Properties

Label 5625.2.a.p.1.6
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $6$
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] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, 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("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.44400625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.68704\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.68704 q^{2} +5.22020 q^{4} +1.68704 q^{7} +8.65280 q^{8} +O(q^{10})\) \(q+2.68704 q^{2} +5.22020 q^{4} +1.68704 q^{7} +8.65280 q^{8} -1.07882 q^{11} +2.67723 q^{13} +4.53315 q^{14} +12.8101 q^{16} +3.93167 q^{17} -1.17755 q^{19} -2.89883 q^{22} +4.06295 q^{23} +7.19384 q^{26} +8.80669 q^{28} -5.95595 q^{29} -7.10310 q^{31} +17.1155 q^{32} +10.5646 q^{34} -4.58187 q^{37} -3.16412 q^{38} +11.2614 q^{41} +2.58587 q^{43} -5.63164 q^{44} +10.9173 q^{46} +1.91782 q^{47} -4.15389 q^{49} +13.9757 q^{52} +2.54861 q^{53} +14.5976 q^{56} -16.0039 q^{58} -1.33599 q^{59} -7.28106 q^{61} -19.0863 q^{62} +20.3701 q^{64} +12.4451 q^{67} +20.5241 q^{68} +5.98480 q^{71} -3.30065 q^{73} -12.3117 q^{74} -6.14702 q^{76} -1.82001 q^{77} -4.00404 q^{79} +30.2600 q^{82} -8.87482 q^{83} +6.94834 q^{86} -9.33480 q^{88} -15.4436 q^{89} +4.51660 q^{91} +21.2094 q^{92} +5.15327 q^{94} +10.7682 q^{97} -11.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.68704 1.90003 0.950013 0.312211i \(-0.101069\pi\)
0.950013 + 0.312211i \(0.101069\pi\)
\(3\) 0 0
\(4\) 5.22020 2.61010
\(5\) 0 0
\(6\) 0 0
\(7\) 1.68704 0.637642 0.318821 0.947815i \(-0.396713\pi\)
0.318821 + 0.947815i \(0.396713\pi\)
\(8\) 8.65280 3.05923
\(9\) 0 0
\(10\) 0 0
\(11\) −1.07882 −0.325276 −0.162638 0.986686i \(-0.552000\pi\)
−0.162638 + 0.986686i \(0.552000\pi\)
\(12\) 0 0
\(13\) 2.67723 0.742531 0.371265 0.928527i \(-0.378924\pi\)
0.371265 + 0.928527i \(0.378924\pi\)
\(14\) 4.53315 1.21154
\(15\) 0 0
\(16\) 12.8101 3.20251
\(17\) 3.93167 0.953570 0.476785 0.879020i \(-0.341802\pi\)
0.476785 + 0.879020i \(0.341802\pi\)
\(18\) 0 0
\(19\) −1.17755 −0.270148 −0.135074 0.990836i \(-0.543127\pi\)
−0.135074 + 0.990836i \(0.543127\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.89883 −0.618032
\(23\) 4.06295 0.847183 0.423591 0.905853i \(-0.360769\pi\)
0.423591 + 0.905853i \(0.360769\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.19384 1.41083
\(27\) 0 0
\(28\) 8.80669 1.66431
\(29\) −5.95595 −1.10599 −0.552996 0.833184i \(-0.686516\pi\)
−0.552996 + 0.833184i \(0.686516\pi\)
\(30\) 0 0
\(31\) −7.10310 −1.27575 −0.637877 0.770138i \(-0.720187\pi\)
−0.637877 + 0.770138i \(0.720187\pi\)
\(32\) 17.1155 3.02563
\(33\) 0 0
\(34\) 10.5646 1.81181
\(35\) 0 0
\(36\) 0 0
\(37\) −4.58187 −0.753254 −0.376627 0.926365i \(-0.622916\pi\)
−0.376627 + 0.926365i \(0.622916\pi\)
\(38\) −3.16412 −0.513287
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2614 1.75874 0.879371 0.476137i \(-0.157963\pi\)
0.879371 + 0.476137i \(0.157963\pi\)
\(42\) 0 0
\(43\) 2.58587 0.394342 0.197171 0.980369i \(-0.436825\pi\)
0.197171 + 0.980369i \(0.436825\pi\)
\(44\) −5.63164 −0.849002
\(45\) 0 0
\(46\) 10.9173 1.60967
\(47\) 1.91782 0.279743 0.139871 0.990170i \(-0.455331\pi\)
0.139871 + 0.990170i \(0.455331\pi\)
\(48\) 0 0
\(49\) −4.15389 −0.593413
\(50\) 0 0
\(51\) 0 0
\(52\) 13.9757 1.93808
\(53\) 2.54861 0.350078 0.175039 0.984561i \(-0.443995\pi\)
0.175039 + 0.984561i \(0.443995\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 14.5976 1.95069
\(57\) 0 0
\(58\) −16.0039 −2.10141
\(59\) −1.33599 −0.173931 −0.0869654 0.996211i \(-0.527717\pi\)
−0.0869654 + 0.996211i \(0.527717\pi\)
\(60\) 0 0
\(61\) −7.28106 −0.932245 −0.466122 0.884720i \(-0.654349\pi\)
−0.466122 + 0.884720i \(0.654349\pi\)
\(62\) −19.0863 −2.42397
\(63\) 0 0
\(64\) 20.3701 2.54626
\(65\) 0 0
\(66\) 0 0
\(67\) 12.4451 1.52041 0.760203 0.649686i \(-0.225100\pi\)
0.760203 + 0.649686i \(0.225100\pi\)
\(68\) 20.5241 2.48891
\(69\) 0 0
\(70\) 0 0
\(71\) 5.98480 0.710266 0.355133 0.934816i \(-0.384436\pi\)
0.355133 + 0.934816i \(0.384436\pi\)
\(72\) 0 0
\(73\) −3.30065 −0.386312 −0.193156 0.981168i \(-0.561872\pi\)
−0.193156 + 0.981168i \(0.561872\pi\)
\(74\) −12.3117 −1.43120
\(75\) 0 0
\(76\) −6.14702 −0.705112
\(77\) −1.82001 −0.207410
\(78\) 0 0
\(79\) −4.00404 −0.450490 −0.225245 0.974302i \(-0.572318\pi\)
−0.225245 + 0.974302i \(0.572318\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 30.2600 3.34166
\(83\) −8.87482 −0.974138 −0.487069 0.873364i \(-0.661934\pi\)
−0.487069 + 0.873364i \(0.661934\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.94834 0.749259
\(87\) 0 0
\(88\) −9.33480 −0.995093
\(89\) −15.4436 −1.63702 −0.818509 0.574493i \(-0.805199\pi\)
−0.818509 + 0.574493i \(0.805199\pi\)
\(90\) 0 0
\(91\) 4.51660 0.473469
\(92\) 21.2094 2.21123
\(93\) 0 0
\(94\) 5.15327 0.531519
\(95\) 0 0
\(96\) 0 0
\(97\) 10.7682 1.09335 0.546673 0.837346i \(-0.315894\pi\)
0.546673 + 0.837346i \(0.315894\pi\)
\(98\) −11.1617 −1.12750
\(99\) 0 0
\(100\) 0 0
\(101\) 13.5654 1.34981 0.674905 0.737905i \(-0.264185\pi\)
0.674905 + 0.737905i \(0.264185\pi\)
\(102\) 0 0
\(103\) 1.36904 0.134895 0.0674476 0.997723i \(-0.478514\pi\)
0.0674476 + 0.997723i \(0.478514\pi\)
\(104\) 23.1656 2.27157
\(105\) 0 0
\(106\) 6.84822 0.665158
\(107\) −11.0115 −1.06452 −0.532262 0.846579i \(-0.678658\pi\)
−0.532262 + 0.846579i \(0.678658\pi\)
\(108\) 0 0
\(109\) 3.44890 0.330344 0.165172 0.986265i \(-0.447182\pi\)
0.165172 + 0.986265i \(0.447182\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 21.6111 2.04206
\(113\) −6.05194 −0.569319 −0.284660 0.958629i \(-0.591881\pi\)
−0.284660 + 0.958629i \(0.591881\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −31.0912 −2.88675
\(117\) 0 0
\(118\) −3.58986 −0.330473
\(119\) 6.63289 0.608036
\(120\) 0 0
\(121\) −9.83615 −0.894196
\(122\) −19.5645 −1.77129
\(123\) 0 0
\(124\) −37.0796 −3.32984
\(125\) 0 0
\(126\) 0 0
\(127\) −20.2362 −1.79567 −0.897835 0.440332i \(-0.854861\pi\)
−0.897835 + 0.440332i \(0.854861\pi\)
\(128\) 20.5042 1.81233
\(129\) 0 0
\(130\) 0 0
\(131\) 3.05011 0.266490 0.133245 0.991083i \(-0.457460\pi\)
0.133245 + 0.991083i \(0.457460\pi\)
\(132\) 0 0
\(133\) −1.98657 −0.172257
\(134\) 33.4404 2.88881
\(135\) 0 0
\(136\) 34.0200 2.91719
\(137\) 20.5100 1.75229 0.876145 0.482048i \(-0.160107\pi\)
0.876145 + 0.482048i \(0.160107\pi\)
\(138\) 0 0
\(139\) 18.3184 1.55374 0.776871 0.629659i \(-0.216806\pi\)
0.776871 + 0.629659i \(0.216806\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0814 1.34952
\(143\) −2.88825 −0.241527
\(144\) 0 0
\(145\) 0 0
\(146\) −8.86899 −0.734003
\(147\) 0 0
\(148\) −23.9182 −1.96607
\(149\) −0.705938 −0.0578327 −0.0289163 0.999582i \(-0.509206\pi\)
−0.0289163 + 0.999582i \(0.509206\pi\)
\(150\) 0 0
\(151\) 7.04538 0.573345 0.286673 0.958029i \(-0.407451\pi\)
0.286673 + 0.958029i \(0.407451\pi\)
\(152\) −10.1891 −0.826443
\(153\) 0 0
\(154\) −4.89045 −0.394083
\(155\) 0 0
\(156\) 0 0
\(157\) −5.12880 −0.409323 −0.204662 0.978833i \(-0.565609\pi\)
−0.204662 + 0.978833i \(0.565609\pi\)
\(158\) −10.7590 −0.855942
\(159\) 0 0
\(160\) 0 0
\(161\) 6.85436 0.540199
\(162\) 0 0
\(163\) −24.2613 −1.90029 −0.950146 0.311806i \(-0.899066\pi\)
−0.950146 + 0.311806i \(0.899066\pi\)
\(164\) 58.7869 4.59049
\(165\) 0 0
\(166\) −23.8470 −1.85089
\(167\) −4.54890 −0.352004 −0.176002 0.984390i \(-0.556317\pi\)
−0.176002 + 0.984390i \(0.556317\pi\)
\(168\) 0 0
\(169\) −5.83243 −0.448648
\(170\) 0 0
\(171\) 0 0
\(172\) 13.4988 1.02927
\(173\) 2.67729 0.203550 0.101775 0.994807i \(-0.467548\pi\)
0.101775 + 0.994807i \(0.467548\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.8197 −1.04170
\(177\) 0 0
\(178\) −41.4976 −3.11038
\(179\) 15.6266 1.16799 0.583995 0.811757i \(-0.301489\pi\)
0.583995 + 0.811757i \(0.301489\pi\)
\(180\) 0 0
\(181\) 0.202843 0.0150772 0.00753859 0.999972i \(-0.497600\pi\)
0.00753859 + 0.999972i \(0.497600\pi\)
\(182\) 12.1363 0.899603
\(183\) 0 0
\(184\) 35.1559 2.59172
\(185\) 0 0
\(186\) 0 0
\(187\) −4.24155 −0.310173
\(188\) 10.0114 0.730156
\(189\) 0 0
\(190\) 0 0
\(191\) 7.91975 0.573053 0.286526 0.958072i \(-0.407499\pi\)
0.286526 + 0.958072i \(0.407499\pi\)
\(192\) 0 0
\(193\) −16.4269 −1.18243 −0.591216 0.806513i \(-0.701352\pi\)
−0.591216 + 0.806513i \(0.701352\pi\)
\(194\) 28.9346 2.07738
\(195\) 0 0
\(196\) −21.6841 −1.54887
\(197\) 17.1369 1.22095 0.610475 0.792035i \(-0.290978\pi\)
0.610475 + 0.792035i \(0.290978\pi\)
\(198\) 0 0
\(199\) −2.01848 −0.143086 −0.0715430 0.997438i \(-0.522792\pi\)
−0.0715430 + 0.997438i \(0.522792\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 36.4509 2.56467
\(203\) −10.0479 −0.705227
\(204\) 0 0
\(205\) 0 0
\(206\) 3.67866 0.256304
\(207\) 0 0
\(208\) 34.2955 2.37796
\(209\) 1.27036 0.0878725
\(210\) 0 0
\(211\) −23.8213 −1.63993 −0.819963 0.572416i \(-0.806006\pi\)
−0.819963 + 0.572416i \(0.806006\pi\)
\(212\) 13.3042 0.913738
\(213\) 0 0
\(214\) −29.5884 −2.02262
\(215\) 0 0
\(216\) 0 0
\(217\) −11.9832 −0.813475
\(218\) 9.26733 0.627663
\(219\) 0 0
\(220\) 0 0
\(221\) 10.5260 0.708055
\(222\) 0 0
\(223\) 13.1112 0.877989 0.438995 0.898490i \(-0.355335\pi\)
0.438995 + 0.898490i \(0.355335\pi\)
\(224\) 28.8746 1.92927
\(225\) 0 0
\(226\) −16.2618 −1.08172
\(227\) 26.2807 1.74431 0.872155 0.489229i \(-0.162722\pi\)
0.872155 + 0.489229i \(0.162722\pi\)
\(228\) 0 0
\(229\) 1.99605 0.131903 0.0659513 0.997823i \(-0.478992\pi\)
0.0659513 + 0.997823i \(0.478992\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −51.5357 −3.38348
\(233\) 0.525548 0.0344298 0.0172149 0.999852i \(-0.494520\pi\)
0.0172149 + 0.999852i \(0.494520\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.97412 −0.453976
\(237\) 0 0
\(238\) 17.8229 1.15528
\(239\) 7.00971 0.453421 0.226710 0.973962i \(-0.427203\pi\)
0.226710 + 0.973962i \(0.427203\pi\)
\(240\) 0 0
\(241\) 2.31554 0.149157 0.0745786 0.997215i \(-0.476239\pi\)
0.0745786 + 0.997215i \(0.476239\pi\)
\(242\) −26.4302 −1.69899
\(243\) 0 0
\(244\) −38.0086 −2.43325
\(245\) 0 0
\(246\) 0 0
\(247\) −3.15257 −0.200593
\(248\) −61.4617 −3.90282
\(249\) 0 0
\(250\) 0 0
\(251\) 6.17885 0.390005 0.195003 0.980803i \(-0.437528\pi\)
0.195003 + 0.980803i \(0.437528\pi\)
\(252\) 0 0
\(253\) −4.38318 −0.275568
\(254\) −54.3755 −3.41182
\(255\) 0 0
\(256\) 14.3555 0.897217
\(257\) −13.2239 −0.824882 −0.412441 0.910984i \(-0.635324\pi\)
−0.412441 + 0.910984i \(0.635324\pi\)
\(258\) 0 0
\(259\) −7.72980 −0.480306
\(260\) 0 0
\(261\) 0 0
\(262\) 8.19578 0.506337
\(263\) 6.97643 0.430185 0.215092 0.976594i \(-0.430995\pi\)
0.215092 + 0.976594i \(0.430995\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.33800 −0.327294
\(267\) 0 0
\(268\) 64.9656 3.96841
\(269\) −23.7199 −1.44623 −0.723113 0.690730i \(-0.757289\pi\)
−0.723113 + 0.690730i \(0.757289\pi\)
\(270\) 0 0
\(271\) 28.9910 1.76108 0.880539 0.473975i \(-0.157181\pi\)
0.880539 + 0.473975i \(0.157181\pi\)
\(272\) 50.3649 3.05382
\(273\) 0 0
\(274\) 55.1113 3.32940
\(275\) 0 0
\(276\) 0 0
\(277\) −8.58667 −0.515923 −0.257962 0.966155i \(-0.583051\pi\)
−0.257962 + 0.966155i \(0.583051\pi\)
\(278\) 49.2222 2.95215
\(279\) 0 0
\(280\) 0 0
\(281\) −17.1661 −1.02405 −0.512023 0.858972i \(-0.671104\pi\)
−0.512023 + 0.858972i \(0.671104\pi\)
\(282\) 0 0
\(283\) −9.62005 −0.571853 −0.285926 0.958252i \(-0.592301\pi\)
−0.285926 + 0.958252i \(0.592301\pi\)
\(284\) 31.2419 1.85386
\(285\) 0 0
\(286\) −7.76084 −0.458908
\(287\) 18.9985 1.12145
\(288\) 0 0
\(289\) −1.54198 −0.0907048
\(290\) 0 0
\(291\) 0 0
\(292\) −17.2301 −1.00831
\(293\) −16.8202 −0.982649 −0.491324 0.870977i \(-0.663487\pi\)
−0.491324 + 0.870977i \(0.663487\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −39.6460 −2.30438
\(297\) 0 0
\(298\) −1.89688 −0.109884
\(299\) 10.8775 0.629059
\(300\) 0 0
\(301\) 4.36247 0.251449
\(302\) 18.9312 1.08937
\(303\) 0 0
\(304\) −15.0844 −0.865151
\(305\) 0 0
\(306\) 0 0
\(307\) 11.3212 0.646134 0.323067 0.946376i \(-0.395286\pi\)
0.323067 + 0.946376i \(0.395286\pi\)
\(308\) −9.50081 −0.541359
\(309\) 0 0
\(310\) 0 0
\(311\) −18.5635 −1.05264 −0.526318 0.850288i \(-0.676428\pi\)
−0.526318 + 0.850288i \(0.676428\pi\)
\(312\) 0 0
\(313\) −9.69951 −0.548249 −0.274124 0.961694i \(-0.588388\pi\)
−0.274124 + 0.961694i \(0.588388\pi\)
\(314\) −13.7813 −0.777724
\(315\) 0 0
\(316\) −20.9019 −1.17582
\(317\) −7.47647 −0.419920 −0.209960 0.977710i \(-0.567333\pi\)
−0.209960 + 0.977710i \(0.567333\pi\)
\(318\) 0 0
\(319\) 6.42538 0.359752
\(320\) 0 0
\(321\) 0 0
\(322\) 18.4180 1.02639
\(323\) −4.62972 −0.257605
\(324\) 0 0
\(325\) 0 0
\(326\) −65.1911 −3.61060
\(327\) 0 0
\(328\) 97.4430 5.38039
\(329\) 3.23544 0.178376
\(330\) 0 0
\(331\) −25.2570 −1.38825 −0.694125 0.719854i \(-0.744209\pi\)
−0.694125 + 0.719854i \(0.744209\pi\)
\(332\) −46.3283 −2.54259
\(333\) 0 0
\(334\) −12.2231 −0.668817
\(335\) 0 0
\(336\) 0 0
\(337\) −13.0240 −0.709463 −0.354731 0.934968i \(-0.615428\pi\)
−0.354731 + 0.934968i \(0.615428\pi\)
\(338\) −15.6720 −0.852443
\(339\) 0 0
\(340\) 0 0
\(341\) 7.66295 0.414972
\(342\) 0 0
\(343\) −18.8171 −1.01603
\(344\) 22.3750 1.20638
\(345\) 0 0
\(346\) 7.19398 0.386751
\(347\) 6.00149 0.322177 0.161089 0.986940i \(-0.448500\pi\)
0.161089 + 0.986940i \(0.448500\pi\)
\(348\) 0 0
\(349\) −22.7183 −1.21608 −0.608042 0.793905i \(-0.708045\pi\)
−0.608042 + 0.793905i \(0.708045\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −18.4646 −0.984164
\(353\) 4.85493 0.258402 0.129201 0.991618i \(-0.458759\pi\)
0.129201 + 0.991618i \(0.458759\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −80.6186 −4.27278
\(357\) 0 0
\(358\) 41.9895 2.21921
\(359\) −28.8910 −1.52481 −0.762403 0.647102i \(-0.775981\pi\)
−0.762403 + 0.647102i \(0.775981\pi\)
\(360\) 0 0
\(361\) −17.6134 −0.927020
\(362\) 0.545047 0.0286470
\(363\) 0 0
\(364\) 23.5776 1.23580
\(365\) 0 0
\(366\) 0 0
\(367\) −14.7089 −0.767800 −0.383900 0.923375i \(-0.625419\pi\)
−0.383900 + 0.923375i \(0.625419\pi\)
\(368\) 52.0465 2.71311
\(369\) 0 0
\(370\) 0 0
\(371\) 4.29961 0.223225
\(372\) 0 0
\(373\) −33.0391 −1.71070 −0.855350 0.518051i \(-0.826658\pi\)
−0.855350 + 0.518051i \(0.826658\pi\)
\(374\) −11.3972 −0.589337
\(375\) 0 0
\(376\) 16.5945 0.855797
\(377\) −15.9455 −0.821233
\(378\) 0 0
\(379\) −15.1556 −0.778493 −0.389247 0.921134i \(-0.627265\pi\)
−0.389247 + 0.921134i \(0.627265\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 21.2807 1.08882
\(383\) 8.00811 0.409195 0.204598 0.978846i \(-0.434411\pi\)
0.204598 + 0.978846i \(0.434411\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −44.1397 −2.24665
\(387\) 0 0
\(388\) 56.2121 2.85374
\(389\) −1.07463 −0.0544858 −0.0272429 0.999629i \(-0.508673\pi\)
−0.0272429 + 0.999629i \(0.508673\pi\)
\(390\) 0 0
\(391\) 15.9742 0.807848
\(392\) −35.9428 −1.81538
\(393\) 0 0
\(394\) 46.0475 2.31984
\(395\) 0 0
\(396\) 0 0
\(397\) −24.2139 −1.21526 −0.607630 0.794220i \(-0.707880\pi\)
−0.607630 + 0.794220i \(0.707880\pi\)
\(398\) −5.42373 −0.271867
\(399\) 0 0
\(400\) 0 0
\(401\) 1.99317 0.0995340 0.0497670 0.998761i \(-0.484152\pi\)
0.0497670 + 0.998761i \(0.484152\pi\)
\(402\) 0 0
\(403\) −19.0166 −0.947287
\(404\) 70.8142 3.52314
\(405\) 0 0
\(406\) −26.9992 −1.33995
\(407\) 4.94300 0.245015
\(408\) 0 0
\(409\) 6.66478 0.329552 0.164776 0.986331i \(-0.447310\pi\)
0.164776 + 0.986331i \(0.447310\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.14664 0.352090
\(413\) −2.25387 −0.110906
\(414\) 0 0
\(415\) 0 0
\(416\) 45.8223 2.24662
\(417\) 0 0
\(418\) 3.41351 0.166960
\(419\) −21.3540 −1.04321 −0.521606 0.853186i \(-0.674667\pi\)
−0.521606 + 0.853186i \(0.674667\pi\)
\(420\) 0 0
\(421\) 22.3367 1.08863 0.544313 0.838882i \(-0.316790\pi\)
0.544313 + 0.838882i \(0.316790\pi\)
\(422\) −64.0088 −3.11590
\(423\) 0 0
\(424\) 22.0526 1.07097
\(425\) 0 0
\(426\) 0 0
\(427\) −12.2835 −0.594438
\(428\) −57.4823 −2.77851
\(429\) 0 0
\(430\) 0 0
\(431\) 30.4129 1.46494 0.732469 0.680800i \(-0.238368\pi\)
0.732469 + 0.680800i \(0.238368\pi\)
\(432\) 0 0
\(433\) 0.576639 0.0277115 0.0138558 0.999904i \(-0.495589\pi\)
0.0138558 + 0.999904i \(0.495589\pi\)
\(434\) −32.1994 −1.54562
\(435\) 0 0
\(436\) 18.0039 0.862231
\(437\) −4.78431 −0.228864
\(438\) 0 0
\(439\) 12.8045 0.611125 0.305563 0.952172i \(-0.401155\pi\)
0.305563 + 0.952172i \(0.401155\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 28.2838 1.34532
\(443\) −14.3147 −0.680110 −0.340055 0.940405i \(-0.610446\pi\)
−0.340055 + 0.940405i \(0.610446\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 35.2303 1.66820
\(447\) 0 0
\(448\) 34.3652 1.62360
\(449\) 11.9663 0.564724 0.282362 0.959308i \(-0.408882\pi\)
0.282362 + 0.959308i \(0.408882\pi\)
\(450\) 0 0
\(451\) −12.1490 −0.572076
\(452\) −31.5923 −1.48598
\(453\) 0 0
\(454\) 70.6173 3.31424
\(455\) 0 0
\(456\) 0 0
\(457\) −8.22154 −0.384587 −0.192294 0.981337i \(-0.561593\pi\)
−0.192294 + 0.981337i \(0.561593\pi\)
\(458\) 5.36346 0.250618
\(459\) 0 0
\(460\) 0 0
\(461\) −1.27216 −0.0592505 −0.0296253 0.999561i \(-0.509431\pi\)
−0.0296253 + 0.999561i \(0.509431\pi\)
\(462\) 0 0
\(463\) 32.6764 1.51860 0.759300 0.650740i \(-0.225541\pi\)
0.759300 + 0.650740i \(0.225541\pi\)
\(464\) −76.2960 −3.54195
\(465\) 0 0
\(466\) 1.41217 0.0654175
\(467\) −14.4347 −0.667958 −0.333979 0.942581i \(-0.608391\pi\)
−0.333979 + 0.942581i \(0.608391\pi\)
\(468\) 0 0
\(469\) 20.9953 0.969474
\(470\) 0 0
\(471\) 0 0
\(472\) −11.5600 −0.532094
\(473\) −2.78968 −0.128270
\(474\) 0 0
\(475\) 0 0
\(476\) 34.6250 1.58703
\(477\) 0 0
\(478\) 18.8354 0.861511
\(479\) 16.8618 0.770436 0.385218 0.922826i \(-0.374126\pi\)
0.385218 + 0.922826i \(0.374126\pi\)
\(480\) 0 0
\(481\) −12.2667 −0.559314
\(482\) 6.22196 0.283403
\(483\) 0 0
\(484\) −51.3466 −2.33394
\(485\) 0 0
\(486\) 0 0
\(487\) 39.8235 1.80457 0.902287 0.431135i \(-0.141887\pi\)
0.902287 + 0.431135i \(0.141887\pi\)
\(488\) −63.0016 −2.85195
\(489\) 0 0
\(490\) 0 0
\(491\) −14.0255 −0.632961 −0.316480 0.948599i \(-0.602501\pi\)
−0.316480 + 0.948599i \(0.602501\pi\)
\(492\) 0 0
\(493\) −23.4168 −1.05464
\(494\) −8.47108 −0.381132
\(495\) 0 0
\(496\) −90.9911 −4.08562
\(497\) 10.0966 0.452895
\(498\) 0 0
\(499\) 15.2315 0.681857 0.340929 0.940089i \(-0.389259\pi\)
0.340929 + 0.940089i \(0.389259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.6028 0.741020
\(503\) −23.7330 −1.05820 −0.529101 0.848559i \(-0.677471\pi\)
−0.529101 + 0.848559i \(0.677471\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −11.7778 −0.523586
\(507\) 0 0
\(508\) −105.637 −4.68688
\(509\) 12.5097 0.554482 0.277241 0.960800i \(-0.410580\pi\)
0.277241 + 0.960800i \(0.410580\pi\)
\(510\) 0 0
\(511\) −5.56834 −0.246329
\(512\) −2.43464 −0.107597
\(513\) 0 0
\(514\) −35.5331 −1.56730
\(515\) 0 0
\(516\) 0 0
\(517\) −2.06898 −0.0909936
\(518\) −20.7703 −0.912595
\(519\) 0 0
\(520\) 0 0
\(521\) 27.9638 1.22512 0.612559 0.790425i \(-0.290140\pi\)
0.612559 + 0.790425i \(0.290140\pi\)
\(522\) 0 0
\(523\) 15.7013 0.686569 0.343285 0.939231i \(-0.388460\pi\)
0.343285 + 0.939231i \(0.388460\pi\)
\(524\) 15.9222 0.695564
\(525\) 0 0
\(526\) 18.7460 0.817362
\(527\) −27.9270 −1.21652
\(528\) 0 0
\(529\) −6.49247 −0.282281
\(530\) 0 0
\(531\) 0 0
\(532\) −10.3703 −0.449609
\(533\) 30.1495 1.30592
\(534\) 0 0
\(535\) 0 0
\(536\) 107.685 4.65127
\(537\) 0 0
\(538\) −63.7363 −2.74787
\(539\) 4.48129 0.193023
\(540\) 0 0
\(541\) −23.9783 −1.03091 −0.515455 0.856917i \(-0.672377\pi\)
−0.515455 + 0.856917i \(0.672377\pi\)
\(542\) 77.9000 3.34609
\(543\) 0 0
\(544\) 67.2927 2.88515
\(545\) 0 0
\(546\) 0 0
\(547\) −5.62681 −0.240585 −0.120293 0.992738i \(-0.538383\pi\)
−0.120293 + 0.992738i \(0.538383\pi\)
\(548\) 107.066 4.57365
\(549\) 0 0
\(550\) 0 0
\(551\) 7.01341 0.298781
\(552\) 0 0
\(553\) −6.75498 −0.287251
\(554\) −23.0728 −0.980268
\(555\) 0 0
\(556\) 95.6254 4.05542
\(557\) 42.4247 1.79759 0.898796 0.438366i \(-0.144443\pi\)
0.898796 + 0.438366i \(0.144443\pi\)
\(558\) 0 0
\(559\) 6.92298 0.292811
\(560\) 0 0
\(561\) 0 0
\(562\) −46.1261 −1.94571
\(563\) −5.78597 −0.243850 −0.121925 0.992539i \(-0.538907\pi\)
−0.121925 + 0.992539i \(0.538907\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −25.8495 −1.08653
\(567\) 0 0
\(568\) 51.7853 2.17286
\(569\) 22.0056 0.922522 0.461261 0.887265i \(-0.347397\pi\)
0.461261 + 0.887265i \(0.347397\pi\)
\(570\) 0 0
\(571\) 33.6143 1.40672 0.703358 0.710835i \(-0.251683\pi\)
0.703358 + 0.710835i \(0.251683\pi\)
\(572\) −15.0772 −0.630410
\(573\) 0 0
\(574\) 51.0499 2.13078
\(575\) 0 0
\(576\) 0 0
\(577\) 40.2350 1.67501 0.837503 0.546433i \(-0.184015\pi\)
0.837503 + 0.546433i \(0.184015\pi\)
\(578\) −4.14337 −0.172341
\(579\) 0 0
\(580\) 0 0
\(581\) −14.9722 −0.621151
\(582\) 0 0
\(583\) −2.74948 −0.113872
\(584\) −28.5599 −1.18182
\(585\) 0 0
\(586\) −45.1967 −1.86706
\(587\) 2.82299 0.116517 0.0582587 0.998302i \(-0.481445\pi\)
0.0582587 + 0.998302i \(0.481445\pi\)
\(588\) 0 0
\(589\) 8.36423 0.344642
\(590\) 0 0
\(591\) 0 0
\(592\) −58.6939 −2.41231
\(593\) 24.6805 1.01351 0.506754 0.862091i \(-0.330845\pi\)
0.506754 + 0.862091i \(0.330845\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.68513 −0.150949
\(597\) 0 0
\(598\) 29.2282 1.19523
\(599\) 7.71547 0.315245 0.157623 0.987499i \(-0.449617\pi\)
0.157623 + 0.987499i \(0.449617\pi\)
\(600\) 0 0
\(601\) 22.6506 0.923938 0.461969 0.886896i \(-0.347143\pi\)
0.461969 + 0.886896i \(0.347143\pi\)
\(602\) 11.7222 0.477759
\(603\) 0 0
\(604\) 36.7783 1.49649
\(605\) 0 0
\(606\) 0 0
\(607\) −24.6423 −1.00020 −0.500099 0.865968i \(-0.666703\pi\)
−0.500099 + 0.865968i \(0.666703\pi\)
\(608\) −20.1543 −0.817367
\(609\) 0 0
\(610\) 0 0
\(611\) 5.13445 0.207718
\(612\) 0 0
\(613\) −41.8034 −1.68842 −0.844211 0.536011i \(-0.819931\pi\)
−0.844211 + 0.536011i \(0.819931\pi\)
\(614\) 30.4205 1.22767
\(615\) 0 0
\(616\) −15.7482 −0.634513
\(617\) −15.9390 −0.641680 −0.320840 0.947133i \(-0.603965\pi\)
−0.320840 + 0.947133i \(0.603965\pi\)
\(618\) 0 0
\(619\) −47.1700 −1.89592 −0.947961 0.318385i \(-0.896859\pi\)
−0.947961 + 0.318385i \(0.896859\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −49.8808 −2.00004
\(623\) −26.0540 −1.04383
\(624\) 0 0
\(625\) 0 0
\(626\) −26.0630 −1.04169
\(627\) 0 0
\(628\) −26.7734 −1.06837
\(629\) −18.0144 −0.718280
\(630\) 0 0
\(631\) 32.3634 1.28836 0.644182 0.764872i \(-0.277198\pi\)
0.644182 + 0.764872i \(0.277198\pi\)
\(632\) −34.6462 −1.37815
\(633\) 0 0
\(634\) −20.0896 −0.797859
\(635\) 0 0
\(636\) 0 0
\(637\) −11.1209 −0.440627
\(638\) 17.2653 0.683539
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0042 −0.711122 −0.355561 0.934653i \(-0.615710\pi\)
−0.355561 + 0.934653i \(0.615710\pi\)
\(642\) 0 0
\(643\) 21.8891 0.863224 0.431612 0.902059i \(-0.357945\pi\)
0.431612 + 0.902059i \(0.357945\pi\)
\(644\) 35.7811 1.40997
\(645\) 0 0
\(646\) −12.4403 −0.489455
\(647\) −19.0263 −0.748003 −0.374001 0.927428i \(-0.622015\pi\)
−0.374001 + 0.927428i \(0.622015\pi\)
\(648\) 0 0
\(649\) 1.44129 0.0565755
\(650\) 0 0
\(651\) 0 0
\(652\) −126.649 −4.95995
\(653\) 2.31971 0.0907774 0.0453887 0.998969i \(-0.485547\pi\)
0.0453887 + 0.998969i \(0.485547\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 144.260 5.63239
\(657\) 0 0
\(658\) 8.69378 0.338919
\(659\) −7.19180 −0.280153 −0.140076 0.990141i \(-0.544735\pi\)
−0.140076 + 0.990141i \(0.544735\pi\)
\(660\) 0 0
\(661\) 43.0214 1.67334 0.836669 0.547709i \(-0.184500\pi\)
0.836669 + 0.547709i \(0.184500\pi\)
\(662\) −67.8666 −2.63771
\(663\) 0 0
\(664\) −76.7920 −2.98011
\(665\) 0 0
\(666\) 0 0
\(667\) −24.1987 −0.936977
\(668\) −23.7461 −0.918765
\(669\) 0 0
\(670\) 0 0
\(671\) 7.85494 0.303237
\(672\) 0 0
\(673\) −29.8031 −1.14882 −0.574412 0.818566i \(-0.694769\pi\)
−0.574412 + 0.818566i \(0.694769\pi\)
\(674\) −34.9961 −1.34800
\(675\) 0 0
\(676\) −30.4464 −1.17102
\(677\) 26.7346 1.02749 0.513746 0.857942i \(-0.328257\pi\)
0.513746 + 0.857942i \(0.328257\pi\)
\(678\) 0 0
\(679\) 18.1664 0.697163
\(680\) 0 0
\(681\) 0 0
\(682\) 20.5907 0.788457
\(683\) −28.4950 −1.09033 −0.545165 0.838329i \(-0.683533\pi\)
−0.545165 + 0.838329i \(0.683533\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −50.5623 −1.93048
\(687\) 0 0
\(688\) 33.1251 1.26288
\(689\) 6.82321 0.259944
\(690\) 0 0
\(691\) 34.6469 1.31803 0.659015 0.752130i \(-0.270973\pi\)
0.659015 + 0.752130i \(0.270973\pi\)
\(692\) 13.9760 0.531286
\(693\) 0 0
\(694\) 16.1263 0.612145
\(695\) 0 0
\(696\) 0 0
\(697\) 44.2763 1.67708
\(698\) −61.0451 −2.31059
\(699\) 0 0
\(700\) 0 0
\(701\) −0.973305 −0.0367612 −0.0183806 0.999831i \(-0.505851\pi\)
−0.0183806 + 0.999831i \(0.505851\pi\)
\(702\) 0 0
\(703\) 5.39536 0.203490
\(704\) −21.9756 −0.828237
\(705\) 0 0
\(706\) 13.0454 0.490971
\(707\) 22.8854 0.860696
\(708\) 0 0
\(709\) −11.3242 −0.425291 −0.212645 0.977129i \(-0.568208\pi\)
−0.212645 + 0.977129i \(0.568208\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −133.630 −5.00801
\(713\) −28.8595 −1.08080
\(714\) 0 0
\(715\) 0 0
\(716\) 81.5742 3.04857
\(717\) 0 0
\(718\) −77.6312 −2.89717
\(719\) 30.7320 1.14611 0.573056 0.819516i \(-0.305758\pi\)
0.573056 + 0.819516i \(0.305758\pi\)
\(720\) 0 0
\(721\) 2.30962 0.0860149
\(722\) −47.3279 −1.76136
\(723\) 0 0
\(724\) 1.05888 0.0393529
\(725\) 0 0
\(726\) 0 0
\(727\) −6.56836 −0.243607 −0.121803 0.992554i \(-0.538868\pi\)
−0.121803 + 0.992554i \(0.538868\pi\)
\(728\) 39.0813 1.44845
\(729\) 0 0
\(730\) 0 0
\(731\) 10.1668 0.376032
\(732\) 0 0
\(733\) −31.9276 −1.17927 −0.589636 0.807669i \(-0.700729\pi\)
−0.589636 + 0.807669i \(0.700729\pi\)
\(734\) −39.5235 −1.45884
\(735\) 0 0
\(736\) 69.5395 2.56326
\(737\) −13.4259 −0.494551
\(738\) 0 0
\(739\) −15.3072 −0.563083 −0.281542 0.959549i \(-0.590846\pi\)
−0.281542 + 0.959549i \(0.590846\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.5532 0.424132
\(743\) −16.5455 −0.606995 −0.303498 0.952832i \(-0.598154\pi\)
−0.303498 + 0.952832i \(0.598154\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −88.7774 −3.25037
\(747\) 0 0
\(748\) −22.1417 −0.809582
\(749\) −18.5769 −0.678786
\(750\) 0 0
\(751\) 46.0279 1.67958 0.839791 0.542911i \(-0.182678\pi\)
0.839791 + 0.542911i \(0.182678\pi\)
\(752\) 24.5674 0.895880
\(753\) 0 0
\(754\) −42.8461 −1.56036
\(755\) 0 0
\(756\) 0 0
\(757\) −26.5282 −0.964184 −0.482092 0.876121i \(-0.660123\pi\)
−0.482092 + 0.876121i \(0.660123\pi\)
\(758\) −40.7239 −1.47916
\(759\) 0 0
\(760\) 0 0
\(761\) 28.4241 1.03037 0.515187 0.857078i \(-0.327723\pi\)
0.515187 + 0.857078i \(0.327723\pi\)
\(762\) 0 0
\(763\) 5.81843 0.210641
\(764\) 41.3426 1.49572
\(765\) 0 0
\(766\) 21.5181 0.777481
\(767\) −3.57675 −0.129149
\(768\) 0 0
\(769\) −8.95952 −0.323089 −0.161544 0.986865i \(-0.551647\pi\)
−0.161544 + 0.986865i \(0.551647\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −85.7515 −3.08626
\(773\) −23.1042 −0.831001 −0.415500 0.909593i \(-0.636393\pi\)
−0.415500 + 0.909593i \(0.636393\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 93.1752 3.34479
\(777\) 0 0
\(778\) −2.88757 −0.103524
\(779\) −13.2609 −0.475120
\(780\) 0 0
\(781\) −6.45651 −0.231032
\(782\) 42.9232 1.53493
\(783\) 0 0
\(784\) −53.2115 −1.90041
\(785\) 0 0
\(786\) 0 0
\(787\) −2.03390 −0.0725009 −0.0362504 0.999343i \(-0.511541\pi\)
−0.0362504 + 0.999343i \(0.511541\pi\)
\(788\) 89.4578 3.18680
\(789\) 0 0
\(790\) 0 0
\(791\) −10.2099 −0.363022
\(792\) 0 0
\(793\) −19.4931 −0.692220
\(794\) −65.0637 −2.30902
\(795\) 0 0
\(796\) −10.5368 −0.373469
\(797\) 12.1923 0.431874 0.215937 0.976407i \(-0.430719\pi\)
0.215937 + 0.976407i \(0.430719\pi\)
\(798\) 0 0
\(799\) 7.54024 0.266754
\(800\) 0 0
\(801\) 0 0
\(802\) 5.35572 0.189117
\(803\) 3.56080 0.125658
\(804\) 0 0
\(805\) 0 0
\(806\) −51.0985 −1.79987
\(807\) 0 0
\(808\) 117.379 4.12938
\(809\) −8.83355 −0.310571 −0.155286 0.987870i \(-0.549630\pi\)
−0.155286 + 0.987870i \(0.549630\pi\)
\(810\) 0 0
\(811\) −13.9184 −0.488741 −0.244370 0.969682i \(-0.578581\pi\)
−0.244370 + 0.969682i \(0.578581\pi\)
\(812\) −52.4522 −1.84071
\(813\) 0 0
\(814\) 13.2820 0.465535
\(815\) 0 0
\(816\) 0 0
\(817\) −3.04498 −0.106530
\(818\) 17.9085 0.626158
\(819\) 0 0
\(820\) 0 0
\(821\) 29.4930 1.02931 0.514656 0.857397i \(-0.327920\pi\)
0.514656 + 0.857397i \(0.327920\pi\)
\(822\) 0 0
\(823\) −41.9661 −1.46285 −0.731423 0.681924i \(-0.761143\pi\)
−0.731423 + 0.681924i \(0.761143\pi\)
\(824\) 11.8460 0.412675
\(825\) 0 0
\(826\) −6.05624 −0.210724
\(827\) −37.4674 −1.30287 −0.651435 0.758704i \(-0.725833\pi\)
−0.651435 + 0.758704i \(0.725833\pi\)
\(828\) 0 0
\(829\) −9.55614 −0.331899 −0.165949 0.986134i \(-0.553069\pi\)
−0.165949 + 0.986134i \(0.553069\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 54.5355 1.89068
\(833\) −16.3317 −0.565860
\(834\) 0 0
\(835\) 0 0
\(836\) 6.63152 0.229356
\(837\) 0 0
\(838\) −57.3792 −1.98213
\(839\) −54.7310 −1.88952 −0.944761 0.327759i \(-0.893707\pi\)
−0.944761 + 0.327759i \(0.893707\pi\)
\(840\) 0 0
\(841\) 6.47334 0.223219
\(842\) 60.0197 2.06842
\(843\) 0 0
\(844\) −124.352 −4.28037
\(845\) 0 0
\(846\) 0 0
\(847\) −16.5940 −0.570177
\(848\) 32.6478 1.12113
\(849\) 0 0
\(850\) 0 0
\(851\) −18.6159 −0.638144
\(852\) 0 0
\(853\) −15.3067 −0.524092 −0.262046 0.965055i \(-0.584397\pi\)
−0.262046 + 0.965055i \(0.584397\pi\)
\(854\) −33.0062 −1.12945
\(855\) 0 0
\(856\) −95.2806 −3.25662
\(857\) 8.66910 0.296131 0.148065 0.988978i \(-0.452695\pi\)
0.148065 + 0.988978i \(0.452695\pi\)
\(858\) 0 0
\(859\) −30.4078 −1.03750 −0.518750 0.854926i \(-0.673603\pi\)
−0.518750 + 0.854926i \(0.673603\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 81.7208 2.78342
\(863\) 18.0425 0.614175 0.307088 0.951681i \(-0.400646\pi\)
0.307088 + 0.951681i \(0.400646\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.54945 0.0526526
\(867\) 0 0
\(868\) −62.5548 −2.12325
\(869\) 4.31963 0.146533
\(870\) 0 0
\(871\) 33.3183 1.12895
\(872\) 29.8426 1.01060
\(873\) 0 0
\(874\) −12.8556 −0.434848
\(875\) 0 0
\(876\) 0 0
\(877\) 22.3057 0.753211 0.376605 0.926374i \(-0.377091\pi\)
0.376605 + 0.926374i \(0.377091\pi\)
\(878\) 34.4062 1.16115
\(879\) 0 0
\(880\) 0 0
\(881\) −7.75956 −0.261426 −0.130713 0.991420i \(-0.541727\pi\)
−0.130713 + 0.991420i \(0.541727\pi\)
\(882\) 0 0
\(883\) −19.9527 −0.671462 −0.335731 0.941958i \(-0.608983\pi\)
−0.335731 + 0.941958i \(0.608983\pi\)
\(884\) 54.9477 1.84809
\(885\) 0 0
\(886\) −38.4641 −1.29223
\(887\) −8.60289 −0.288857 −0.144428 0.989515i \(-0.546134\pi\)
−0.144428 + 0.989515i \(0.546134\pi\)
\(888\) 0 0
\(889\) −34.1393 −1.14499
\(890\) 0 0
\(891\) 0 0
\(892\) 68.4429 2.29164
\(893\) −2.25832 −0.0755719
\(894\) 0 0
\(895\) 0 0
\(896\) 34.5915 1.15562
\(897\) 0 0
\(898\) 32.1539 1.07299
\(899\) 42.3057 1.41097
\(900\) 0 0
\(901\) 10.0203 0.333824
\(902\) −32.6450 −1.08696
\(903\) 0 0
\(904\) −52.3663 −1.74168
\(905\) 0 0
\(906\) 0 0
\(907\) 6.26125 0.207902 0.103951 0.994582i \(-0.466852\pi\)
0.103951 + 0.994582i \(0.466852\pi\)
\(908\) 137.190 4.55282
\(909\) 0 0
\(910\) 0 0
\(911\) −0.905575 −0.0300030 −0.0150015 0.999887i \(-0.504775\pi\)
−0.0150015 + 0.999887i \(0.504775\pi\)
\(912\) 0 0
\(913\) 9.57431 0.316863
\(914\) −22.0916 −0.730726
\(915\) 0 0
\(916\) 10.4198 0.344279
\(917\) 5.14567 0.169925
\(918\) 0 0
\(919\) 43.9933 1.45120 0.725602 0.688114i \(-0.241561\pi\)
0.725602 + 0.688114i \(0.241561\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.41836 −0.112578
\(923\) 16.0227 0.527394
\(924\) 0 0
\(925\) 0 0
\(926\) 87.8029 2.88538
\(927\) 0 0
\(928\) −101.939 −3.34632
\(929\) 36.7447 1.20555 0.602777 0.797910i \(-0.294061\pi\)
0.602777 + 0.797910i \(0.294061\pi\)
\(930\) 0 0
\(931\) 4.89140 0.160309
\(932\) 2.74347 0.0898652
\(933\) 0 0
\(934\) −38.7866 −1.26914
\(935\) 0 0
\(936\) 0 0
\(937\) 24.6490 0.805248 0.402624 0.915366i \(-0.368098\pi\)
0.402624 + 0.915366i \(0.368098\pi\)
\(938\) 56.4153 1.84203
\(939\) 0 0
\(940\) 0 0
\(941\) 59.1479 1.92817 0.964083 0.265602i \(-0.0855706\pi\)
0.964083 + 0.265602i \(0.0855706\pi\)
\(942\) 0 0
\(943\) 45.7546 1.48998
\(944\) −17.1141 −0.557016
\(945\) 0 0
\(946\) −7.49600 −0.243716
\(947\) 30.0960 0.977989 0.488995 0.872287i \(-0.337364\pi\)
0.488995 + 0.872287i \(0.337364\pi\)
\(948\) 0 0
\(949\) −8.83661 −0.286849
\(950\) 0 0
\(951\) 0 0
\(952\) 57.3931 1.86012
\(953\) 20.6393 0.668572 0.334286 0.942472i \(-0.391505\pi\)
0.334286 + 0.942472i \(0.391505\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 36.5921 1.18347
\(957\) 0 0
\(958\) 45.3084 1.46385
\(959\) 34.6013 1.11733
\(960\) 0 0
\(961\) 19.4540 0.627549
\(962\) −32.9612 −1.06271
\(963\) 0 0
\(964\) 12.0876 0.389315
\(965\) 0 0
\(966\) 0 0
\(967\) −22.6016 −0.726817 −0.363408 0.931630i \(-0.618387\pi\)
−0.363408 + 0.931630i \(0.618387\pi\)
\(968\) −85.1103 −2.73555
\(969\) 0 0
\(970\) 0 0
\(971\) −24.1381 −0.774628 −0.387314 0.921948i \(-0.626597\pi\)
−0.387314 + 0.921948i \(0.626597\pi\)
\(972\) 0 0
\(973\) 30.9038 0.990732
\(974\) 107.007 3.42874
\(975\) 0 0
\(976\) −93.2708 −2.98553
\(977\) −23.0506 −0.737454 −0.368727 0.929538i \(-0.620206\pi\)
−0.368727 + 0.929538i \(0.620206\pi\)
\(978\) 0 0
\(979\) 16.6608 0.532483
\(980\) 0 0
\(981\) 0 0
\(982\) −37.6870 −1.20264
\(983\) −29.8713 −0.952745 −0.476373 0.879243i \(-0.658049\pi\)
−0.476373 + 0.879243i \(0.658049\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −62.9220 −2.00384
\(987\) 0 0
\(988\) −16.4570 −0.523567
\(989\) 10.5063 0.334079
\(990\) 0 0
\(991\) 16.6560 0.529096 0.264548 0.964372i \(-0.414777\pi\)
0.264548 + 0.964372i \(0.414777\pi\)
\(992\) −121.573 −3.85996
\(993\) 0 0
\(994\) 27.1300 0.860513
\(995\) 0 0
\(996\) 0 0
\(997\) 12.4934 0.395669 0.197835 0.980235i \(-0.436609\pi\)
0.197835 + 0.980235i \(0.436609\pi\)
\(998\) 40.9278 1.29555
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5625.2.a.p.1.6 6
3.2 odd 2 1875.2.a.j.1.1 6
5.4 even 2 5625.2.a.q.1.1 6
15.2 even 4 1875.2.b.f.1249.1 12
15.8 even 4 1875.2.b.f.1249.12 12
15.14 odd 2 1875.2.a.k.1.6 6
25.11 even 5 225.2.h.d.46.3 12
25.16 even 5 225.2.h.d.181.3 12
75.2 even 20 375.2.i.d.274.1 24
75.11 odd 10 75.2.g.c.46.1 yes 12
75.14 odd 10 375.2.g.c.226.3 12
75.23 even 20 375.2.i.d.274.6 24
75.38 even 20 375.2.i.d.349.1 24
75.41 odd 10 75.2.g.c.31.1 12
75.59 odd 10 375.2.g.c.151.3 12
75.62 even 20 375.2.i.d.349.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.31.1 12 75.41 odd 10
75.2.g.c.46.1 yes 12 75.11 odd 10
225.2.h.d.46.3 12 25.11 even 5
225.2.h.d.181.3 12 25.16 even 5
375.2.g.c.151.3 12 75.59 odd 10
375.2.g.c.226.3 12 75.14 odd 10
375.2.i.d.274.1 24 75.2 even 20
375.2.i.d.274.6 24 75.23 even 20
375.2.i.d.349.1 24 75.38 even 20
375.2.i.d.349.6 24 75.62 even 20
1875.2.a.j.1.1 6 3.2 odd 2
1875.2.a.k.1.6 6 15.14 odd 2
1875.2.b.f.1249.1 12 15.2 even 4
1875.2.b.f.1249.12 12 15.8 even 4
5625.2.a.p.1.6 6 1.1 even 1 trivial
5625.2.a.q.1.1 6 5.4 even 2