721 0.375581
\(934\) 23.0344 0.753710
\(935\) 0 0
\(936\) −13.9443 −0.455783
\(937\) −32.5967 −1.06489 −0.532445 0.846465i \(-0.678727\pi\)
−0.532445 + 0.846465i \(0.678727\pi\)
\(938\) 51.2705 1.67404
\(939\) −27.8328 −0.908290
\(940\) 0.145898 0.00475867
\(941\) −33.6312 −1.09635 −0.548173 0.836365i \(-0.684676\pi\)
−0.548173 + 0.836365i \(0.684676\pi\)
\(942\) −15.7082 −0.511801
\(943\) 32.5279 1.05925
\(944\) 25.8541 0.841479
\(945\) 1.14590 0.0372761
\(946\) 0 0
\(947\) 2.67376 0.0868856 0.0434428 0.999056i \(-0.486167\pi\)
0.0434428 + 0.999056i \(0.486167\pi\)
\(948\) −0.326238 −0.0105957
\(949\) 7.70820 0.250219
\(950\) 6.70820 0.217643
\(951\) 25.3607 0.822376
\(952\) −4.14590 −0.134369
\(953\) −60.1803 −1.94943 −0.974716 0.223446i \(-0.928269\pi\)
−0.974716 + 0.223446i \(0.928269\pi\)
\(954\) 11.9443 0.386710
\(955\) −0.562306 −0.0181958
\(956\) 10.8541 0.351047
\(957\) 0 0
\(958\) −27.3607 −0.883983
\(959\) 4.41641 0.142613
\(960\) −1.61803 −0.0522218
\(961\) −16.1459 −0.520835
\(962\) −42.7426 −1.37808
\(963\) −0.236068 −0.00760718
\(964\) −10.5836 −0.340875
\(965\) 0.596748 0.0192100
\(966\) −26.5623 −0.854628
\(967\) −25.6869 −0.826036 −0.413018 0.910723i \(-0.635525\pi\)
−0.413018 + 0.910723i \(0.635525\pi\)
\(968\) 0 0
\(969\) 0.527864 0.0169574
\(970\) 9.29180 0.298342
\(971\) −33.7771 −1.08396 −0.541979 0.840392i \(-0.682325\pi\)
−0.541979 + 0.840392i \(0.682325\pi\)
\(972\) 0.618034 0.0198234
\(973\) 17.5623 0.563022
\(974\) −63.3951 −2.03131
\(975\) 30.2705 0.969432
\(976\) 5.56231 0.178045
\(977\) −49.3607 −1.57919 −0.789594 0.613630i \(-0.789709\pi\)
−0.789594 + 0.613630i \(0.789709\pi\)
\(978\) 24.7082 0.790081
\(979\) 0 0
\(980\) −0.472136 −0.0150818
\(981\) 0 0
\(982\) −42.4164 −1.35356
\(983\) 35.3050 1.12605 0.563027 0.826439i \(-0.309637\pi\)
0.563027 + 0.826439i \(0.309637\pi\)
\(984\) −13.2918 −0.423727
\(985\) 10.1672 0.323953
\(986\) −4.47214 −0.142422
\(987\) 1.85410 0.0590167
\(988\) −3.29180 −0.104726
\(989\) 9.65248 0.306931
\(990\) 0 0
\(991\) −12.2705 −0.389786 −0.194893 0.980825i \(-0.562436\pi\)
−0.194893 + 0.980825i \(0.562436\pi\)
\(992\) −13.0344 −0.413844
\(993\) −22.5967 −0.717086
\(994\) 70.6869 2.24205
\(995\) 1.25735 0.0398608
\(996\) 7.85410 0.248867
\(997\) −33.9787 −1.07612 −0.538058 0.842908i \(-0.680842\pi\)
−0.538058 + 0.842908i \(0.680842\pi\)
\(998\) −4.14590 −0.131236
\(999\) −4.23607 −0.134023
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 363.2.a.e.1.1 2
3.2 odd 2 1089.2.a.s.1.2 2
4.3 odd 2 5808.2.a.bm.1.2 2
5.4 even 2 9075.2.a.bv.1.2 2
11.2 odd 10 33.2.e.a.4.1 4
11.3 even 5 363.2.e.c.130.1 4
11.4 even 5 363.2.e.c.148.1 4
11.5 even 5 363.2.e.j.124.1 4
11.6 odd 10 33.2.e.a.25.1 yes 4
11.7 odd 10 363.2.e.h.148.1 4
11.8 odd 10 363.2.e.h.130.1 4
11.9 even 5 363.2.e.j.202.1 4
11.10 odd 2 363.2.a.h.1.2 2
33.2 even 10 99.2.f.b.37.1 4
33.17 even 10 99.2.f.b.91.1 4
33.32 even 2 1089.2.a.m.1.1 2
44.35 even 10 528.2.y.f.433.1 4
44.39 even 10 528.2.y.f.289.1 4
44.43 even 2 5808.2.a.bl.1.2 2
55.2 even 20 825.2.bx.b.499.2 8
55.13 even 20 825.2.bx.b.499.1 8
55.17 even 20 825.2.bx.b.124.1 8
55.24 odd 10 825.2.n.f.301.1 4
55.28 even 20 825.2.bx.b.124.2 8
55.39 odd 10 825.2.n.f.751.1 4
55.54 odd 2 9075.2.a.x.1.1 2
99.2 even 30 891.2.n.a.433.1 8
99.13 odd 30 891.2.n.d.136.1 8
99.50 even 30 891.2.n.a.784.1 8
99.61 odd 30 891.2.n.d.190.1 8
99.68 even 30 891.2.n.a.136.1 8
99.79 odd 30 891.2.n.d.433.1 8
99.83 even 30 891.2.n.a.190.1 8
99.94 odd 30 891.2.n.d.784.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.4.1 4 11.2 odd 10
33.2.e.a.25.1 yes 4 11.6 odd 10
99.2.f.b.37.1 4 33.2 even 10
99.2.f.b.91.1 4 33.17 even 10
363.2.a.e.1.1 2 1.1 even 1 trivial
363.2.a.h.1.2 2 11.10 odd 2
363.2.e.c.130.1 4 11.3 even 5
363.2.e.c.148.1 4 11.4 even 5
363.2.e.h.130.1 4 11.8 odd 10
363.2.e.h.148.1 4 11.7 odd 10
363.2.e.j.124.1 4 11.5 even 5
363.2.e.j.202.1 4 11.9 even 5
528.2.y.f.289.1 4 44.39 even 10
528.2.y.f.433.1 4 44.35 even 10
825.2.n.f.301.1 4 55.24 odd 10
825.2.n.f.751.1 4 55.39 odd 10
825.2.bx.b.124.1 8 55.17 even 20
825.2.bx.b.124.2 8 55.28 even 20
825.2.bx.b.499.1 8 55.13 even 20
825.2.bx.b.499.2 8 55.2 even 20
891.2.n.a.136.1 8 99.68 even 30
891.2.n.a.190.1 8 99.83 even 30
891.2.n.a.433.1 8 99.2 even 30
891.2.n.a.784.1 8 99.50 even 30
891.2.n.d.136.1 8 99.13 odd 30
891.2.n.d.190.1 8 99.61 odd 30
891.2.n.d.433.1 8 99.79 odd 30
891.2.n.d.784.1 8 99.94 odd 30
1089.2.a.m.1.1 2 33.32 even 2
1089.2.a.s.1.2 2 3.2 odd 2
5808.2.a.bl.1.2 2 44.43 even 2
5808.2.a.bm.1.2 2 4.3 odd 2
9075.2.a.x.1.1 2 55.54 odd 2
9075.2.a.bv.1.2 2 5.4 even 2