Properties

Label 2320.2.a.s.1.2
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $3$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.806063 q^{3} -1.00000 q^{5} +4.15633 q^{7} -2.35026 q^{9} +O(q^{10})\) \(q+0.806063 q^{3} -1.00000 q^{5} +4.15633 q^{7} -2.35026 q^{9} -2.80606 q^{11} +1.35026 q^{13} -0.806063 q^{15} -7.11871 q^{17} -3.76845 q^{19} +3.35026 q^{21} -4.80606 q^{23} +1.00000 q^{25} -4.31265 q^{27} +1.00000 q^{29} -0.231548 q^{31} -2.26187 q^{33} -4.15633 q^{35} -5.50659 q^{37} +1.08840 q^{39} -6.96239 q^{41} +3.19394 q^{43} +2.35026 q^{45} -6.41819 q^{47} +10.2750 q^{49} -5.73813 q^{51} +6.96239 q^{53} +2.80606 q^{55} -3.03761 q^{57} +2.57452 q^{59} +5.35026 q^{61} -9.76845 q^{63} -1.35026 q^{65} +3.19394 q^{67} -3.87399 q^{69} -11.3503 q^{71} -11.2447 q^{73} +0.806063 q^{75} -11.6629 q^{77} +4.73084 q^{79} +3.57452 q^{81} -2.54420 q^{83} +7.11871 q^{85} +0.806063 q^{87} +14.3127 q^{89} +5.61213 q^{91} -0.186642 q^{93} +3.76845 q^{95} -1.53102 q^{97} +6.59498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} - 6 q^{13} - 2 q^{15} - 14 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 12 q^{31} - 16 q^{33} - 2 q^{35} + 4 q^{37} - 16 q^{39} - 10 q^{41} + 10 q^{43} - 3 q^{45} - 18 q^{47} - q^{49} - 8 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 4 q^{59} + 6 q^{61} - 18 q^{63} + 6 q^{65} + 10 q^{67} - 20 q^{69} - 24 q^{71} - 4 q^{73} + 2 q^{75} - 4 q^{77} - 8 q^{79} - q^{81} + 2 q^{83} + 2 q^{87} + 22 q^{89} + 16 q^{91} + 12 q^{93} - 36 q^{97} - 20 q^{99}+O(q^{100}) \) 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
\(3\) 0.806063 0.465381 0.232690 0.972551i \(-0.425247\pi\)
0.232690 + 0.972551i \(0.425247\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.15633 1.57094 0.785472 0.618898i \(-0.212420\pi\)
0.785472 + 0.618898i \(0.212420\pi\)
\(8\) 0 0
\(9\) −2.35026 −0.783421
\(10\) 0 0
\(11\) −2.80606 −0.846060 −0.423030 0.906116i \(-0.639034\pi\)
−0.423030 + 0.906116i \(0.639034\pi\)
\(12\) 0 0
\(13\) 1.35026 0.374495 0.187248 0.982313i \(-0.440043\pi\)
0.187248 + 0.982313i \(0.440043\pi\)
\(14\) 0 0
\(15\) −0.806063 −0.208125
\(16\) 0 0
\(17\) −7.11871 −1.72654 −0.863271 0.504741i \(-0.831588\pi\)
−0.863271 + 0.504741i \(0.831588\pi\)
\(18\) 0 0
\(19\) −3.76845 −0.864542 −0.432271 0.901744i \(-0.642288\pi\)
−0.432271 + 0.901744i \(0.642288\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) 0 0
\(23\) −4.80606 −1.00213 −0.501067 0.865409i \(-0.667059\pi\)
−0.501067 + 0.865409i \(0.667059\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.31265 −0.829970
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.231548 −0.0415872 −0.0207936 0.999784i \(-0.506619\pi\)
−0.0207936 + 0.999784i \(0.506619\pi\)
\(32\) 0 0
\(33\) −2.26187 −0.393740
\(34\) 0 0
\(35\) −4.15633 −0.702547
\(36\) 0 0
\(37\) −5.50659 −0.905277 −0.452639 0.891694i \(-0.649517\pi\)
−0.452639 + 0.891694i \(0.649517\pi\)
\(38\) 0 0
\(39\) 1.08840 0.174283
\(40\) 0 0
\(41\) −6.96239 −1.08734 −0.543671 0.839298i \(-0.682966\pi\)
−0.543671 + 0.839298i \(0.682966\pi\)
\(42\) 0 0
\(43\) 3.19394 0.487071 0.243535 0.969892i \(-0.421693\pi\)
0.243535 + 0.969892i \(0.421693\pi\)
\(44\) 0 0
\(45\) 2.35026 0.350356
\(46\) 0 0
\(47\) −6.41819 −0.936189 −0.468095 0.883678i \(-0.655059\pi\)
−0.468095 + 0.883678i \(0.655059\pi\)
\(48\) 0 0
\(49\) 10.2750 1.46786
\(50\) 0 0
\(51\) −5.73813 −0.803500
\(52\) 0 0
\(53\) 6.96239 0.956358 0.478179 0.878263i \(-0.341297\pi\)
0.478179 + 0.878263i \(0.341297\pi\)
\(54\) 0 0
\(55\) 2.80606 0.378370
\(56\) 0 0
\(57\) −3.03761 −0.402341
\(58\) 0 0
\(59\) 2.57452 0.335173 0.167587 0.985857i \(-0.446403\pi\)
0.167587 + 0.985857i \(0.446403\pi\)
\(60\) 0 0
\(61\) 5.35026 0.685031 0.342515 0.939512i \(-0.388721\pi\)
0.342515 + 0.939512i \(0.388721\pi\)
\(62\) 0 0
\(63\) −9.76845 −1.23071
\(64\) 0 0
\(65\) −1.35026 −0.167479
\(66\) 0 0
\(67\) 3.19394 0.390201 0.195101 0.980783i \(-0.437497\pi\)
0.195101 + 0.980783i \(0.437497\pi\)
\(68\) 0 0
\(69\) −3.87399 −0.466374
\(70\) 0 0
\(71\) −11.3503 −1.34703 −0.673514 0.739174i \(-0.735216\pi\)
−0.673514 + 0.739174i \(0.735216\pi\)
\(72\) 0 0
\(73\) −11.2447 −1.31610 −0.658048 0.752976i \(-0.728617\pi\)
−0.658048 + 0.752976i \(0.728617\pi\)
\(74\) 0 0
\(75\) 0.806063 0.0930762
\(76\) 0 0
\(77\) −11.6629 −1.32911
\(78\) 0 0
\(79\) 4.73084 0.532261 0.266131 0.963937i \(-0.414255\pi\)
0.266131 + 0.963937i \(0.414255\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 0 0
\(83\) −2.54420 −0.279262 −0.139631 0.990204i \(-0.544592\pi\)
−0.139631 + 0.990204i \(0.544592\pi\)
\(84\) 0 0
\(85\) 7.11871 0.772133
\(86\) 0 0
\(87\) 0.806063 0.0864191
\(88\) 0 0
\(89\) 14.3127 1.51714 0.758569 0.651593i \(-0.225899\pi\)
0.758569 + 0.651593i \(0.225899\pi\)
\(90\) 0 0
\(91\) 5.61213 0.588311
\(92\) 0 0
\(93\) −0.186642 −0.0193539
\(94\) 0 0
\(95\) 3.76845 0.386635
\(96\) 0 0
\(97\) −1.53102 −0.155452 −0.0777260 0.996975i \(-0.524766\pi\)
−0.0777260 + 0.996975i \(0.524766\pi\)
\(98\) 0 0
\(99\) 6.59498 0.662821
\(100\) 0 0
\(101\) 2.83638 0.282230 0.141115 0.989993i \(-0.454931\pi\)
0.141115 + 0.989993i \(0.454931\pi\)
\(102\) 0 0
\(103\) 9.89446 0.974930 0.487465 0.873142i \(-0.337922\pi\)
0.487465 + 0.873142i \(0.337922\pi\)
\(104\) 0 0
\(105\) −3.35026 −0.326952
\(106\) 0 0
\(107\) 11.6932 1.13043 0.565214 0.824945i \(-0.308794\pi\)
0.565214 + 0.824945i \(0.308794\pi\)
\(108\) 0 0
\(109\) −14.4993 −1.38878 −0.694390 0.719599i \(-0.744326\pi\)
−0.694390 + 0.719599i \(0.744326\pi\)
\(110\) 0 0
\(111\) −4.43866 −0.421299
\(112\) 0 0
\(113\) −16.3938 −1.54219 −0.771097 0.636717i \(-0.780292\pi\)
−0.771097 + 0.636717i \(0.780292\pi\)
\(114\) 0 0
\(115\) 4.80606 0.448168
\(116\) 0 0
\(117\) −3.17347 −0.293387
\(118\) 0 0
\(119\) −29.5877 −2.71230
\(120\) 0 0
\(121\) −3.12601 −0.284183
\(122\) 0 0
\(123\) −5.61213 −0.506028
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.3561 −1.36264 −0.681319 0.731987i \(-0.738593\pi\)
−0.681319 + 0.731987i \(0.738593\pi\)
\(128\) 0 0
\(129\) 2.57452 0.226673
\(130\) 0 0
\(131\) −1.38058 −0.120622 −0.0603109 0.998180i \(-0.519209\pi\)
−0.0603109 + 0.998180i \(0.519209\pi\)
\(132\) 0 0
\(133\) −15.6629 −1.35815
\(134\) 0 0
\(135\) 4.31265 0.371174
\(136\) 0 0
\(137\) −6.49341 −0.554770 −0.277385 0.960759i \(-0.589468\pi\)
−0.277385 + 0.960759i \(0.589468\pi\)
\(138\) 0 0
\(139\) 19.0132 1.61268 0.806338 0.591455i \(-0.201446\pi\)
0.806338 + 0.591455i \(0.201446\pi\)
\(140\) 0 0
\(141\) −5.17347 −0.435685
\(142\) 0 0
\(143\) −3.78892 −0.316845
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 8.28233 0.683115
\(148\) 0 0
\(149\) 6.62530 0.542766 0.271383 0.962471i \(-0.412519\pi\)
0.271383 + 0.962471i \(0.412519\pi\)
\(150\) 0 0
\(151\) 9.27504 0.754792 0.377396 0.926052i \(-0.376820\pi\)
0.377396 + 0.926052i \(0.376820\pi\)
\(152\) 0 0
\(153\) 16.7308 1.35261
\(154\) 0 0
\(155\) 0.231548 0.0185984
\(156\) 0 0
\(157\) −5.00729 −0.399626 −0.199813 0.979834i \(-0.564033\pi\)
−0.199813 + 0.979834i \(0.564033\pi\)
\(158\) 0 0
\(159\) 5.61213 0.445071
\(160\) 0 0
\(161\) −19.9756 −1.57429
\(162\) 0 0
\(163\) −7.50659 −0.587961 −0.293981 0.955811i \(-0.594980\pi\)
−0.293981 + 0.955811i \(0.594980\pi\)
\(164\) 0 0
\(165\) 2.26187 0.176086
\(166\) 0 0
\(167\) −21.8945 −1.69424 −0.847122 0.531398i \(-0.821667\pi\)
−0.847122 + 0.531398i \(0.821667\pi\)
\(168\) 0 0
\(169\) −11.1768 −0.859753
\(170\) 0 0
\(171\) 8.85685 0.677300
\(172\) 0 0
\(173\) 7.02302 0.533951 0.266975 0.963703i \(-0.413976\pi\)
0.266975 + 0.963703i \(0.413976\pi\)
\(174\) 0 0
\(175\) 4.15633 0.314189
\(176\) 0 0
\(177\) 2.07522 0.155983
\(178\) 0 0
\(179\) −4.77575 −0.356956 −0.178478 0.983944i \(-0.557117\pi\)
−0.178478 + 0.983944i \(0.557117\pi\)
\(180\) 0 0
\(181\) 1.87399 0.139293 0.0696464 0.997572i \(-0.477813\pi\)
0.0696464 + 0.997572i \(0.477813\pi\)
\(182\) 0 0
\(183\) 4.31265 0.318800
\(184\) 0 0
\(185\) 5.50659 0.404852
\(186\) 0 0
\(187\) 19.9756 1.46076
\(188\) 0 0
\(189\) −17.9248 −1.30384
\(190\) 0 0
\(191\) −19.1187 −1.38338 −0.691691 0.722194i \(-0.743134\pi\)
−0.691691 + 0.722194i \(0.743134\pi\)
\(192\) 0 0
\(193\) 19.8945 1.43203 0.716017 0.698083i \(-0.245963\pi\)
0.716017 + 0.698083i \(0.245963\pi\)
\(194\) 0 0
\(195\) −1.08840 −0.0779417
\(196\) 0 0
\(197\) −13.5369 −0.964464 −0.482232 0.876043i \(-0.660174\pi\)
−0.482232 + 0.876043i \(0.660174\pi\)
\(198\) 0 0
\(199\) −2.57452 −0.182503 −0.0912513 0.995828i \(-0.529087\pi\)
−0.0912513 + 0.995828i \(0.529087\pi\)
\(200\) 0 0
\(201\) 2.57452 0.181592
\(202\) 0 0
\(203\) 4.15633 0.291717
\(204\) 0 0
\(205\) 6.96239 0.486274
\(206\) 0 0
\(207\) 11.2955 0.785092
\(208\) 0 0
\(209\) 10.5745 0.731455
\(210\) 0 0
\(211\) −11.8945 −0.818848 −0.409424 0.912344i \(-0.634270\pi\)
−0.409424 + 0.912344i \(0.634270\pi\)
\(212\) 0 0
\(213\) −9.14903 −0.626881
\(214\) 0 0
\(215\) −3.19394 −0.217825
\(216\) 0 0
\(217\) −0.962389 −0.0653312
\(218\) 0 0
\(219\) −9.06396 −0.612486
\(220\) 0 0
\(221\) −9.61213 −0.646582
\(222\) 0 0
\(223\) 1.11871 0.0749146 0.0374573 0.999298i \(-0.488074\pi\)
0.0374573 + 0.999298i \(0.488074\pi\)
\(224\) 0 0
\(225\) −2.35026 −0.156684
\(226\) 0 0
\(227\) −0.0303172 −0.00201222 −0.00100611 0.999999i \(-0.500320\pi\)
−0.00100611 + 0.999999i \(0.500320\pi\)
\(228\) 0 0
\(229\) −5.84955 −0.386549 −0.193275 0.981145i \(-0.561911\pi\)
−0.193275 + 0.981145i \(0.561911\pi\)
\(230\) 0 0
\(231\) −9.40105 −0.618543
\(232\) 0 0
\(233\) 26.1016 1.70997 0.854985 0.518652i \(-0.173566\pi\)
0.854985 + 0.518652i \(0.173566\pi\)
\(234\) 0 0
\(235\) 6.41819 0.418677
\(236\) 0 0
\(237\) 3.81336 0.247704
\(238\) 0 0
\(239\) −1.42548 −0.0922069 −0.0461035 0.998937i \(-0.514680\pi\)
−0.0461035 + 0.998937i \(0.514680\pi\)
\(240\) 0 0
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) 0 0
\(243\) 15.8192 1.01480
\(244\) 0 0
\(245\) −10.2750 −0.656448
\(246\) 0 0
\(247\) −5.08840 −0.323767
\(248\) 0 0
\(249\) −2.05079 −0.129963
\(250\) 0 0
\(251\) −16.9829 −1.07195 −0.535974 0.844234i \(-0.680056\pi\)
−0.535974 + 0.844234i \(0.680056\pi\)
\(252\) 0 0
\(253\) 13.4861 0.847865
\(254\) 0 0
\(255\) 5.73813 0.359336
\(256\) 0 0
\(257\) 25.1998 1.57192 0.785961 0.618276i \(-0.212169\pi\)
0.785961 + 0.618276i \(0.212169\pi\)
\(258\) 0 0
\(259\) −22.8872 −1.42214
\(260\) 0 0
\(261\) −2.35026 −0.145478
\(262\) 0 0
\(263\) 16.1319 0.994735 0.497367 0.867540i \(-0.334300\pi\)
0.497367 + 0.867540i \(0.334300\pi\)
\(264\) 0 0
\(265\) −6.96239 −0.427696
\(266\) 0 0
\(267\) 11.5369 0.706047
\(268\) 0 0
\(269\) 5.28963 0.322514 0.161257 0.986912i \(-0.448445\pi\)
0.161257 + 0.986912i \(0.448445\pi\)
\(270\) 0 0
\(271\) −1.13330 −0.0688432 −0.0344216 0.999407i \(-0.510959\pi\)
−0.0344216 + 0.999407i \(0.510959\pi\)
\(272\) 0 0
\(273\) 4.52373 0.273789
\(274\) 0 0
\(275\) −2.80606 −0.169212
\(276\) 0 0
\(277\) −16.3634 −0.983184 −0.491592 0.870826i \(-0.663585\pi\)
−0.491592 + 0.870826i \(0.663585\pi\)
\(278\) 0 0
\(279\) 0.544198 0.0325803
\(280\) 0 0
\(281\) −24.8265 −1.48103 −0.740513 0.672042i \(-0.765418\pi\)
−0.740513 + 0.672042i \(0.765418\pi\)
\(282\) 0 0
\(283\) 4.18076 0.248521 0.124260 0.992250i \(-0.460344\pi\)
0.124260 + 0.992250i \(0.460344\pi\)
\(284\) 0 0
\(285\) 3.03761 0.179933
\(286\) 0 0
\(287\) −28.9380 −1.70815
\(288\) 0 0
\(289\) 33.6761 1.98095
\(290\) 0 0
\(291\) −1.23410 −0.0723444
\(292\) 0 0
\(293\) 23.6180 1.37978 0.689889 0.723915i \(-0.257659\pi\)
0.689889 + 0.723915i \(0.257659\pi\)
\(294\) 0 0
\(295\) −2.57452 −0.149894
\(296\) 0 0
\(297\) 12.1016 0.702204
\(298\) 0 0
\(299\) −6.48944 −0.375294
\(300\) 0 0
\(301\) 13.2750 0.765161
\(302\) 0 0
\(303\) 2.28630 0.131345
\(304\) 0 0
\(305\) −5.35026 −0.306355
\(306\) 0 0
\(307\) −32.5052 −1.85517 −0.927584 0.373614i \(-0.878118\pi\)
−0.927584 + 0.373614i \(0.878118\pi\)
\(308\) 0 0
\(309\) 7.97556 0.453714
\(310\) 0 0
\(311\) 9.31994 0.528486 0.264243 0.964456i \(-0.414878\pi\)
0.264243 + 0.964456i \(0.414878\pi\)
\(312\) 0 0
\(313\) 9.60228 0.542753 0.271376 0.962473i \(-0.412521\pi\)
0.271376 + 0.962473i \(0.412521\pi\)
\(314\) 0 0
\(315\) 9.76845 0.550390
\(316\) 0 0
\(317\) 19.3707 1.08797 0.543984 0.839095i \(-0.316915\pi\)
0.543984 + 0.839095i \(0.316915\pi\)
\(318\) 0 0
\(319\) −2.80606 −0.157109
\(320\) 0 0
\(321\) 9.42548 0.526079
\(322\) 0 0
\(323\) 26.8265 1.49267
\(324\) 0 0
\(325\) 1.35026 0.0748990
\(326\) 0 0
\(327\) −11.6873 −0.646312
\(328\) 0 0
\(329\) −26.6761 −1.47070
\(330\) 0 0
\(331\) 29.5428 1.62382 0.811909 0.583784i \(-0.198428\pi\)
0.811909 + 0.583784i \(0.198428\pi\)
\(332\) 0 0
\(333\) 12.9419 0.709213
\(334\) 0 0
\(335\) −3.19394 −0.174503
\(336\) 0 0
\(337\) 12.5442 0.683326 0.341663 0.939823i \(-0.389010\pi\)
0.341663 + 0.939823i \(0.389010\pi\)
\(338\) 0 0
\(339\) −13.2144 −0.717708
\(340\) 0 0
\(341\) 0.649738 0.0351853
\(342\) 0 0
\(343\) 13.6121 0.734986
\(344\) 0 0
\(345\) 3.87399 0.208569
\(346\) 0 0
\(347\) 25.0943 1.34713 0.673566 0.739127i \(-0.264762\pi\)
0.673566 + 0.739127i \(0.264762\pi\)
\(348\) 0 0
\(349\) −17.0738 −0.913940 −0.456970 0.889482i \(-0.651065\pi\)
−0.456970 + 0.889482i \(0.651065\pi\)
\(350\) 0 0
\(351\) −5.82321 −0.310820
\(352\) 0 0
\(353\) 5.66291 0.301406 0.150703 0.988579i \(-0.451846\pi\)
0.150703 + 0.988579i \(0.451846\pi\)
\(354\) 0 0
\(355\) 11.3503 0.602409
\(356\) 0 0
\(357\) −23.8496 −1.26225
\(358\) 0 0
\(359\) −0.755278 −0.0398621 −0.0199310 0.999801i \(-0.506345\pi\)
−0.0199310 + 0.999801i \(0.506345\pi\)
\(360\) 0 0
\(361\) −4.79877 −0.252567
\(362\) 0 0
\(363\) −2.51976 −0.132253
\(364\) 0 0
\(365\) 11.2447 0.588576
\(366\) 0 0
\(367\) −11.4460 −0.597474 −0.298737 0.954335i \(-0.596565\pi\)
−0.298737 + 0.954335i \(0.596565\pi\)
\(368\) 0 0
\(369\) 16.3634 0.851846
\(370\) 0 0
\(371\) 28.9380 1.50238
\(372\) 0 0
\(373\) 3.86414 0.200078 0.100039 0.994984i \(-0.468103\pi\)
0.100039 + 0.994984i \(0.468103\pi\)
\(374\) 0 0
\(375\) −0.806063 −0.0416249
\(376\) 0 0
\(377\) 1.35026 0.0695420
\(378\) 0 0
\(379\) −12.1055 −0.621820 −0.310910 0.950439i \(-0.600634\pi\)
−0.310910 + 0.950439i \(0.600634\pi\)
\(380\) 0 0
\(381\) −12.3780 −0.634145
\(382\) 0 0
\(383\) 10.0205 0.512022 0.256011 0.966674i \(-0.417592\pi\)
0.256011 + 0.966674i \(0.417592\pi\)
\(384\) 0 0
\(385\) 11.6629 0.594397
\(386\) 0 0
\(387\) −7.50659 −0.381581
\(388\) 0 0
\(389\) −25.6629 −1.30116 −0.650581 0.759437i \(-0.725474\pi\)
−0.650581 + 0.759437i \(0.725474\pi\)
\(390\) 0 0
\(391\) 34.2130 1.73023
\(392\) 0 0
\(393\) −1.11283 −0.0561351
\(394\) 0 0
\(395\) −4.73084 −0.238034
\(396\) 0 0
\(397\) 27.7137 1.39091 0.695455 0.718569i \(-0.255203\pi\)
0.695455 + 0.718569i \(0.255203\pi\)
\(398\) 0 0
\(399\) −12.6253 −0.632056
\(400\) 0 0
\(401\) 7.42548 0.370811 0.185406 0.982662i \(-0.440640\pi\)
0.185406 + 0.982662i \(0.440640\pi\)
\(402\) 0 0
\(403\) −0.312650 −0.0155742
\(404\) 0 0
\(405\) −3.57452 −0.177619
\(406\) 0 0
\(407\) 15.4518 0.765919
\(408\) 0 0
\(409\) −33.1998 −1.64163 −0.820813 0.571198i \(-0.806479\pi\)
−0.820813 + 0.571198i \(0.806479\pi\)
\(410\) 0 0
\(411\) −5.23410 −0.258179
\(412\) 0 0
\(413\) 10.7005 0.526538
\(414\) 0 0
\(415\) 2.54420 0.124890
\(416\) 0 0
\(417\) 15.3258 0.750509
\(418\) 0 0
\(419\) −16.5599 −0.809005 −0.404503 0.914537i \(-0.632555\pi\)
−0.404503 + 0.914537i \(0.632555\pi\)
\(420\) 0 0
\(421\) −8.82653 −0.430179 −0.215089 0.976594i \(-0.569004\pi\)
−0.215089 + 0.976594i \(0.569004\pi\)
\(422\) 0 0
\(423\) 15.0844 0.733430
\(424\) 0 0
\(425\) −7.11871 −0.345308
\(426\) 0 0
\(427\) 22.2374 1.07614
\(428\) 0 0
\(429\) −3.05411 −0.147454
\(430\) 0 0
\(431\) −4.25202 −0.204812 −0.102406 0.994743i \(-0.532654\pi\)
−0.102406 + 0.994743i \(0.532654\pi\)
\(432\) 0 0
\(433\) −1.81924 −0.0874270 −0.0437135 0.999044i \(-0.513919\pi\)
−0.0437135 + 0.999044i \(0.513919\pi\)
\(434\) 0 0
\(435\) −0.806063 −0.0386478
\(436\) 0 0
\(437\) 18.1114 0.866387
\(438\) 0 0
\(439\) −14.1114 −0.673501 −0.336751 0.941594i \(-0.609328\pi\)
−0.336751 + 0.941594i \(0.609328\pi\)
\(440\) 0 0
\(441\) −24.1490 −1.14995
\(442\) 0 0
\(443\) 17.2809 0.821041 0.410521 0.911851i \(-0.365347\pi\)
0.410521 + 0.911851i \(0.365347\pi\)
\(444\) 0 0
\(445\) −14.3127 −0.678485
\(446\) 0 0
\(447\) 5.34041 0.252593
\(448\) 0 0
\(449\) 9.35026 0.441266 0.220633 0.975357i \(-0.429188\pi\)
0.220633 + 0.975357i \(0.429188\pi\)
\(450\) 0 0
\(451\) 19.5369 0.919957
\(452\) 0 0
\(453\) 7.47627 0.351266
\(454\) 0 0
\(455\) −5.61213 −0.263101
\(456\) 0 0
\(457\) −17.6629 −0.826236 −0.413118 0.910677i \(-0.635560\pi\)
−0.413118 + 0.910677i \(0.635560\pi\)
\(458\) 0 0
\(459\) 30.7005 1.43298
\(460\) 0 0
\(461\) 15.5633 0.724853 0.362426 0.932012i \(-0.381948\pi\)
0.362426 + 0.932012i \(0.381948\pi\)
\(462\) 0 0
\(463\) −2.98286 −0.138625 −0.0693126 0.997595i \(-0.522081\pi\)
−0.0693126 + 0.997595i \(0.522081\pi\)
\(464\) 0 0
\(465\) 0.186642 0.00865533
\(466\) 0 0
\(467\) −34.5804 −1.60019 −0.800095 0.599873i \(-0.795218\pi\)
−0.800095 + 0.599873i \(0.795218\pi\)
\(468\) 0 0
\(469\) 13.2750 0.612984
\(470\) 0 0
\(471\) −4.03620 −0.185978
\(472\) 0 0
\(473\) −8.96239 −0.412091
\(474\) 0 0
\(475\) −3.76845 −0.172908
\(476\) 0 0
\(477\) −16.3634 −0.749230
\(478\) 0 0
\(479\) 34.1925 1.56230 0.781148 0.624346i \(-0.214634\pi\)
0.781148 + 0.624346i \(0.214634\pi\)
\(480\) 0 0
\(481\) −7.43533 −0.339022
\(482\) 0 0
\(483\) −16.1016 −0.732647
\(484\) 0 0
\(485\) 1.53102 0.0695202
\(486\) 0 0
\(487\) −38.4953 −1.74439 −0.872195 0.489159i \(-0.837304\pi\)
−0.872195 + 0.489159i \(0.837304\pi\)
\(488\) 0 0
\(489\) −6.05079 −0.273626
\(490\) 0 0
\(491\) 27.4676 1.23959 0.619797 0.784762i \(-0.287215\pi\)
0.619797 + 0.784762i \(0.287215\pi\)
\(492\) 0 0
\(493\) −7.11871 −0.320611
\(494\) 0 0
\(495\) −6.59498 −0.296422
\(496\) 0 0
\(497\) −47.1754 −2.11610
\(498\) 0 0
\(499\) 32.1016 1.43706 0.718532 0.695494i \(-0.244814\pi\)
0.718532 + 0.695494i \(0.244814\pi\)
\(500\) 0 0
\(501\) −17.6483 −0.788469
\(502\) 0 0
\(503\) −9.74401 −0.434464 −0.217232 0.976120i \(-0.569703\pi\)
−0.217232 + 0.976120i \(0.569703\pi\)
\(504\) 0 0
\(505\) −2.83638 −0.126217
\(506\) 0 0
\(507\) −9.00920 −0.400113
\(508\) 0 0
\(509\) −8.57452 −0.380059 −0.190029 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(510\) 0 0
\(511\) −46.7367 −2.06751
\(512\) 0 0
\(513\) 16.2520 0.717544
\(514\) 0 0
\(515\) −9.89446 −0.436002
\(516\) 0 0
\(517\) 18.0098 0.792072
\(518\) 0 0
\(519\) 5.66100 0.248490
\(520\) 0 0
\(521\) −37.8251 −1.65715 −0.828574 0.559879i \(-0.810848\pi\)
−0.828574 + 0.559879i \(0.810848\pi\)
\(522\) 0 0
\(523\) 13.5818 0.593891 0.296946 0.954894i \(-0.404032\pi\)
0.296946 + 0.954894i \(0.404032\pi\)
\(524\) 0 0
\(525\) 3.35026 0.146217
\(526\) 0 0
\(527\) 1.64832 0.0718021
\(528\) 0 0
\(529\) 0.0982457 0.00427155
\(530\) 0 0
\(531\) −6.05079 −0.262582
\(532\) 0 0
\(533\) −9.40105 −0.407205
\(534\) 0 0
\(535\) −11.6932 −0.505542
\(536\) 0 0
\(537\) −3.84955 −0.166121
\(538\) 0 0
\(539\) −28.8324 −1.24190
\(540\) 0 0
\(541\) 42.3127 1.81916 0.909581 0.415526i \(-0.136402\pi\)
0.909581 + 0.415526i \(0.136402\pi\)
\(542\) 0 0
\(543\) 1.51056 0.0648242
\(544\) 0 0
\(545\) 14.4993 0.621081
\(546\) 0 0
\(547\) 36.4690 1.55930 0.779650 0.626215i \(-0.215397\pi\)
0.779650 + 0.626215i \(0.215397\pi\)
\(548\) 0 0
\(549\) −12.5745 −0.536667
\(550\) 0 0
\(551\) −3.76845 −0.160541
\(552\) 0 0
\(553\) 19.6629 0.836152
\(554\) 0 0
\(555\) 4.43866 0.188411
\(556\) 0 0
\(557\) 19.5223 0.827187 0.413594 0.910462i \(-0.364273\pi\)
0.413594 + 0.910462i \(0.364273\pi\)
\(558\) 0 0
\(559\) 4.31265 0.182406
\(560\) 0 0
\(561\) 16.1016 0.679809
\(562\) 0 0
\(563\) 2.94192 0.123987 0.0619936 0.998077i \(-0.480254\pi\)
0.0619936 + 0.998077i \(0.480254\pi\)
\(564\) 0 0
\(565\) 16.3938 0.689690
\(566\) 0 0
\(567\) 14.8568 0.623929
\(568\) 0 0
\(569\) 2.49929 0.104776 0.0523879 0.998627i \(-0.483317\pi\)
0.0523879 + 0.998627i \(0.483317\pi\)
\(570\) 0 0
\(571\) −43.8007 −1.83300 −0.916501 0.400033i \(-0.868999\pi\)
−0.916501 + 0.400033i \(0.868999\pi\)
\(572\) 0 0
\(573\) −15.4109 −0.643799
\(574\) 0 0
\(575\) −4.80606 −0.200427
\(576\) 0 0
\(577\) −23.9062 −0.995229 −0.497614 0.867398i \(-0.665791\pi\)
−0.497614 + 0.867398i \(0.665791\pi\)
\(578\) 0 0
\(579\) 16.0362 0.666442
\(580\) 0 0
\(581\) −10.5745 −0.438705
\(582\) 0 0
\(583\) −19.5369 −0.809136
\(584\) 0 0
\(585\) 3.17347 0.131207
\(586\) 0 0
\(587\) 3.71767 0.153445 0.0767223 0.997053i \(-0.475555\pi\)
0.0767223 + 0.997053i \(0.475555\pi\)
\(588\) 0 0
\(589\) 0.872577 0.0359539
\(590\) 0 0
\(591\) −10.9116 −0.448843
\(592\) 0 0
\(593\) −25.5125 −1.04767 −0.523836 0.851819i \(-0.675499\pi\)
−0.523836 + 0.851819i \(0.675499\pi\)
\(594\) 0 0
\(595\) 29.5877 1.21298
\(596\) 0 0
\(597\) −2.07522 −0.0849332
\(598\) 0 0
\(599\) 15.6834 0.640806 0.320403 0.947281i \(-0.396182\pi\)
0.320403 + 0.947281i \(0.396182\pi\)
\(600\) 0 0
\(601\) 13.1392 0.535958 0.267979 0.963425i \(-0.413644\pi\)
0.267979 + 0.963425i \(0.413644\pi\)
\(602\) 0 0
\(603\) −7.50659 −0.305692
\(604\) 0 0
\(605\) 3.12601 0.127090
\(606\) 0 0
\(607\) −18.1465 −0.736543 −0.368271 0.929718i \(-0.620050\pi\)
−0.368271 + 0.929718i \(0.620050\pi\)
\(608\) 0 0
\(609\) 3.35026 0.135759
\(610\) 0 0
\(611\) −8.66624 −0.350598
\(612\) 0 0
\(613\) −32.5501 −1.31469 −0.657343 0.753592i \(-0.728320\pi\)
−0.657343 + 0.753592i \(0.728320\pi\)
\(614\) 0 0
\(615\) 5.61213 0.226303
\(616\) 0 0
\(617\) 20.2433 0.814965 0.407482 0.913213i \(-0.366407\pi\)
0.407482 + 0.913213i \(0.366407\pi\)
\(618\) 0 0
\(619\) 16.2071 0.651419 0.325709 0.945470i \(-0.394397\pi\)
0.325709 + 0.945470i \(0.394397\pi\)
\(620\) 0 0
\(621\) 20.7269 0.831741
\(622\) 0 0
\(623\) 59.4880 2.38334
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 8.52373 0.340405
\(628\) 0 0
\(629\) 39.1998 1.56300
\(630\) 0 0
\(631\) −2.13586 −0.0850271 −0.0425136 0.999096i \(-0.513537\pi\)
−0.0425136 + 0.999096i \(0.513537\pi\)
\(632\) 0 0
\(633\) −9.58769 −0.381076
\(634\) 0 0
\(635\) 15.3561 0.609390
\(636\) 0 0
\(637\) 13.8740 0.549708
\(638\) 0 0
\(639\) 26.6761 1.05529
\(640\) 0 0
\(641\) −14.0362 −0.554396 −0.277198 0.960813i \(-0.589406\pi\)
−0.277198 + 0.960813i \(0.589406\pi\)
\(642\) 0 0
\(643\) 43.8799 1.73045 0.865227 0.501381i \(-0.167174\pi\)
0.865227 + 0.501381i \(0.167174\pi\)
\(644\) 0 0
\(645\) −2.57452 −0.101371
\(646\) 0 0
\(647\) 22.5560 0.886766 0.443383 0.896332i \(-0.353778\pi\)
0.443383 + 0.896332i \(0.353778\pi\)
\(648\) 0 0
\(649\) −7.22425 −0.283577
\(650\) 0 0
\(651\) −0.775746 −0.0304039
\(652\) 0 0
\(653\) 27.3054 1.06854 0.534271 0.845314i \(-0.320586\pi\)
0.534271 + 0.845314i \(0.320586\pi\)
\(654\) 0 0
\(655\) 1.38058 0.0539437
\(656\) 0 0
\(657\) 26.4280 1.03106
\(658\) 0 0
\(659\) −14.0665 −0.547954 −0.273977 0.961736i \(-0.588339\pi\)
−0.273977 + 0.961736i \(0.588339\pi\)
\(660\) 0 0
\(661\) 38.9741 1.51592 0.757959 0.652302i \(-0.226197\pi\)
0.757959 + 0.652302i \(0.226197\pi\)
\(662\) 0 0
\(663\) −7.74798 −0.300907
\(664\) 0 0
\(665\) 15.6629 0.607382
\(666\) 0 0
\(667\) −4.80606 −0.186092
\(668\) 0 0
\(669\) 0.901754 0.0348638
\(670\) 0 0
\(671\) −15.0132 −0.579577
\(672\) 0 0
\(673\) 20.3390 0.784011 0.392005 0.919963i \(-0.371781\pi\)
0.392005 + 0.919963i \(0.371781\pi\)
\(674\) 0 0
\(675\) −4.31265 −0.165994
\(676\) 0 0
\(677\) −19.1841 −0.737304 −0.368652 0.929567i \(-0.620181\pi\)
−0.368652 + 0.929567i \(0.620181\pi\)
\(678\) 0 0
\(679\) −6.36344 −0.244206
\(680\) 0 0
\(681\) −0.0244376 −0.000936449 0
\(682\) 0 0
\(683\) 24.1319 0.923381 0.461691 0.887041i \(-0.347243\pi\)
0.461691 + 0.887041i \(0.347243\pi\)
\(684\) 0 0
\(685\) 6.49341 0.248101
\(686\) 0 0
\(687\) −4.71511 −0.179893
\(688\) 0 0
\(689\) 9.40105 0.358151
\(690\) 0 0
\(691\) 4.28821 0.163131 0.0815657 0.996668i \(-0.474008\pi\)
0.0815657 + 0.996668i \(0.474008\pi\)
\(692\) 0 0
\(693\) 27.4109 1.04125
\(694\) 0 0
\(695\) −19.0132 −0.721211
\(696\) 0 0
\(697\) 49.5633 1.87734
\(698\) 0 0
\(699\) 21.0395 0.795788
\(700\) 0 0
\(701\) −14.1260 −0.533532 −0.266766 0.963761i \(-0.585955\pi\)
−0.266766 + 0.963761i \(0.585955\pi\)
\(702\) 0 0
\(703\) 20.7513 0.782650
\(704\) 0 0
\(705\) 5.17347 0.194844
\(706\) 0 0
\(707\) 11.7889 0.443368
\(708\) 0 0
\(709\) 6.75131 0.253551 0.126775 0.991931i \(-0.459537\pi\)
0.126775 + 0.991931i \(0.459537\pi\)
\(710\) 0 0
\(711\) −11.1187 −0.416984
\(712\) 0 0
\(713\) 1.11283 0.0416760
\(714\) 0 0
\(715\) 3.78892 0.141698
\(716\) 0 0
\(717\) −1.14903 −0.0429113
\(718\) 0 0
\(719\) 43.8251 1.63440 0.817201 0.576353i \(-0.195525\pi\)
0.817201 + 0.576353i \(0.195525\pi\)
\(720\) 0 0
\(721\) 41.1246 1.53156
\(722\) 0 0
\(723\) −0.0606343 −0.00225502
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 14.8813 0.551916 0.275958 0.961170i \(-0.411005\pi\)
0.275958 + 0.961170i \(0.411005\pi\)
\(728\) 0 0
\(729\) 2.02776 0.0751023
\(730\) 0 0
\(731\) −22.7367 −0.840948
\(732\) 0 0
\(733\) 7.17935 0.265175 0.132588 0.991171i \(-0.457671\pi\)
0.132588 + 0.991171i \(0.457671\pi\)
\(734\) 0 0
\(735\) −8.28233 −0.305498
\(736\) 0 0
\(737\) −8.96239 −0.330134
\(738\) 0 0
\(739\) −16.1709 −0.594857 −0.297428 0.954744i \(-0.596129\pi\)
−0.297428 + 0.954744i \(0.596129\pi\)
\(740\) 0 0
\(741\) −4.10157 −0.150675
\(742\) 0 0
\(743\) −27.8192 −1.02059 −0.510294 0.860000i \(-0.670464\pi\)
−0.510294 + 0.860000i \(0.670464\pi\)
\(744\) 0 0
\(745\) −6.62530 −0.242732
\(746\) 0 0
\(747\) 5.97953 0.218780
\(748\) 0 0
\(749\) 48.6009 1.77584
\(750\) 0 0
\(751\) 17.6326 0.643423 0.321711 0.946838i \(-0.395742\pi\)
0.321711 + 0.946838i \(0.395742\pi\)
\(752\) 0 0
\(753\) −13.6893 −0.498864
\(754\) 0 0
\(755\) −9.27504 −0.337553
\(756\) 0 0
\(757\) 1.53102 0.0556460 0.0278230 0.999613i \(-0.491143\pi\)
0.0278230 + 0.999613i \(0.491143\pi\)
\(758\) 0 0
\(759\) 10.8707 0.394580
\(760\) 0 0
\(761\) −34.4749 −1.24971 −0.624856 0.780740i \(-0.714842\pi\)
−0.624856 + 0.780740i \(0.714842\pi\)
\(762\) 0 0
\(763\) −60.2638 −2.18170
\(764\) 0 0
\(765\) −16.7308 −0.604905
\(766\) 0 0
\(767\) 3.47627 0.125521
\(768\) 0 0
\(769\) −19.3404 −0.697433 −0.348717 0.937228i \(-0.613382\pi\)
−0.348717 + 0.937228i \(0.613382\pi\)
\(770\) 0 0
\(771\) 20.3127 0.731542
\(772\) 0 0
\(773\) 36.2677 1.30446 0.652230 0.758021i \(-0.273834\pi\)
0.652230 + 0.758021i \(0.273834\pi\)
\(774\) 0 0
\(775\) −0.231548 −0.00831745
\(776\) 0 0
\(777\) −18.4485 −0.661837
\(778\) 0 0
\(779\) 26.2374 0.940053
\(780\) 0 0
\(781\) 31.8496 1.13967
\(782\) 0 0
\(783\) −4.31265 −0.154122
\(784\) 0 0
\(785\) 5.00729 0.178718
\(786\) 0 0
\(787\) −32.0059 −1.14089 −0.570443 0.821337i \(-0.693229\pi\)
−0.570443 + 0.821337i \(0.693229\pi\)
\(788\) 0 0
\(789\) 13.0033 0.462931
\(790\) 0 0
\(791\) −68.1378 −2.42270
\(792\) 0 0
\(793\) 7.22425 0.256541
\(794\) 0 0
\(795\) −5.61213 −0.199042
\(796\) 0 0
\(797\) −41.6932 −1.47685 −0.738425 0.674336i \(-0.764430\pi\)
−0.738425 + 0.674336i \(0.764430\pi\)
\(798\) 0 0
\(799\) 45.6893 1.61637
\(800\) 0 0
\(801\) −33.6385 −1.18856
\(802\) 0 0
\(803\) 31.5534 1.11350
\(804\) 0 0
\(805\) 19.9756 0.704046
\(806\) 0 0
\(807\) 4.26378 0.150092
\(808\) 0 0
\(809\) 30.6371 1.07714 0.538571 0.842580i \(-0.318964\pi\)
0.538571 + 0.842580i \(0.318964\pi\)
\(810\) 0 0
\(811\) −25.4617 −0.894081 −0.447040 0.894514i \(-0.647522\pi\)
−0.447040 + 0.894514i \(0.647522\pi\)
\(812\) 0 0
\(813\) −0.913513 −0.0320383
\(814\) 0 0
\(815\) 7.50659 0.262944
\(816\) 0 0
\(817\) −12.0362 −0.421093
\(818\) 0 0
\(819\) −13.1900 −0.460895
\(820\) 0 0
\(821\) −32.7005 −1.14126 −0.570628 0.821209i \(-0.693300\pi\)
−0.570628 + 0.821209i \(0.693300\pi\)
\(822\) 0 0
\(823\) 31.1041 1.08422 0.542111 0.840307i \(-0.317625\pi\)
0.542111 + 0.840307i \(0.317625\pi\)
\(824\) 0 0
\(825\) −2.26187 −0.0787480
\(826\) 0 0
\(827\) 1.58181 0.0550049 0.0275025 0.999622i \(-0.491245\pi\)
0.0275025 + 0.999622i \(0.491245\pi\)
\(828\) 0 0
\(829\) −0.111420 −0.00386976 −0.00193488 0.999998i \(-0.500616\pi\)
−0.00193488 + 0.999998i \(0.500616\pi\)
\(830\) 0 0
\(831\) −13.1900 −0.457555
\(832\) 0 0
\(833\) −73.1451 −2.53433
\(834\) 0 0
\(835\) 21.8945 0.757689
\(836\) 0 0
\(837\) 0.998585 0.0345162
\(838\) 0 0
\(839\) −28.9829 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −20.0118 −0.689242
\(844\) 0 0
\(845\) 11.1768 0.384493
\(846\) 0 0
\(847\) −12.9927 −0.446435
\(848\) 0 0
\(849\) 3.36996 0.115657
\(850\) 0 0
\(851\) 26.4650 0.907209
\(852\) 0 0
\(853\) −7.77319 −0.266149 −0.133075 0.991106i \(-0.542485\pi\)
−0.133075 + 0.991106i \(0.542485\pi\)
\(854\) 0 0
\(855\) −8.85685 −0.302898
\(856\) 0 0
\(857\) −13.8740 −0.473927 −0.236963 0.971519i \(-0.576152\pi\)
−0.236963 + 0.971519i \(0.576152\pi\)
\(858\) 0 0
\(859\) 15.2809 0.521378 0.260689 0.965423i \(-0.416050\pi\)
0.260689 + 0.965423i \(0.416050\pi\)
\(860\) 0 0
\(861\) −23.3258 −0.794942
\(862\) 0 0
\(863\) −31.2301 −1.06309 −0.531543 0.847031i \(-0.678388\pi\)
−0.531543 + 0.847031i \(0.678388\pi\)
\(864\) 0 0
\(865\) −7.02302 −0.238790
\(866\) 0 0
\(867\) 27.1451 0.921895
\(868\) 0 0
\(869\) −13.2750 −0.450325
\(870\) 0 0
\(871\) 4.31265 0.146129
\(872\) 0 0
\(873\) 3.59831 0.121784
\(874\) 0 0
\(875\) −4.15633 −0.140509
\(876\) 0 0
\(877\) −4.26187 −0.143913 −0.0719565 0.997408i \(-0.522924\pi\)
−0.0719565 + 0.997408i \(0.522924\pi\)
\(878\) 0 0
\(879\) 19.0376 0.642123
\(880\) 0 0
\(881\) 15.2144 0.512586 0.256293 0.966599i \(-0.417499\pi\)
0.256293 + 0.966599i \(0.417499\pi\)
\(882\) 0 0
\(883\) 13.7078 0.461305 0.230652 0.973036i \(-0.425914\pi\)
0.230652 + 0.973036i \(0.425914\pi\)
\(884\) 0 0
\(885\) −2.07522 −0.0697579
\(886\) 0 0
\(887\) −12.6556 −0.424934 −0.212467 0.977168i \(-0.568150\pi\)
−0.212467 + 0.977168i \(0.568150\pi\)
\(888\) 0 0
\(889\) −63.8251 −2.14063
\(890\) 0 0
\(891\) −10.0303 −0.336028
\(892\) 0 0
\(893\) 24.1866 0.809375
\(894\) 0 0
\(895\) 4.77575 0.159636
\(896\) 0 0
\(897\) −5.23090 −0.174655
\(898\) 0 0
\(899\) −0.231548 −0.00772256
\(900\) 0 0
\(901\) −49.5633 −1.65119
\(902\) 0 0
\(903\) 10.7005 0.356091
\(904\) 0 0
\(905\) −1.87399 −0.0622936
\(906\) 0 0
\(907\) 12.5540 0.416850 0.208425 0.978038i \(-0.433166\pi\)
0.208425 + 0.978038i \(0.433166\pi\)
\(908\) 0 0
\(909\) −6.66624 −0.221105
\(910\) 0 0
\(911\) −22.8714 −0.757765 −0.378882 0.925445i \(-0.623691\pi\)
−0.378882 + 0.925445i \(0.623691\pi\)
\(912\) 0 0
\(913\) 7.13918 0.236272
\(914\) 0 0
\(915\) −4.31265 −0.142572
\(916\) 0 0
\(917\) −5.73813 −0.189490
\(918\) 0 0
\(919\) −9.67750 −0.319231 −0.159616 0.987179i \(-0.551025\pi\)
−0.159616 + 0.987179i \(0.551025\pi\)
\(920\) 0 0
\(921\) −26.2012 −0.863360
\(922\) 0 0
\(923\) −15.3258 −0.504456
\(924\) 0 0
\(925\) −5.50659 −0.181055
\(926\) 0 0
\(927\) −23.2546 −0.763780
\(928\) 0 0
\(929\) 51.9248 1.70360 0.851798 0.523870i \(-0.175512\pi\)
0.851798 + 0.523870i \(0.175512\pi\)
\(930\) 0 0
\(931\) −38.7210 −1.26903
\(932\) 0 0
\(933\) 7.51247 0.245947
\(934\) 0 0
\(935\) −19.9756 −0.653271
\(936\) 0 0
\(937\) 3.58769 0.117205 0.0586024 0.998281i \(-0.481336\pi\)
0.0586024 + 0.998281i \(0.481336\pi\)
\(938\) 0 0
\(939\) 7.74004 0.252587
\(940\) 0 0
\(941\) −18.6253 −0.607167 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(942\) 0 0
\(943\) 33.4617 1.08966
\(944\) 0 0
\(945\) 17.9248 0.583093
\(946\) 0 0
\(947\) 16.5950 0.539265 0.269632 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(948\) 0 0
\(949\) −15.1833 −0.492871
\(950\) 0 0
\(951\) 15.6140 0.506320
\(952\) 0 0
\(953\) −12.7005 −0.411410 −0.205705 0.978614i \(-0.565949\pi\)
−0.205705 + 0.978614i \(0.565949\pi\)
\(954\) 0 0
\(955\) 19.1187 0.618667
\(956\) 0 0
\(957\) −2.26187 −0.0731157
\(958\) 0 0
\(959\) −26.9887 −0.871512
\(960\) 0 0
\(961\) −30.9464 −0.998271
\(962\) 0 0
\(963\) −27.4821 −0.885600
\(964\) 0 0
\(965\) −19.8945 −0.640425
\(966\) 0 0
\(967\) −37.4314 −1.20371 −0.601856 0.798605i \(-0.705572\pi\)
−0.601856 + 0.798605i \(0.705572\pi\)
\(968\) 0 0
\(969\) 21.6239 0.694659
\(970\) 0 0
\(971\) −7.51644 −0.241214 −0.120607 0.992700i \(-0.538484\pi\)
−0.120607 + 0.992700i \(0.538484\pi\)
\(972\) 0 0
\(973\) 79.0249 2.53342
\(974\) 0 0
\(975\) 1.08840 0.0348566
\(976\) 0 0
\(977\) −2.52847 −0.0808929 −0.0404465 0.999182i \(-0.512878\pi\)
−0.0404465 + 0.999182i \(0.512878\pi\)
\(978\) 0 0
\(979\) −40.1622 −1.28359
\(980\) 0 0
\(981\) 34.0771 1.08800
\(982\) 0 0
\(983\) 9.32979 0.297574 0.148787 0.988869i \(-0.452463\pi\)
0.148787 + 0.988869i \(0.452463\pi\)
\(984\) 0 0
\(985\) 13.5369 0.431322
\(986\) 0 0
\(987\) −21.5026 −0.684436
\(988\) 0 0
\(989\) −15.3503 −0.488110
\(990\) 0 0
\(991\) −38.4241 −1.22058 −0.610290 0.792178i \(-0.708947\pi\)
−0.610290 + 0.792178i \(0.708947\pi\)
\(992\) 0 0
\(993\) 23.8134 0.755694
\(994\) 0 0
\(995\) 2.57452 0.0816176
\(996\) 0 0
\(997\) −18.8423 −0.596740 −0.298370 0.954450i \(-0.596443\pi\)
−0.298370 + 0.954450i \(0.596443\pi\)
\(998\) 0 0
\(999\) 23.7480 0.751353
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 2320.2.a.s.1.2 3
4.3 odd 2 145.2.a.d.1.3 3
8.3 odd 2 9280.2.a.bu.1.2 3
8.5 even 2 9280.2.a.bm.1.2 3
12.11 even 2 1305.2.a.o.1.1 3
20.3 even 4 725.2.b.d.349.1 6
20.7 even 4 725.2.b.d.349.6 6
20.19 odd 2 725.2.a.d.1.1 3
28.27 even 2 7105.2.a.p.1.3 3
60.59 even 2 6525.2.a.bh.1.3 3
116.115 odd 2 4205.2.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.3 3 4.3 odd 2
725.2.a.d.1.1 3 20.19 odd 2
725.2.b.d.349.1 6 20.3 even 4
725.2.b.d.349.6 6 20.7 even 4
1305.2.a.o.1.1 3 12.11 even 2
2320.2.a.s.1.2 3 1.1 even 1 trivial
4205.2.a.e.1.1 3 116.115 odd 2
6525.2.a.bh.1.3 3 60.59 even 2
7105.2.a.p.1.3 3 28.27 even 2
9280.2.a.bm.1.2 3 8.5 even 2
9280.2.a.bu.1.2 3 8.3 odd 2