Properties

Label 6080.2.a.cc.1.4
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $0$
Dimension $4$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) 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 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.552409\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +O(q^{10})\) \(q+2.87834 q^{3} +1.00000 q^{5} +3.10482 q^{7} +5.28487 q^{9} +1.10482 q^{11} -1.77353 q^{13} +2.87834 q^{15} +7.75669 q^{17} -1.00000 q^{19} +8.93674 q^{21} -6.65187 q^{23} +1.00000 q^{25} +6.57664 q^{27} -7.75669 q^{29} +6.57664 q^{31} +3.18005 q^{33} +3.10482 q^{35} +1.40652 q^{37} -5.10482 q^{39} +2.81995 q^{41} -3.10482 q^{43} +5.28487 q^{45} +1.46492 q^{47} +2.63990 q^{49} +22.3264 q^{51} +2.59348 q^{53} +1.10482 q^{55} -2.87834 q^{57} +5.38969 q^{59} -4.07523 q^{61} +16.4086 q^{63} -1.77353 q^{65} +15.8151 q^{67} -19.1464 q^{69} +7.14638 q^{71} +0.243310 q^{73} +2.87834 q^{75} +3.43026 q^{77} -9.38969 q^{79} +3.07523 q^{81} -8.86151 q^{83} +7.75669 q^{85} -22.3264 q^{87} +0.813048 q^{89} -5.50648 q^{91} +18.9298 q^{93} -1.00000 q^{95} +3.19689 q^{97} +5.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{21} - 8 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} + 6 q^{37} - 12 q^{39} + 16 q^{41} - 4 q^{43} + 8 q^{45} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 4 q^{55} + 2 q^{57} - 20 q^{61} + 20 q^{63} - 2 q^{65} + 18 q^{67} - 28 q^{69} - 20 q^{71} + 28 q^{73} - 2 q^{75} + 40 q^{77} - 16 q^{79} + 16 q^{81} + 4 q^{85} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.87834 1.66181 0.830907 0.556412i \(-0.187822\pi\)
0.830907 + 0.556412i \(0.187822\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.10482 1.17351 0.586756 0.809764i \(-0.300405\pi\)
0.586756 + 0.809764i \(0.300405\pi\)
\(8\) 0 0
\(9\) 5.28487 1.76162
\(10\) 0 0
\(11\) 1.10482 0.333115 0.166558 0.986032i \(-0.446735\pi\)
0.166558 + 0.986032i \(0.446735\pi\)
\(12\) 0 0
\(13\) −1.77353 −0.491888 −0.245944 0.969284i \(-0.579098\pi\)
−0.245944 + 0.969284i \(0.579098\pi\)
\(14\) 0 0
\(15\) 2.87834 0.743185
\(16\) 0 0
\(17\) 7.75669 1.88127 0.940637 0.339415i \(-0.110229\pi\)
0.940637 + 0.339415i \(0.110229\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 8.93674 1.95016
\(22\) 0 0
\(23\) −6.65187 −1.38701 −0.693505 0.720451i \(-0.743935\pi\)
−0.693505 + 0.720451i \(0.743935\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 6.57664 1.26567
\(28\) 0 0
\(29\) −7.75669 −1.44038 −0.720191 0.693776i \(-0.755946\pi\)
−0.720191 + 0.693776i \(0.755946\pi\)
\(30\) 0 0
\(31\) 6.57664 1.18120 0.590600 0.806965i \(-0.298891\pi\)
0.590600 + 0.806965i \(0.298891\pi\)
\(32\) 0 0
\(33\) 3.18005 0.553576
\(34\) 0 0
\(35\) 3.10482 0.524810
\(36\) 0 0
\(37\) 1.40652 0.231231 0.115616 0.993294i \(-0.463116\pi\)
0.115616 + 0.993294i \(0.463116\pi\)
\(38\) 0 0
\(39\) −5.10482 −0.817425
\(40\) 0 0
\(41\) 2.81995 0.440402 0.220201 0.975454i \(-0.429329\pi\)
0.220201 + 0.975454i \(0.429329\pi\)
\(42\) 0 0
\(43\) −3.10482 −0.473480 −0.236740 0.971573i \(-0.576079\pi\)
−0.236740 + 0.971573i \(0.576079\pi\)
\(44\) 0 0
\(45\) 5.28487 0.787822
\(46\) 0 0
\(47\) 1.46492 0.213680 0.106840 0.994276i \(-0.465927\pi\)
0.106840 + 0.994276i \(0.465927\pi\)
\(48\) 0 0
\(49\) 2.63990 0.377129
\(50\) 0 0
\(51\) 22.3264 3.12633
\(52\) 0 0
\(53\) 2.59348 0.356241 0.178121 0.984009i \(-0.442998\pi\)
0.178121 + 0.984009i \(0.442998\pi\)
\(54\) 0 0
\(55\) 1.10482 0.148974
\(56\) 0 0
\(57\) −2.87834 −0.381246
\(58\) 0 0
\(59\) 5.38969 0.701678 0.350839 0.936436i \(-0.385897\pi\)
0.350839 + 0.936436i \(0.385897\pi\)
\(60\) 0 0
\(61\) −4.07523 −0.521780 −0.260890 0.965369i \(-0.584016\pi\)
−0.260890 + 0.965369i \(0.584016\pi\)
\(62\) 0 0
\(63\) 16.4086 2.06728
\(64\) 0 0
\(65\) −1.77353 −0.219979
\(66\) 0 0
\(67\) 15.8151 1.93212 0.966060 0.258318i \(-0.0831681\pi\)
0.966060 + 0.258318i \(0.0831681\pi\)
\(68\) 0 0
\(69\) −19.1464 −2.30495
\(70\) 0 0
\(71\) 7.14638 0.848119 0.424059 0.905634i \(-0.360605\pi\)
0.424059 + 0.905634i \(0.360605\pi\)
\(72\) 0 0
\(73\) 0.243310 0.0284773 0.0142387 0.999899i \(-0.495468\pi\)
0.0142387 + 0.999899i \(0.495468\pi\)
\(74\) 0 0
\(75\) 2.87834 0.332363
\(76\) 0 0
\(77\) 3.43026 0.390915
\(78\) 0 0
\(79\) −9.38969 −1.05642 −0.528211 0.849113i \(-0.677137\pi\)
−0.528211 + 0.849113i \(0.677137\pi\)
\(80\) 0 0
\(81\) 3.07523 0.341692
\(82\) 0 0
\(83\) −8.86151 −0.972677 −0.486338 0.873771i \(-0.661668\pi\)
−0.486338 + 0.873771i \(0.661668\pi\)
\(84\) 0 0
\(85\) 7.75669 0.841331
\(86\) 0 0
\(87\) −22.3264 −2.39364
\(88\) 0 0
\(89\) 0.813048 0.0861829 0.0430914 0.999071i \(-0.486279\pi\)
0.0430914 + 0.999071i \(0.486279\pi\)
\(90\) 0 0
\(91\) −5.50648 −0.577236
\(92\) 0 0
\(93\) 18.9298 1.96293
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 3.19689 0.324595 0.162297 0.986742i \(-0.448110\pi\)
0.162297 + 0.986742i \(0.448110\pi\)
\(98\) 0 0
\(99\) 5.83882 0.586824
\(100\) 0 0
\(101\) 3.01887 0.300389 0.150195 0.988656i \(-0.452010\pi\)
0.150195 + 0.988656i \(0.452010\pi\)
\(102\) 0 0
\(103\) 6.05839 0.596951 0.298476 0.954417i \(-0.403522\pi\)
0.298476 + 0.954417i \(0.403522\pi\)
\(104\) 0 0
\(105\) 8.93674 0.872136
\(106\) 0 0
\(107\) −20.3848 −1.97068 −0.985338 0.170616i \(-0.945424\pi\)
−0.985338 + 0.170616i \(0.945424\pi\)
\(108\) 0 0
\(109\) −4.72710 −0.452774 −0.226387 0.974037i \(-0.572691\pi\)
−0.226387 + 0.974037i \(0.572691\pi\)
\(110\) 0 0
\(111\) 4.04846 0.384263
\(112\) 0 0
\(113\) −7.98316 −0.750993 −0.375496 0.926824i \(-0.622528\pi\)
−0.375496 + 0.926824i \(0.622528\pi\)
\(114\) 0 0
\(115\) −6.65187 −0.620290
\(116\) 0 0
\(117\) −9.37285 −0.866520
\(118\) 0 0
\(119\) 24.0831 2.20770
\(120\) 0 0
\(121\) −9.77938 −0.889034
\(122\) 0 0
\(123\) 8.11679 0.731866
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.63503 0.411293 0.205646 0.978626i \(-0.434070\pi\)
0.205646 + 0.978626i \(0.434070\pi\)
\(128\) 0 0
\(129\) −8.93674 −0.786836
\(130\) 0 0
\(131\) −15.5134 −1.35541 −0.677705 0.735334i \(-0.737025\pi\)
−0.677705 + 0.735334i \(0.737025\pi\)
\(132\) 0 0
\(133\) −3.10482 −0.269222
\(134\) 0 0
\(135\) 6.57664 0.566027
\(136\) 0 0
\(137\) −0.813048 −0.0694634 −0.0347317 0.999397i \(-0.511058\pi\)
−0.0347317 + 0.999397i \(0.511058\pi\)
\(138\) 0 0
\(139\) 12.7687 1.08302 0.541512 0.840693i \(-0.317852\pi\)
0.541512 + 0.840693i \(0.317852\pi\)
\(140\) 0 0
\(141\) 4.21654 0.355097
\(142\) 0 0
\(143\) −1.95942 −0.163855
\(144\) 0 0
\(145\) −7.75669 −0.644158
\(146\) 0 0
\(147\) 7.59854 0.626717
\(148\) 0 0
\(149\) 13.7982 1.13040 0.565198 0.824955i \(-0.308800\pi\)
0.565198 + 0.824955i \(0.308800\pi\)
\(150\) 0 0
\(151\) 5.02269 0.408740 0.204370 0.978894i \(-0.434485\pi\)
0.204370 + 0.978894i \(0.434485\pi\)
\(152\) 0 0
\(153\) 40.9931 3.31409
\(154\) 0 0
\(155\) 6.57664 0.528248
\(156\) 0 0
\(157\) −7.18695 −0.573581 −0.286791 0.957993i \(-0.592588\pi\)
−0.286791 + 0.957993i \(0.592588\pi\)
\(158\) 0 0
\(159\) 7.46492 0.592007
\(160\) 0 0
\(161\) −20.6529 −1.62767
\(162\) 0 0
\(163\) −7.25528 −0.568277 −0.284139 0.958783i \(-0.591708\pi\)
−0.284139 + 0.958783i \(0.591708\pi\)
\(164\) 0 0
\(165\) 3.18005 0.247567
\(166\) 0 0
\(167\) −7.33129 −0.567312 −0.283656 0.958926i \(-0.591547\pi\)
−0.283656 + 0.958926i \(0.591547\pi\)
\(168\) 0 0
\(169\) −9.85461 −0.758047
\(170\) 0 0
\(171\) −5.28487 −0.404144
\(172\) 0 0
\(173\) 5.77353 0.438953 0.219477 0.975618i \(-0.429565\pi\)
0.219477 + 0.975618i \(0.429565\pi\)
\(174\) 0 0
\(175\) 3.10482 0.234702
\(176\) 0 0
\(177\) 15.5134 1.16606
\(178\) 0 0
\(179\) 4.48379 0.335134 0.167567 0.985861i \(-0.446409\pi\)
0.167567 + 0.985861i \(0.446409\pi\)
\(180\) 0 0
\(181\) 2.73400 0.203217 0.101608 0.994824i \(-0.467601\pi\)
0.101608 + 0.994824i \(0.467601\pi\)
\(182\) 0 0
\(183\) −11.7299 −0.867101
\(184\) 0 0
\(185\) 1.40652 0.103410
\(186\) 0 0
\(187\) 8.56974 0.626681
\(188\) 0 0
\(189\) 20.4193 1.48528
\(190\) 0 0
\(191\) −17.7230 −1.28239 −0.641196 0.767377i \(-0.721562\pi\)
−0.641196 + 0.767377i \(0.721562\pi\)
\(192\) 0 0
\(193\) −13.5371 −0.974423 −0.487212 0.873284i \(-0.661986\pi\)
−0.487212 + 0.873284i \(0.661986\pi\)
\(194\) 0 0
\(195\) −5.10482 −0.365564
\(196\) 0 0
\(197\) 21.5134 1.53276 0.766382 0.642385i \(-0.222055\pi\)
0.766382 + 0.642385i \(0.222055\pi\)
\(198\) 0 0
\(199\) 7.09410 0.502888 0.251444 0.967872i \(-0.419095\pi\)
0.251444 + 0.967872i \(0.419095\pi\)
\(200\) 0 0
\(201\) 45.5213 3.21082
\(202\) 0 0
\(203\) −24.0831 −1.69030
\(204\) 0 0
\(205\) 2.81995 0.196954
\(206\) 0 0
\(207\) −35.1543 −2.44339
\(208\) 0 0
\(209\) −1.10482 −0.0764219
\(210\) 0 0
\(211\) −11.0604 −0.761431 −0.380716 0.924692i \(-0.624322\pi\)
−0.380716 + 0.924692i \(0.624322\pi\)
\(212\) 0 0
\(213\) 20.5697 1.40942
\(214\) 0 0
\(215\) −3.10482 −0.211747
\(216\) 0 0
\(217\) 20.4193 1.38615
\(218\) 0 0
\(219\) 0.700331 0.0473240
\(220\) 0 0
\(221\) −13.7567 −0.925375
\(222\) 0 0
\(223\) −12.6350 −0.846104 −0.423052 0.906105i \(-0.639041\pi\)
−0.423052 + 0.906105i \(0.639041\pi\)
\(224\) 0 0
\(225\) 5.28487 0.352325
\(226\) 0 0
\(227\) 6.62813 0.439925 0.219962 0.975508i \(-0.429407\pi\)
0.219962 + 0.975508i \(0.429407\pi\)
\(228\) 0 0
\(229\) 0.0752308 0.00497139 0.00248570 0.999997i \(-0.499209\pi\)
0.00248570 + 0.999997i \(0.499209\pi\)
\(230\) 0 0
\(231\) 9.87348 0.649627
\(232\) 0 0
\(233\) 9.51338 0.623242 0.311621 0.950206i \(-0.399128\pi\)
0.311621 + 0.950206i \(0.399128\pi\)
\(234\) 0 0
\(235\) 1.46492 0.0955607
\(236\) 0 0
\(237\) −27.0268 −1.75558
\(238\) 0 0
\(239\) −2.92984 −0.189515 −0.0947577 0.995500i \(-0.530208\pi\)
−0.0947577 + 0.995500i \(0.530208\pi\)
\(240\) 0 0
\(241\) −9.42743 −0.607274 −0.303637 0.952788i \(-0.598201\pi\)
−0.303637 + 0.952788i \(0.598201\pi\)
\(242\) 0 0
\(243\) −10.8783 −0.697846
\(244\) 0 0
\(245\) 2.63990 0.168657
\(246\) 0 0
\(247\) 1.77353 0.112847
\(248\) 0 0
\(249\) −25.5065 −1.61641
\(250\) 0 0
\(251\) 14.9298 0.942363 0.471181 0.882036i \(-0.343828\pi\)
0.471181 + 0.882036i \(0.343828\pi\)
\(252\) 0 0
\(253\) −7.34911 −0.462035
\(254\) 0 0
\(255\) 22.3264 1.39814
\(256\) 0 0
\(257\) −15.1295 −0.943755 −0.471877 0.881664i \(-0.656424\pi\)
−0.471877 + 0.881664i \(0.656424\pi\)
\(258\) 0 0
\(259\) 4.36700 0.271352
\(260\) 0 0
\(261\) −40.9931 −2.53741
\(262\) 0 0
\(263\) −15.2216 −0.938605 −0.469302 0.883038i \(-0.655495\pi\)
−0.469302 + 0.883038i \(0.655495\pi\)
\(264\) 0 0
\(265\) 2.59348 0.159316
\(266\) 0 0
\(267\) 2.34023 0.143220
\(268\) 0 0
\(269\) −11.5203 −0.702404 −0.351202 0.936300i \(-0.614227\pi\)
−0.351202 + 0.936300i \(0.614227\pi\)
\(270\) 0 0
\(271\) 2.16118 0.131282 0.0656411 0.997843i \(-0.479091\pi\)
0.0656411 + 0.997843i \(0.479091\pi\)
\(272\) 0 0
\(273\) −15.8495 −0.959258
\(274\) 0 0
\(275\) 1.10482 0.0666231
\(276\) 0 0
\(277\) −23.4205 −1.40720 −0.703602 0.710595i \(-0.748426\pi\)
−0.703602 + 0.710595i \(0.748426\pi\)
\(278\) 0 0
\(279\) 34.7567 2.08083
\(280\) 0 0
\(281\) 25.6824 1.53209 0.766043 0.642789i \(-0.222223\pi\)
0.766043 + 0.642789i \(0.222223\pi\)
\(282\) 0 0
\(283\) −7.38587 −0.439045 −0.219522 0.975607i \(-0.570450\pi\)
−0.219522 + 0.975607i \(0.570450\pi\)
\(284\) 0 0
\(285\) −2.87834 −0.170498
\(286\) 0 0
\(287\) 8.75543 0.516817
\(288\) 0 0
\(289\) 43.1662 2.53919
\(290\) 0 0
\(291\) 9.20174 0.539416
\(292\) 0 0
\(293\) 24.1522 1.41099 0.705494 0.708716i \(-0.250725\pi\)
0.705494 + 0.708716i \(0.250725\pi\)
\(294\) 0 0
\(295\) 5.38969 0.313800
\(296\) 0 0
\(297\) 7.26600 0.421616
\(298\) 0 0
\(299\) 11.7973 0.682253
\(300\) 0 0
\(301\) −9.63990 −0.555635
\(302\) 0 0
\(303\) 8.68936 0.499190
\(304\) 0 0
\(305\) −4.07523 −0.233347
\(306\) 0 0
\(307\) −4.38482 −0.250255 −0.125127 0.992141i \(-0.539934\pi\)
−0.125127 + 0.992141i \(0.539934\pi\)
\(308\) 0 0
\(309\) 17.4381 0.992022
\(310\) 0 0
\(311\) −15.7123 −0.890963 −0.445481 0.895291i \(-0.646967\pi\)
−0.445481 + 0.895291i \(0.646967\pi\)
\(312\) 0 0
\(313\) 24.7794 1.40061 0.700307 0.713842i \(-0.253047\pi\)
0.700307 + 0.713842i \(0.253047\pi\)
\(314\) 0 0
\(315\) 16.4086 0.924518
\(316\) 0 0
\(317\) 27.3798 1.53780 0.768900 0.639369i \(-0.220804\pi\)
0.768900 + 0.639369i \(0.220804\pi\)
\(318\) 0 0
\(319\) −8.56974 −0.479813
\(320\) 0 0
\(321\) −58.6745 −3.27489
\(322\) 0 0
\(323\) −7.75669 −0.431594
\(324\) 0 0
\(325\) −1.77353 −0.0983775
\(326\) 0 0
\(327\) −13.6062 −0.752426
\(328\) 0 0
\(329\) 4.54831 0.250756
\(330\) 0 0
\(331\) −4.70033 −0.258354 −0.129177 0.991622i \(-0.541233\pi\)
−0.129177 + 0.991622i \(0.541233\pi\)
\(332\) 0 0
\(333\) 7.43329 0.407342
\(334\) 0 0
\(335\) 15.8151 0.864070
\(336\) 0 0
\(337\) −20.6360 −1.12412 −0.562058 0.827098i \(-0.689990\pi\)
−0.562058 + 0.827098i \(0.689990\pi\)
\(338\) 0 0
\(339\) −22.9783 −1.24801
\(340\) 0 0
\(341\) 7.26600 0.393476
\(342\) 0 0
\(343\) −13.5373 −0.730947
\(344\) 0 0
\(345\) −19.1464 −1.03081
\(346\) 0 0
\(347\) −30.0148 −1.61128 −0.805639 0.592407i \(-0.798178\pi\)
−0.805639 + 0.592407i \(0.798178\pi\)
\(348\) 0 0
\(349\) 14.7340 0.788693 0.394347 0.918962i \(-0.370971\pi\)
0.394347 + 0.918962i \(0.370971\pi\)
\(350\) 0 0
\(351\) −11.6638 −0.622570
\(352\) 0 0
\(353\) −29.6302 −1.57705 −0.788527 0.615000i \(-0.789156\pi\)
−0.788527 + 0.615000i \(0.789156\pi\)
\(354\) 0 0
\(355\) 7.14638 0.379290
\(356\) 0 0
\(357\) 69.3195 3.66878
\(358\) 0 0
\(359\) 21.7577 1.14833 0.574163 0.818741i \(-0.305328\pi\)
0.574163 + 0.818741i \(0.305328\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −28.1484 −1.47741
\(364\) 0 0
\(365\) 0.243310 0.0127355
\(366\) 0 0
\(367\) 30.0347 1.56780 0.783898 0.620889i \(-0.213228\pi\)
0.783898 + 0.620889i \(0.213228\pi\)
\(368\) 0 0
\(369\) 14.9031 0.775823
\(370\) 0 0
\(371\) 8.05227 0.418053
\(372\) 0 0
\(373\) 37.3055 1.93161 0.965803 0.259277i \(-0.0834843\pi\)
0.965803 + 0.259277i \(0.0834843\pi\)
\(374\) 0 0
\(375\) 2.87834 0.148637
\(376\) 0 0
\(377\) 13.7567 0.708506
\(378\) 0 0
\(379\) 14.7794 0.759166 0.379583 0.925158i \(-0.376068\pi\)
0.379583 + 0.925158i \(0.376068\pi\)
\(380\) 0 0
\(381\) 13.3412 0.683492
\(382\) 0 0
\(383\) −29.0020 −1.48193 −0.740967 0.671541i \(-0.765633\pi\)
−0.740967 + 0.671541i \(0.765633\pi\)
\(384\) 0 0
\(385\) 3.43026 0.174822
\(386\) 0 0
\(387\) −16.4086 −0.834094
\(388\) 0 0
\(389\) −21.1987 −1.07481 −0.537407 0.843323i \(-0.680596\pi\)
−0.537407 + 0.843323i \(0.680596\pi\)
\(390\) 0 0
\(391\) −51.5965 −2.60935
\(392\) 0 0
\(393\) −44.6529 −2.25244
\(394\) 0 0
\(395\) −9.38969 −0.472446
\(396\) 0 0
\(397\) −10.3856 −0.521238 −0.260619 0.965442i \(-0.583927\pi\)
−0.260619 + 0.965442i \(0.583927\pi\)
\(398\) 0 0
\(399\) −8.93674 −0.447397
\(400\) 0 0
\(401\) 25.6638 1.28159 0.640796 0.767712i \(-0.278605\pi\)
0.640796 + 0.767712i \(0.278605\pi\)
\(402\) 0 0
\(403\) −11.6638 −0.581017
\(404\) 0 0
\(405\) 3.07523 0.152809
\(406\) 0 0
\(407\) 1.55395 0.0770266
\(408\) 0 0
\(409\) 5.71611 0.282644 0.141322 0.989964i \(-0.454865\pi\)
0.141322 + 0.989964i \(0.454865\pi\)
\(410\) 0 0
\(411\) −2.34023 −0.115435
\(412\) 0 0
\(413\) 16.7340 0.823427
\(414\) 0 0
\(415\) −8.86151 −0.434994
\(416\) 0 0
\(417\) 36.7526 1.79978
\(418\) 0 0
\(419\) −15.4303 −0.753818 −0.376909 0.926250i \(-0.623013\pi\)
−0.376909 + 0.926250i \(0.623013\pi\)
\(420\) 0 0
\(421\) 1.18005 0.0575121 0.0287561 0.999586i \(-0.490845\pi\)
0.0287561 + 0.999586i \(0.490845\pi\)
\(422\) 0 0
\(423\) 7.74190 0.376424
\(424\) 0 0
\(425\) 7.75669 0.376255
\(426\) 0 0
\(427\) −12.6529 −0.612315
\(428\) 0 0
\(429\) −5.63990 −0.272297
\(430\) 0 0
\(431\) −14.0255 −0.675585 −0.337792 0.941221i \(-0.609680\pi\)
−0.337792 + 0.941221i \(0.609680\pi\)
\(432\) 0 0
\(433\) 25.5894 1.22975 0.614874 0.788625i \(-0.289207\pi\)
0.614874 + 0.788625i \(0.289207\pi\)
\(434\) 0 0
\(435\) −22.3264 −1.07047
\(436\) 0 0
\(437\) 6.65187 0.318202
\(438\) 0 0
\(439\) −21.9732 −1.04873 −0.524363 0.851495i \(-0.675696\pi\)
−0.524363 + 0.851495i \(0.675696\pi\)
\(440\) 0 0
\(441\) 13.9515 0.664358
\(442\) 0 0
\(443\) 19.3721 0.920395 0.460197 0.887817i \(-0.347779\pi\)
0.460197 + 0.887817i \(0.347779\pi\)
\(444\) 0 0
\(445\) 0.813048 0.0385422
\(446\) 0 0
\(447\) 39.7161 1.87851
\(448\) 0 0
\(449\) −19.1841 −0.905355 −0.452677 0.891674i \(-0.649531\pi\)
−0.452677 + 0.891674i \(0.649531\pi\)
\(450\) 0 0
\(451\) 3.11553 0.146705
\(452\) 0 0
\(453\) 14.4570 0.679250
\(454\) 0 0
\(455\) −5.50648 −0.258148
\(456\) 0 0
\(457\) 8.36010 0.391069 0.195534 0.980697i \(-0.437356\pi\)
0.195534 + 0.980697i \(0.437356\pi\)
\(458\) 0 0
\(459\) 51.0130 2.38108
\(460\) 0 0
\(461\) 25.0268 1.16561 0.582806 0.812611i \(-0.301955\pi\)
0.582806 + 0.812611i \(0.301955\pi\)
\(462\) 0 0
\(463\) 32.3749 1.50459 0.752294 0.658827i \(-0.228947\pi\)
0.752294 + 0.658827i \(0.228947\pi\)
\(464\) 0 0
\(465\) 18.9298 0.877850
\(466\) 0 0
\(467\) −15.2216 −0.704372 −0.352186 0.935930i \(-0.614562\pi\)
−0.352186 + 0.935930i \(0.614562\pi\)
\(468\) 0 0
\(469\) 49.1030 2.26736
\(470\) 0 0
\(471\) −20.6865 −0.953185
\(472\) 0 0
\(473\) −3.43026 −0.157724
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 13.7062 0.627563
\(478\) 0 0
\(479\) −20.0485 −0.916038 −0.458019 0.888943i \(-0.651441\pi\)
−0.458019 + 0.888943i \(0.651441\pi\)
\(480\) 0 0
\(481\) −2.49451 −0.113740
\(482\) 0 0
\(483\) −59.4460 −2.70489
\(484\) 0 0
\(485\) 3.19689 0.145163
\(486\) 0 0
\(487\) −31.6508 −1.43424 −0.717118 0.696952i \(-0.754539\pi\)
−0.717118 + 0.696952i \(0.754539\pi\)
\(488\) 0 0
\(489\) −20.8832 −0.944371
\(490\) 0 0
\(491\) −24.8033 −1.11936 −0.559679 0.828710i \(-0.689076\pi\)
−0.559679 + 0.828710i \(0.689076\pi\)
\(492\) 0 0
\(493\) −60.1662 −2.70975
\(494\) 0 0
\(495\) 5.83882 0.262436
\(496\) 0 0
\(497\) 22.1882 0.995277
\(498\) 0 0
\(499\) 33.0237 1.47834 0.739171 0.673518i \(-0.235217\pi\)
0.739171 + 0.673518i \(0.235217\pi\)
\(500\) 0 0
\(501\) −21.1020 −0.942767
\(502\) 0 0
\(503\) −3.54396 −0.158017 −0.0790087 0.996874i \(-0.525175\pi\)
−0.0790087 + 0.996874i \(0.525175\pi\)
\(504\) 0 0
\(505\) 3.01887 0.134338
\(506\) 0 0
\(507\) −28.3650 −1.25973
\(508\) 0 0
\(509\) 2.11679 0.0938250 0.0469125 0.998899i \(-0.485062\pi\)
0.0469125 + 0.998899i \(0.485062\pi\)
\(510\) 0 0
\(511\) 0.755435 0.0334185
\(512\) 0 0
\(513\) −6.57664 −0.290366
\(514\) 0 0
\(515\) 6.05839 0.266965
\(516\) 0 0
\(517\) 1.61847 0.0711802
\(518\) 0 0
\(519\) 16.6182 0.729458
\(520\) 0 0
\(521\) −8.60748 −0.377101 −0.188550 0.982064i \(-0.560379\pi\)
−0.188550 + 0.982064i \(0.560379\pi\)
\(522\) 0 0
\(523\) −36.8349 −1.61068 −0.805340 0.592814i \(-0.798017\pi\)
−0.805340 + 0.592814i \(0.798017\pi\)
\(524\) 0 0
\(525\) 8.93674 0.390031
\(526\) 0 0
\(527\) 51.0130 2.22216
\(528\) 0 0
\(529\) 21.2474 0.923799
\(530\) 0 0
\(531\) 28.4838 1.23609
\(532\) 0 0
\(533\) −5.00125 −0.216628
\(534\) 0 0
\(535\) −20.3848 −0.881313
\(536\) 0 0
\(537\) 12.9059 0.556931
\(538\) 0 0
\(539\) 2.91661 0.125627
\(540\) 0 0
\(541\) 32.0079 1.37613 0.688063 0.725651i \(-0.258461\pi\)
0.688063 + 0.725651i \(0.258461\pi\)
\(542\) 0 0
\(543\) 7.86941 0.337709
\(544\) 0 0
\(545\) −4.72710 −0.202487
\(546\) 0 0
\(547\) −17.6240 −0.753550 −0.376775 0.926305i \(-0.622967\pi\)
−0.376775 + 0.926305i \(0.622967\pi\)
\(548\) 0 0
\(549\) −21.5371 −0.919179
\(550\) 0 0
\(551\) 7.75669 0.330446
\(552\) 0 0
\(553\) −29.1533 −1.23972
\(554\) 0 0
\(555\) 4.04846 0.171848
\(556\) 0 0
\(557\) 34.6865 1.46972 0.734858 0.678221i \(-0.237249\pi\)
0.734858 + 0.678221i \(0.237249\pi\)
\(558\) 0 0
\(559\) 5.50648 0.232899
\(560\) 0 0
\(561\) 24.6667 1.04143
\(562\) 0 0
\(563\) −13.5073 −0.569263 −0.284632 0.958637i \(-0.591871\pi\)
−0.284632 + 0.958637i \(0.591871\pi\)
\(564\) 0 0
\(565\) −7.98316 −0.335854
\(566\) 0 0
\(567\) 9.54803 0.400980
\(568\) 0 0
\(569\) 32.3588 1.35655 0.678276 0.734807i \(-0.262727\pi\)
0.678276 + 0.734807i \(0.262727\pi\)
\(570\) 0 0
\(571\) 8.93293 0.373831 0.186916 0.982376i \(-0.440151\pi\)
0.186916 + 0.982376i \(0.440151\pi\)
\(572\) 0 0
\(573\) −51.0130 −2.13110
\(574\) 0 0
\(575\) −6.65187 −0.277402
\(576\) 0 0
\(577\) 14.9059 0.620541 0.310270 0.950648i \(-0.399580\pi\)
0.310270 + 0.950648i \(0.399580\pi\)
\(578\) 0 0
\(579\) −38.9645 −1.61931
\(580\) 0 0
\(581\) −27.5134 −1.14145
\(582\) 0 0
\(583\) 2.86532 0.118669
\(584\) 0 0
\(585\) −9.37285 −0.387520
\(586\) 0 0
\(587\) −7.37207 −0.304278 −0.152139 0.988359i \(-0.548616\pi\)
−0.152139 + 0.988359i \(0.548616\pi\)
\(588\) 0 0
\(589\) −6.57664 −0.270986
\(590\) 0 0
\(591\) 61.9229 2.54717
\(592\) 0 0
\(593\) −17.6853 −0.726247 −0.363124 0.931741i \(-0.618290\pi\)
−0.363124 + 0.931741i \(0.618290\pi\)
\(594\) 0 0
\(595\) 24.0831 0.987312
\(596\) 0 0
\(597\) 20.4193 0.835705
\(598\) 0 0
\(599\) 5.30657 0.216821 0.108410 0.994106i \(-0.465424\pi\)
0.108410 + 0.994106i \(0.465424\pi\)
\(600\) 0 0
\(601\) 43.7875 1.78613 0.893065 0.449927i \(-0.148550\pi\)
0.893065 + 0.449927i \(0.148550\pi\)
\(602\) 0 0
\(603\) 83.5806 3.40367
\(604\) 0 0
\(605\) −9.77938 −0.397588
\(606\) 0 0
\(607\) −7.24535 −0.294080 −0.147040 0.989131i \(-0.546975\pi\)
−0.147040 + 0.989131i \(0.546975\pi\)
\(608\) 0 0
\(609\) −69.3195 −2.80897
\(610\) 0 0
\(611\) −2.59807 −0.105107
\(612\) 0 0
\(613\) −20.8962 −0.843988 −0.421994 0.906599i \(-0.638670\pi\)
−0.421994 + 0.906599i \(0.638670\pi\)
\(614\) 0 0
\(615\) 8.11679 0.327301
\(616\) 0 0
\(617\) −22.0494 −0.887677 −0.443839 0.896107i \(-0.646384\pi\)
−0.443839 + 0.896107i \(0.646384\pi\)
\(618\) 0 0
\(619\) 0.0484607 0.00194780 0.000973900 1.00000i \(-0.499690\pi\)
0.000973900 1.00000i \(0.499690\pi\)
\(620\) 0 0
\(621\) −43.7470 −1.75550
\(622\) 0 0
\(623\) 2.52437 0.101137
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.18005 −0.126999
\(628\) 0 0
\(629\) 10.9100 0.435009
\(630\) 0 0
\(631\) −32.6182 −1.29851 −0.649255 0.760571i \(-0.724919\pi\)
−0.649255 + 0.760571i \(0.724919\pi\)
\(632\) 0 0
\(633\) −31.8357 −1.26536
\(634\) 0 0
\(635\) 4.63503 0.183936
\(636\) 0 0
\(637\) −4.68193 −0.185505
\(638\) 0 0
\(639\) 37.7677 1.49407
\(640\) 0 0
\(641\) −13.4136 −0.529806 −0.264903 0.964275i \(-0.585340\pi\)
−0.264903 + 0.964275i \(0.585340\pi\)
\(642\) 0 0
\(643\) −23.8052 −0.938783 −0.469392 0.882990i \(-0.655527\pi\)
−0.469392 + 0.882990i \(0.655527\pi\)
\(644\) 0 0
\(645\) −8.93674 −0.351884
\(646\) 0 0
\(647\) −48.4580 −1.90508 −0.952540 0.304412i \(-0.901540\pi\)
−0.952540 + 0.304412i \(0.901540\pi\)
\(648\) 0 0
\(649\) 5.95463 0.233740
\(650\) 0 0
\(651\) 58.7737 2.30352
\(652\) 0 0
\(653\) −22.2351 −0.870128 −0.435064 0.900399i \(-0.643274\pi\)
−0.435064 + 0.900399i \(0.643274\pi\)
\(654\) 0 0
\(655\) −15.5134 −0.606158
\(656\) 0 0
\(657\) 1.28586 0.0501663
\(658\) 0 0
\(659\) 10.7072 0.417095 0.208547 0.978012i \(-0.433126\pi\)
0.208547 + 0.978012i \(0.433126\pi\)
\(660\) 0 0
\(661\) 44.7980 1.74244 0.871220 0.490893i \(-0.163330\pi\)
0.871220 + 0.490893i \(0.163330\pi\)
\(662\) 0 0
\(663\) −39.5965 −1.53780
\(664\) 0 0
\(665\) −3.10482 −0.120400
\(666\) 0 0
\(667\) 51.5965 1.99782
\(668\) 0 0
\(669\) −36.3680 −1.40607
\(670\) 0 0
\(671\) −4.50239 −0.173813
\(672\) 0 0
\(673\) −43.2772 −1.66821 −0.834106 0.551604i \(-0.814016\pi\)
−0.834106 + 0.551604i \(0.814016\pi\)
\(674\) 0 0
\(675\) 6.57664 0.253135
\(676\) 0 0
\(677\) −11.5330 −0.443251 −0.221625 0.975132i \(-0.571136\pi\)
−0.221625 + 0.975132i \(0.571136\pi\)
\(678\) 0 0
\(679\) 9.92575 0.380915
\(680\) 0 0
\(681\) 19.0780 0.731072
\(682\) 0 0
\(683\) −0.916090 −0.0350532 −0.0175266 0.999846i \(-0.505579\pi\)
−0.0175266 + 0.999846i \(0.505579\pi\)
\(684\) 0 0
\(685\) −0.813048 −0.0310650
\(686\) 0 0
\(687\) 0.216540 0.00826153
\(688\) 0 0
\(689\) −4.59960 −0.175231
\(690\) 0 0
\(691\) −14.8278 −0.564077 −0.282039 0.959403i \(-0.591011\pi\)
−0.282039 + 0.959403i \(0.591011\pi\)
\(692\) 0 0
\(693\) 18.1285 0.688644
\(694\) 0 0
\(695\) 12.7687 0.484343
\(696\) 0 0
\(697\) 21.8735 0.828517
\(698\) 0 0
\(699\) 27.3828 1.03571
\(700\) 0 0
\(701\) −36.1722 −1.36620 −0.683102 0.730323i \(-0.739369\pi\)
−0.683102 + 0.730323i \(0.739369\pi\)
\(702\) 0 0
\(703\) −1.40652 −0.0530481
\(704\) 0 0
\(705\) 4.21654 0.158804
\(706\) 0 0
\(707\) 9.37305 0.352510
\(708\) 0 0
\(709\) −10.4331 −0.391823 −0.195911 0.980622i \(-0.562766\pi\)
−0.195911 + 0.980622i \(0.562766\pi\)
\(710\) 0 0
\(711\) −49.6233 −1.86102
\(712\) 0 0
\(713\) −43.7470 −1.63834
\(714\) 0 0
\(715\) −1.95942 −0.0732783
\(716\) 0 0
\(717\) −8.43308 −0.314939
\(718\) 0 0
\(719\) 7.73373 0.288420 0.144210 0.989547i \(-0.453936\pi\)
0.144210 + 0.989547i \(0.453936\pi\)
\(720\) 0 0
\(721\) 18.8102 0.700529
\(722\) 0 0
\(723\) −27.1354 −1.00918
\(724\) 0 0
\(725\) −7.75669 −0.288076
\(726\) 0 0
\(727\) −39.5241 −1.46587 −0.732934 0.680300i \(-0.761849\pi\)
−0.732934 + 0.680300i \(0.761849\pi\)
\(728\) 0 0
\(729\) −40.5373 −1.50138
\(730\) 0 0
\(731\) −24.0831 −0.890746
\(732\) 0 0
\(733\) −50.7498 −1.87449 −0.937243 0.348677i \(-0.886631\pi\)
−0.937243 + 0.348677i \(0.886631\pi\)
\(734\) 0 0
\(735\) 7.59854 0.280277
\(736\) 0 0
\(737\) 17.4728 0.643619
\(738\) 0 0
\(739\) −0.419276 −0.0154233 −0.00771165 0.999970i \(-0.502455\pi\)
−0.00771165 + 0.999970i \(0.502455\pi\)
\(740\) 0 0
\(741\) 5.10482 0.187530
\(742\) 0 0
\(743\) 19.5511 0.717259 0.358630 0.933480i \(-0.383244\pi\)
0.358630 + 0.933480i \(0.383244\pi\)
\(744\) 0 0
\(745\) 13.7982 0.505529
\(746\) 0 0
\(747\) −46.8319 −1.71349
\(748\) 0 0
\(749\) −63.2912 −2.31261
\(750\) 0 0
\(751\) 8.76768 0.319937 0.159969 0.987122i \(-0.448861\pi\)
0.159969 + 0.987122i \(0.448861\pi\)
\(752\) 0 0
\(753\) 42.9732 1.56603
\(754\) 0 0
\(755\) 5.02269 0.182794
\(756\) 0 0
\(757\) 48.0535 1.74653 0.873267 0.487241i \(-0.161997\pi\)
0.873267 + 0.487241i \(0.161997\pi\)
\(758\) 0 0
\(759\) −21.1533 −0.767815
\(760\) 0 0
\(761\) 29.3947 1.06556 0.532779 0.846254i \(-0.321148\pi\)
0.532779 + 0.846254i \(0.321148\pi\)
\(762\) 0 0
\(763\) −14.6768 −0.531336
\(764\) 0 0
\(765\) 40.9931 1.48211
\(766\) 0 0
\(767\) −9.55875 −0.345146
\(768\) 0 0
\(769\) −13.3790 −0.482458 −0.241229 0.970468i \(-0.577551\pi\)
−0.241229 + 0.970468i \(0.577551\pi\)
\(770\) 0 0
\(771\) −43.5480 −1.56834
\(772\) 0 0
\(773\) −0.133625 −0.00480617 −0.00240309 0.999997i \(-0.500765\pi\)
−0.00240309 + 0.999997i \(0.500765\pi\)
\(774\) 0 0
\(775\) 6.57664 0.236240
\(776\) 0 0
\(777\) 12.5697 0.450937
\(778\) 0 0
\(779\) −2.81995 −0.101035
\(780\) 0 0
\(781\) 7.89545 0.282522
\(782\) 0 0
\(783\) −51.0130 −1.82305
\(784\) 0 0
\(785\) −7.18695 −0.256513
\(786\) 0 0
\(787\) 4.84060 0.172549 0.0862744 0.996271i \(-0.472504\pi\)
0.0862744 + 0.996271i \(0.472504\pi\)
\(788\) 0 0
\(789\) −43.8130 −1.55979
\(790\) 0 0
\(791\) −24.7863 −0.881299
\(792\) 0 0
\(793\) 7.22753 0.256657
\(794\) 0 0
\(795\) 7.46492 0.264753
\(796\) 0 0
\(797\) −11.6993 −0.414410 −0.207205 0.978298i \(-0.566437\pi\)
−0.207205 + 0.978298i \(0.566437\pi\)
\(798\) 0 0
\(799\) 11.3629 0.401991
\(800\) 0 0
\(801\) 4.29685 0.151822
\(802\) 0 0
\(803\) 0.268814 0.00948624
\(804\) 0 0
\(805\) −20.6529 −0.727917
\(806\) 0 0
\(807\) −33.1593 −1.16726
\(808\) 0 0
\(809\) −33.1987 −1.16720 −0.583601 0.812040i \(-0.698357\pi\)
−0.583601 + 0.812040i \(0.698357\pi\)
\(810\) 0 0
\(811\) 26.5766 0.933232 0.466616 0.884460i \(-0.345473\pi\)
0.466616 + 0.884460i \(0.345473\pi\)
\(812\) 0 0
\(813\) 6.22061 0.218166
\(814\) 0 0
\(815\) −7.25528 −0.254141
\(816\) 0 0
\(817\) 3.10482 0.108624
\(818\) 0 0
\(819\) −29.1010 −1.01687
\(820\) 0 0
\(821\) −25.5807 −0.892773 −0.446387 0.894840i \(-0.647289\pi\)
−0.446387 + 0.894840i \(0.647289\pi\)
\(822\) 0 0
\(823\) 12.9783 0.452395 0.226198 0.974081i \(-0.427371\pi\)
0.226198 + 0.974081i \(0.427371\pi\)
\(824\) 0 0
\(825\) 3.18005 0.110715
\(826\) 0 0
\(827\) 12.5167 0.435248 0.217624 0.976033i \(-0.430169\pi\)
0.217624 + 0.976033i \(0.430169\pi\)
\(828\) 0 0
\(829\) 19.4391 0.675149 0.337574 0.941299i \(-0.390394\pi\)
0.337574 + 0.941299i \(0.390394\pi\)
\(830\) 0 0
\(831\) −67.4124 −2.33851
\(832\) 0 0
\(833\) 20.4769 0.709482
\(834\) 0 0
\(835\) −7.33129 −0.253710
\(836\) 0 0
\(837\) 43.2522 1.49501
\(838\) 0 0
\(839\) −43.9972 −1.51895 −0.759475 0.650536i \(-0.774544\pi\)
−0.759475 + 0.650536i \(0.774544\pi\)
\(840\) 0 0
\(841\) 31.1662 1.07470
\(842\) 0 0
\(843\) 73.9229 2.54604
\(844\) 0 0
\(845\) −9.85461 −0.339009
\(846\) 0 0
\(847\) −30.3632 −1.04329
\(848\) 0 0
\(849\) −21.2591 −0.729610
\(850\) 0 0
\(851\) −9.35601 −0.320720
\(852\) 0 0
\(853\) −3.30374 −0.113118 −0.0565590 0.998399i \(-0.518013\pi\)
−0.0565590 + 0.998399i \(0.518013\pi\)
\(854\) 0 0
\(855\) −5.28487 −0.180739
\(856\) 0 0
\(857\) 6.02374 0.205767 0.102883 0.994693i \(-0.467193\pi\)
0.102883 + 0.994693i \(0.467193\pi\)
\(858\) 0 0
\(859\) 36.0158 1.22884 0.614421 0.788978i \(-0.289390\pi\)
0.614421 + 0.788978i \(0.289390\pi\)
\(860\) 0 0
\(861\) 25.2012 0.858853
\(862\) 0 0
\(863\) 29.9250 1.01866 0.509329 0.860572i \(-0.329894\pi\)
0.509329 + 0.860572i \(0.329894\pi\)
\(864\) 0 0
\(865\) 5.77353 0.196306
\(866\) 0 0
\(867\) 124.247 4.21966
\(868\) 0 0
\(869\) −10.3739 −0.351911
\(870\) 0 0
\(871\) −28.0485 −0.950386
\(872\) 0 0
\(873\) 16.8951 0.571813
\(874\) 0 0
\(875\) 3.10482 0.104962
\(876\) 0 0
\(877\) −0.618980 −0.0209015 −0.0104507 0.999945i \(-0.503327\pi\)
−0.0104507 + 0.999945i \(0.503327\pi\)
\(878\) 0 0
\(879\) 69.5184 2.34480
\(880\) 0 0
\(881\) −21.2423 −0.715672 −0.357836 0.933784i \(-0.616485\pi\)
−0.357836 + 0.933784i \(0.616485\pi\)
\(882\) 0 0
\(883\) 48.8971 1.64552 0.822760 0.568389i \(-0.192433\pi\)
0.822760 + 0.568389i \(0.192433\pi\)
\(884\) 0 0
\(885\) 15.5134 0.521477
\(886\) 0 0
\(887\) 58.2797 1.95684 0.978421 0.206622i \(-0.0662469\pi\)
0.978421 + 0.206622i \(0.0662469\pi\)
\(888\) 0 0
\(889\) 14.3909 0.482657
\(890\) 0 0
\(891\) 3.39757 0.113823
\(892\) 0 0
\(893\) −1.46492 −0.0490216
\(894\) 0 0
\(895\) 4.48379 0.149877
\(896\) 0 0
\(897\) 33.9566 1.13378
\(898\) 0 0
\(899\) −51.0130 −1.70138
\(900\) 0 0
\(901\) 20.1168 0.670187
\(902\) 0 0
\(903\) −27.7470 −0.923361
\(904\) 0 0
\(905\) 2.73400 0.0908814
\(906\) 0 0
\(907\) 36.5154 1.21247 0.606237 0.795284i \(-0.292678\pi\)
0.606237 + 0.795284i \(0.292678\pi\)
\(908\) 0 0
\(909\) 15.9543 0.529172
\(910\) 0 0
\(911\) 6.62735 0.219574 0.109787 0.993955i \(-0.464983\pi\)
0.109787 + 0.993955i \(0.464983\pi\)
\(912\) 0 0
\(913\) −9.79036 −0.324014
\(914\) 0 0
\(915\) −11.7299 −0.387779
\(916\) 0 0
\(917\) −48.1662 −1.59059
\(918\) 0 0
\(919\) 7.91688 0.261154 0.130577 0.991438i \(-0.458317\pi\)
0.130577 + 0.991438i \(0.458317\pi\)
\(920\) 0 0
\(921\) −12.6210 −0.415877
\(922\) 0 0
\(923\) −12.6743 −0.417179
\(924\) 0 0
\(925\) 1.40652 0.0462462
\(926\) 0 0
\(927\) 32.0178 1.05160
\(928\) 0 0
\(929\) 35.8597 1.17652 0.588259 0.808673i \(-0.299814\pi\)
0.588259 + 0.808673i \(0.299814\pi\)
\(930\) 0 0
\(931\) −2.63990 −0.0865192
\(932\) 0 0
\(933\) −45.2254 −1.48061
\(934\) 0 0
\(935\) 8.56974 0.280260
\(936\) 0 0
\(937\) −11.4420 −0.373793 −0.186896 0.982380i \(-0.559843\pi\)
−0.186896 + 0.982380i \(0.559843\pi\)
\(938\) 0 0
\(939\) 71.3236 2.32756
\(940\) 0 0
\(941\) 0.360100 0.0117389 0.00586945 0.999983i \(-0.498132\pi\)
0.00586945 + 0.999983i \(0.498132\pi\)
\(942\) 0 0
\(943\) −18.7579 −0.610843
\(944\) 0 0
\(945\) 20.4193 0.664239
\(946\) 0 0
\(947\) −2.63807 −0.0857256 −0.0428628 0.999081i \(-0.513648\pi\)
−0.0428628 + 0.999081i \(0.513648\pi\)
\(948\) 0 0
\(949\) −0.431517 −0.0140076
\(950\) 0 0
\(951\) 78.8084 2.55554
\(952\) 0 0
\(953\) 10.4430 0.338282 0.169141 0.985592i \(-0.445901\pi\)
0.169141 + 0.985592i \(0.445901\pi\)
\(954\) 0 0
\(955\) −17.7230 −0.573503
\(956\) 0 0
\(957\) −24.6667 −0.797360
\(958\) 0 0
\(959\) −2.52437 −0.0815160
\(960\) 0 0
\(961\) 12.2522 0.395232
\(962\) 0 0
\(963\) −107.731 −3.47159
\(964\) 0 0
\(965\) −13.5371 −0.435775
\(966\) 0 0
\(967\) 33.2732 1.06999 0.534996 0.844854i \(-0.320313\pi\)
0.534996 + 0.844854i \(0.320313\pi\)
\(968\) 0 0
\(969\) −22.3264 −0.717228
\(970\) 0 0
\(971\) −23.5657 −0.756258 −0.378129 0.925753i \(-0.623432\pi\)
−0.378129 + 0.925753i \(0.623432\pi\)
\(972\) 0 0
\(973\) 39.6444 1.27094
\(974\) 0 0
\(975\) −5.10482 −0.163485
\(976\) 0 0
\(977\) 0.402439 0.0128752 0.00643759 0.999979i \(-0.497951\pi\)
0.00643759 + 0.999979i \(0.497951\pi\)
\(978\) 0 0
\(979\) 0.898271 0.0287089
\(980\) 0 0
\(981\) −24.9821 −0.797617
\(982\) 0 0
\(983\) 11.4927 0.366561 0.183281 0.983061i \(-0.441328\pi\)
0.183281 + 0.983061i \(0.441328\pi\)
\(984\) 0 0
\(985\) 21.5134 0.685473
\(986\) 0 0
\(987\) 13.0916 0.416710
\(988\) 0 0
\(989\) 20.6529 0.656723
\(990\) 0 0
\(991\) 6.05761 0.192426 0.0962132 0.995361i \(-0.469327\pi\)
0.0962132 + 0.995361i \(0.469327\pi\)
\(992\) 0 0
\(993\) −13.5292 −0.429335
\(994\) 0 0
\(995\) 7.09410 0.224898
\(996\) 0 0
\(997\) 48.1086 1.52362 0.761808 0.647803i \(-0.224312\pi\)
0.761808 + 0.647803i \(0.224312\pi\)
\(998\) 0 0
\(999\) 9.25020 0.292663
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 6080.2.a.cc.1.4 4
4.3 odd 2 6080.2.a.ch.1.1 4
8.3 odd 2 1520.2.a.t.1.4 4
8.5 even 2 95.2.a.b.1.2 4
24.5 odd 2 855.2.a.m.1.3 4
40.13 odd 4 475.2.b.e.324.7 8
40.19 odd 2 7600.2.a.cf.1.1 4
40.29 even 2 475.2.a.i.1.3 4
40.37 odd 4 475.2.b.e.324.2 8
56.13 odd 2 4655.2.a.y.1.2 4
120.29 odd 2 4275.2.a.bo.1.2 4
152.37 odd 2 1805.2.a.p.1.3 4
760.189 odd 2 9025.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.2 4 8.5 even 2
475.2.a.i.1.3 4 40.29 even 2
475.2.b.e.324.2 8 40.37 odd 4
475.2.b.e.324.7 8 40.13 odd 4
855.2.a.m.1.3 4 24.5 odd 2
1520.2.a.t.1.4 4 8.3 odd 2
1805.2.a.p.1.3 4 152.37 odd 2
4275.2.a.bo.1.2 4 120.29 odd 2
4655.2.a.y.1.2 4 56.13 odd 2
6080.2.a.cc.1.4 4 1.1 even 1 trivial
6080.2.a.ch.1.1 4 4.3 odd 2
7600.2.a.cf.1.1 4 40.19 odd 2
9025.2.a.bf.1.2 4 760.189 odd 2