Properties

Label 1875.2.a.j.1.1
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(1\)
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 1.1
Root \(2.68704\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.68704 q^{2} -1.00000 q^{3} +5.22020 q^{4} +2.68704 q^{6} +1.68704 q^{7} -8.65280 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.68704 q^{2} -1.00000 q^{3} +5.22020 q^{4} +2.68704 q^{6} +1.68704 q^{7} -8.65280 q^{8} +1.00000 q^{9} +1.07882 q^{11} -5.22020 q^{12} +2.67723 q^{13} -4.53315 q^{14} +12.8101 q^{16} -3.93167 q^{17} -2.68704 q^{18} -1.17755 q^{19} -1.68704 q^{21} -2.89883 q^{22} -4.06295 q^{23} +8.65280 q^{24} -7.19384 q^{26} -1.00000 q^{27} +8.80669 q^{28} +5.95595 q^{29} -7.10310 q^{31} -17.1155 q^{32} -1.07882 q^{33} +10.5646 q^{34} +5.22020 q^{36} -4.58187 q^{37} +3.16412 q^{38} -2.67723 q^{39} -11.2614 q^{41} +4.53315 q^{42} +2.58587 q^{43} +5.63164 q^{44} +10.9173 q^{46} -1.91782 q^{47} -12.8101 q^{48} -4.15389 q^{49} +3.93167 q^{51} +13.9757 q^{52} -2.54861 q^{53} +2.68704 q^{54} -14.5976 q^{56} +1.17755 q^{57} -16.0039 q^{58} +1.33599 q^{59} -7.28106 q^{61} +19.0863 q^{62} +1.68704 q^{63} +20.3701 q^{64} +2.89883 q^{66} +12.4451 q^{67} -20.5241 q^{68} +4.06295 q^{69} -5.98480 q^{71} -8.65280 q^{72} -3.30065 q^{73} +12.3117 q^{74} -6.14702 q^{76} +1.82001 q^{77} +7.19384 q^{78} -4.00404 q^{79} +1.00000 q^{81} +30.2600 q^{82} +8.87482 q^{83} -8.80669 q^{84} -6.94834 q^{86} -5.95595 q^{87} -9.33480 q^{88} +15.4436 q^{89} +4.51660 q^{91} -21.2094 q^{92} +7.10310 q^{93} +5.15327 q^{94} +17.1155 q^{96} +10.7682 q^{97} +11.1617 q^{98} +1.07882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 10 q^{4} - 6 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 10 q^{4} - 6 q^{7} - 3 q^{8} + 6 q^{9} + 3 q^{11} - 10 q^{12} - 6 q^{13} - 22 q^{14} + 18 q^{16} - 13 q^{17} + 11 q^{19} + 6 q^{21} - 16 q^{22} - 13 q^{23} + 3 q^{24} - 28 q^{26} - 6 q^{27} - 7 q^{28} - 3 q^{29} - 11 q^{31} - 16 q^{32} - 3 q^{33} + 15 q^{34} + 10 q^{36} - 21 q^{37} + 9 q^{38} + 6 q^{39} - q^{41} + 22 q^{42} - 2 q^{43} + 9 q^{44} + 19 q^{46} - 14 q^{47} - 18 q^{48} - 14 q^{49} + 13 q^{51} - 13 q^{52} - 23 q^{53} - 35 q^{56} - 11 q^{57} - 22 q^{58} + 9 q^{59} + 11 q^{61} + 23 q^{62} - 6 q^{63} - 23 q^{64} + 16 q^{66} - 8 q^{67} - 50 q^{68} + 13 q^{69} - 8 q^{71} - 3 q^{72} - 13 q^{73} - 22 q^{74} - 26 q^{76} + 13 q^{77} + 28 q^{78} - 5 q^{79} + 6 q^{81} + 13 q^{82} + 20 q^{83} + 7 q^{84} - 37 q^{86} + 3 q^{87} - 28 q^{88} - 4 q^{89} + 34 q^{91} - 61 q^{92} + 11 q^{93} + 41 q^{94} + 16 q^{96} + 7 q^{97} + 41 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.68704 −1.90003 −0.950013 0.312211i \(-0.898931\pi\)
−0.950013 + 0.312211i \(0.898931\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.22020 2.61010
\(5\) 0 0
\(6\) 2.68704 1.09698
\(7\) 1.68704 0.637642 0.318821 0.947815i \(-0.396713\pi\)
0.318821 + 0.947815i \(0.396713\pi\)
\(8\) −8.65280 −3.05923
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.07882 0.325276 0.162638 0.986686i \(-0.448000\pi\)
0.162638 + 0.986686i \(0.448000\pi\)
\(12\) −5.22020 −1.50694
\(13\) 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 \(-0.658198\pi\)
−0.476785 + 0.879020i \(0.658198\pi\)
\(18\) −2.68704 −0.633342
\(19\) −1.17755 −0.270148 −0.135074 0.990836i \(-0.543127\pi\)
−0.135074 + 0.990836i \(0.543127\pi\)
\(20\) 0 0
\(21\) −1.68704 −0.368143
\(22\) −2.89883 −0.618032
\(23\) −4.06295 −0.847183 −0.423591 0.905853i \(-0.639231\pi\)
−0.423591 + 0.905853i \(0.639231\pi\)
\(24\) 8.65280 1.76625
\(25\) 0 0
\(26\) −7.19384 −1.41083
\(27\) −1.00000 −0.192450
\(28\) 8.80669 1.66431
\(29\) 5.95595 1.10599 0.552996 0.833184i \(-0.313484\pi\)
0.552996 + 0.833184i \(0.313484\pi\)
\(30\) 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
\(33\) −1.07882 −0.187798
\(34\) 10.5646 1.81181
\(35\) 0 0
\(36\) 5.22020 0.870033
\(37\) −4.58187 −0.753254 −0.376627 0.926365i \(-0.622916\pi\)
−0.376627 + 0.926365i \(0.622916\pi\)
\(38\) 3.16412 0.513287
\(39\) −2.67723 −0.428700
\(40\) 0 0
\(41\) −11.2614 −1.75874 −0.879371 0.476137i \(-0.842037\pi\)
−0.879371 + 0.476137i \(0.842037\pi\)
\(42\) 4.53315 0.699481
\(43\) 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 \(-0.544669\pi\)
−0.139871 + 0.990170i \(0.544669\pi\)
\(48\) −12.8101 −1.84897
\(49\) −4.15389 −0.593413
\(50\) 0 0
\(51\) 3.93167 0.550544
\(52\) 13.9757 1.93808
\(53\) −2.54861 −0.350078 −0.175039 0.984561i \(-0.556005\pi\)
−0.175039 + 0.984561i \(0.556005\pi\)
\(54\) 2.68704 0.365660
\(55\) 0 0
\(56\) −14.5976 −1.95069
\(57\) 1.17755 0.155970
\(58\) −16.0039 −2.10141
\(59\) 1.33599 0.173931 0.0869654 0.996211i \(-0.472283\pi\)
0.0869654 + 0.996211i \(0.472283\pi\)
\(60\) 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
\(63\) 1.68704 0.212547
\(64\) 20.3701 2.54626
\(65\) 0 0
\(66\) 2.89883 0.356821
\(67\) 12.4451 1.52041 0.760203 0.649686i \(-0.225100\pi\)
0.760203 + 0.649686i \(0.225100\pi\)
\(68\) −20.5241 −2.48891
\(69\) 4.06295 0.489121
\(70\) 0 0
\(71\) −5.98480 −0.710266 −0.355133 0.934816i \(-0.615564\pi\)
−0.355133 + 0.934816i \(0.615564\pi\)
\(72\) −8.65280 −1.01974
\(73\) −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
\(78\) 7.19384 0.814542
\(79\) −4.00404 −0.450490 −0.225245 0.974302i \(-0.572318\pi\)
−0.225245 + 0.974302i \(0.572318\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 30.2600 3.34166
\(83\) 8.87482 0.974138 0.487069 0.873364i \(-0.338066\pi\)
0.487069 + 0.873364i \(0.338066\pi\)
\(84\) −8.80669 −0.960889
\(85\) 0 0
\(86\) −6.94834 −0.749259
\(87\) −5.95595 −0.638545
\(88\) −9.33480 −0.995093
\(89\) 15.4436 1.63702 0.818509 0.574493i \(-0.194801\pi\)
0.818509 + 0.574493i \(0.194801\pi\)
\(90\) 0 0
\(91\) 4.51660 0.473469
\(92\) −21.2094 −2.21123
\(93\) 7.10310 0.736557
\(94\) 5.15327 0.531519
\(95\) 0 0
\(96\) 17.1155 1.74685
\(97\) 10.7682 1.09335 0.546673 0.837346i \(-0.315894\pi\)
0.546673 + 0.837346i \(0.315894\pi\)
\(98\) 11.1617 1.12750
\(99\) 1.07882 0.108425
\(100\) 0 0
\(101\) −13.5654 −1.34981 −0.674905 0.737905i \(-0.735815\pi\)
−0.674905 + 0.737905i \(0.735815\pi\)
\(102\) −10.5646 −1.04605
\(103\) 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 \(-0.321342\pi\)
0.532262 + 0.846579i \(0.321342\pi\)
\(108\) −5.22020 −0.502314
\(109\) 3.44890 0.330344 0.165172 0.986265i \(-0.447182\pi\)
0.165172 + 0.986265i \(0.447182\pi\)
\(110\) 0 0
\(111\) 4.58187 0.434891
\(112\) 21.6111 2.04206
\(113\) 6.05194 0.569319 0.284660 0.958629i \(-0.408119\pi\)
0.284660 + 0.958629i \(0.408119\pi\)
\(114\) −3.16412 −0.296347
\(115\) 0 0
\(116\) 31.0912 2.88675
\(117\) 2.67723 0.247510
\(118\) −3.58986 −0.330473
\(119\) −6.63289 −0.608036
\(120\) 0 0
\(121\) −9.83615 −0.894196
\(122\) 19.5645 1.77129
\(123\) 11.2614 1.01541
\(124\) −37.0796 −3.32984
\(125\) 0 0
\(126\) −4.53315 −0.403845
\(127\) −20.2362 −1.79567 −0.897835 0.440332i \(-0.854861\pi\)
−0.897835 + 0.440332i \(0.854861\pi\)
\(128\) −20.5042 −1.81233
\(129\) −2.58587 −0.227673
\(130\) 0 0
\(131\) −3.05011 −0.266490 −0.133245 0.991083i \(-0.542540\pi\)
−0.133245 + 0.991083i \(0.542540\pi\)
\(132\) −5.63164 −0.490171
\(133\) −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 \(-0.839893\pi\)
−0.876145 + 0.482048i \(0.839893\pi\)
\(138\) −10.9173 −0.929343
\(139\) 18.3184 1.55374 0.776871 0.629659i \(-0.216806\pi\)
0.776871 + 0.629659i \(0.216806\pi\)
\(140\) 0 0
\(141\) 1.91782 0.161510
\(142\) 16.0814 1.34952
\(143\) 2.88825 0.241527
\(144\) 12.8101 1.06750
\(145\) 0 0
\(146\) 8.86899 0.734003
\(147\) 4.15389 0.342607
\(148\) −23.9182 −1.96607
\(149\) 0.705938 0.0578327 0.0289163 0.999582i \(-0.490794\pi\)
0.0289163 + 0.999582i \(0.490794\pi\)
\(150\) 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
\(153\) −3.93167 −0.317857
\(154\) −4.89045 −0.394083
\(155\) 0 0
\(156\) −13.9757 −1.11895
\(157\) −5.12880 −0.409323 −0.204662 0.978833i \(-0.565609\pi\)
−0.204662 + 0.978833i \(0.565609\pi\)
\(158\) 10.7590 0.855942
\(159\) 2.54861 0.202118
\(160\) 0 0
\(161\) −6.85436 −0.540199
\(162\) −2.68704 −0.211114
\(163\) −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 \(-0.443683\pi\)
0.176002 + 0.984390i \(0.443683\pi\)
\(168\) 14.5976 1.12623
\(169\) −5.83243 −0.448648
\(170\) 0 0
\(171\) −1.17755 −0.0900492
\(172\) 13.4988 1.02927
\(173\) −2.67729 −0.203550 −0.101775 0.994807i \(-0.532452\pi\)
−0.101775 + 0.994807i \(0.532452\pi\)
\(174\) 16.0039 1.21325
\(175\) 0 0
\(176\) 13.8197 1.04170
\(177\) −1.33599 −0.100419
\(178\) −41.4976 −3.11038
\(179\) −15.6266 −1.16799 −0.583995 0.811757i \(-0.698511\pi\)
−0.583995 + 0.811757i \(0.698511\pi\)
\(180\) 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
\(183\) 7.28106 0.538232
\(184\) 35.1559 2.59172
\(185\) 0 0
\(186\) −19.0863 −1.39948
\(187\) −4.24155 −0.310173
\(188\) −10.0114 −0.730156
\(189\) −1.68704 −0.122714
\(190\) 0 0
\(191\) −7.91975 −0.573053 −0.286526 0.958072i \(-0.592501\pi\)
−0.286526 + 0.958072i \(0.592501\pi\)
\(192\) −20.3701 −1.47008
\(193\) −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 \(-0.709022\pi\)
−0.610475 + 0.792035i \(0.709022\pi\)
\(198\) −2.89883 −0.206011
\(199\) −2.01848 −0.143086 −0.0715430 0.997438i \(-0.522792\pi\)
−0.0715430 + 0.997438i \(0.522792\pi\)
\(200\) 0 0
\(201\) −12.4451 −0.877806
\(202\) 36.4509 2.56467
\(203\) 10.0479 0.705227
\(204\) 20.5241 1.43697
\(205\) 0 0
\(206\) −3.67866 −0.256304
\(207\) −4.06295 −0.282394
\(208\) 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
\(213\) 5.98480 0.410072
\(214\) −29.5884 −2.02262
\(215\) 0 0
\(216\) 8.65280 0.588749
\(217\) −11.9832 −0.813475
\(218\) −9.26733 −0.627663
\(219\) 3.30065 0.223037
\(220\) 0 0
\(221\) −10.5260 −0.708055
\(222\) −12.3117 −0.826305
\(223\) 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 \(-0.837278\pi\)
−0.872155 + 0.489229i \(0.837278\pi\)
\(228\) 6.14702 0.407096
\(229\) 1.99605 0.131903 0.0659513 0.997823i \(-0.478992\pi\)
0.0659513 + 0.997823i \(0.478992\pi\)
\(230\) 0 0
\(231\) −1.82001 −0.119748
\(232\) −51.5357 −3.38348
\(233\) −0.525548 −0.0344298 −0.0172149 0.999852i \(-0.505480\pi\)
−0.0172149 + 0.999852i \(0.505480\pi\)
\(234\) −7.19384 −0.470276
\(235\) 0 0
\(236\) 6.97412 0.453976
\(237\) 4.00404 0.260090
\(238\) 17.8229 1.15528
\(239\) −7.00971 −0.453421 −0.226710 0.973962i \(-0.572797\pi\)
−0.226710 + 0.973962i \(0.572797\pi\)
\(240\) 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
\(243\) −1.00000 −0.0641500
\(244\) −38.0086 −2.43325
\(245\) 0 0
\(246\) −30.2600 −1.92931
\(247\) −3.15257 −0.200593
\(248\) 61.4617 3.90282
\(249\) −8.87482 −0.562419
\(250\) 0 0
\(251\) −6.17885 −0.390005 −0.195003 0.980803i \(-0.562472\pi\)
−0.195003 + 0.980803i \(0.562472\pi\)
\(252\) 8.80669 0.554769
\(253\) −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 \(-0.364676\pi\)
0.412441 + 0.910984i \(0.364676\pi\)
\(258\) 6.94834 0.432585
\(259\) −7.72980 −0.480306
\(260\) 0 0
\(261\) 5.95595 0.368664
\(262\) 8.19578 0.506337
\(263\) −6.97643 −0.430185 −0.215092 0.976594i \(-0.569005\pi\)
−0.215092 + 0.976594i \(0.569005\pi\)
\(264\) 9.33480 0.574517
\(265\) 0 0
\(266\) 5.33800 0.327294
\(267\) −15.4436 −0.945133
\(268\) 64.9656 3.96841
\(269\) 23.7199 1.44623 0.723113 0.690730i \(-0.242711\pi\)
0.723113 + 0.690730i \(0.242711\pi\)
\(270\) 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
\(273\) −4.51660 −0.273357
\(274\) 55.1113 3.32940
\(275\) 0 0
\(276\) 21.2094 1.27665
\(277\) −8.58667 −0.515923 −0.257962 0.966155i \(-0.583051\pi\)
−0.257962 + 0.966155i \(0.583051\pi\)
\(278\) −49.2222 −2.95215
\(279\) −7.10310 −0.425251
\(280\) 0 0
\(281\) 17.1661 1.02405 0.512023 0.858972i \(-0.328896\pi\)
0.512023 + 0.858972i \(0.328896\pi\)
\(282\) −5.15327 −0.306872
\(283\) −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
\(288\) −17.1155 −1.00854
\(289\) −1.54198 −0.0907048
\(290\) 0 0
\(291\) −10.7682 −0.631243
\(292\) −17.2301 −1.00831
\(293\) 16.8202 0.982649 0.491324 0.870977i \(-0.336513\pi\)
0.491324 + 0.870977i \(0.336513\pi\)
\(294\) −11.1617 −0.650962
\(295\) 0 0
\(296\) 39.6460 2.30438
\(297\) −1.07882 −0.0625994
\(298\) −1.89688 −0.109884
\(299\) −10.8775 −0.629059
\(300\) 0 0
\(301\) 4.36247 0.251449
\(302\) −18.9312 −1.08937
\(303\) 13.5654 0.779313
\(304\) −15.0844 −0.865151
\(305\) 0 0
\(306\) 10.5646 0.603936
\(307\) 11.3212 0.646134 0.323067 0.946376i \(-0.395286\pi\)
0.323067 + 0.946376i \(0.395286\pi\)
\(308\) 9.50081 0.541359
\(309\) −1.36904 −0.0778818
\(310\) 0 0
\(311\) 18.5635 1.05264 0.526318 0.850288i \(-0.323572\pi\)
0.526318 + 0.850288i \(0.323572\pi\)
\(312\) 23.1656 1.31149
\(313\) −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 \(-0.432667\pi\)
0.209960 + 0.977710i \(0.432667\pi\)
\(318\) −6.84822 −0.384029
\(319\) 6.42538 0.359752
\(320\) 0 0
\(321\) −11.0115 −0.614604
\(322\) 18.4180 1.02639
\(323\) 4.62972 0.257605
\(324\) 5.22020 0.290011
\(325\) 0 0
\(326\) 65.1911 3.61060
\(327\) −3.44890 −0.190724
\(328\) 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
\(333\) −4.58187 −0.251085
\(334\) −12.2231 −0.668817
\(335\) 0 0
\(336\) −21.6111 −1.17898
\(337\) −13.0240 −0.709463 −0.354731 0.934968i \(-0.615428\pi\)
−0.354731 + 0.934968i \(0.615428\pi\)
\(338\) 15.6720 0.852443
\(339\) −6.05194 −0.328697
\(340\) 0 0
\(341\) −7.66295 −0.414972
\(342\) 3.16412 0.171096
\(343\) −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 \(-0.551500\pi\)
−0.161089 + 0.986940i \(0.551500\pi\)
\(348\) −31.0912 −1.66666
\(349\) −22.7183 −1.21608 −0.608042 0.793905i \(-0.708045\pi\)
−0.608042 + 0.793905i \(0.708045\pi\)
\(350\) 0 0
\(351\) −2.67723 −0.142900
\(352\) −18.4646 −0.984164
\(353\) −4.85493 −0.258402 −0.129201 0.991618i \(-0.541241\pi\)
−0.129201 + 0.991618i \(0.541241\pi\)
\(354\) 3.58986 0.190799
\(355\) 0 0
\(356\) 80.6186 4.27278
\(357\) 6.63289 0.351050
\(358\) 41.9895 2.21921
\(359\) 28.8910 1.52481 0.762403 0.647102i \(-0.224019\pi\)
0.762403 + 0.647102i \(0.224019\pi\)
\(360\) 0 0
\(361\) −17.6134 −0.927020
\(362\) −0.545047 −0.0286470
\(363\) 9.83615 0.516264
\(364\) 23.5776 1.23580
\(365\) 0 0
\(366\) −19.5645 −1.02265
\(367\) −14.7089 −0.767800 −0.383900 0.923375i \(-0.625419\pi\)
−0.383900 + 0.923375i \(0.625419\pi\)
\(368\) −52.0465 −2.71311
\(369\) −11.2614 −0.586247
\(370\) 0 0
\(371\) −4.29961 −0.223225
\(372\) 37.0796 1.92249
\(373\) −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
\(378\) 4.53315 0.233160
\(379\) −15.1556 −0.778493 −0.389247 0.921134i \(-0.627265\pi\)
−0.389247 + 0.921134i \(0.627265\pi\)
\(380\) 0 0
\(381\) 20.2362 1.03673
\(382\) 21.2807 1.08882
\(383\) −8.00811 −0.409195 −0.204598 0.978846i \(-0.565589\pi\)
−0.204598 + 0.978846i \(0.565589\pi\)
\(384\) 20.5042 1.04635
\(385\) 0 0
\(386\) 44.1397 2.24665
\(387\) 2.58587 0.131447
\(388\) 56.2121 2.85374
\(389\) 1.07463 0.0544858 0.0272429 0.999629i \(-0.491327\pi\)
0.0272429 + 0.999629i \(0.491327\pi\)
\(390\) 0 0
\(391\) 15.9742 0.807848
\(392\) 35.9428 1.81538
\(393\) 3.05011 0.153858
\(394\) 46.0475 2.31984
\(395\) 0 0
\(396\) 5.63164 0.283001
\(397\) −24.2139 −1.21526 −0.607630 0.794220i \(-0.707880\pi\)
−0.607630 + 0.794220i \(0.707880\pi\)
\(398\) 5.42373 0.271867
\(399\) 1.98657 0.0994529
\(400\) 0 0
\(401\) −1.99317 −0.0995340 −0.0497670 0.998761i \(-0.515848\pi\)
−0.0497670 + 0.998761i \(0.515848\pi\)
\(402\) 33.4404 1.66785
\(403\) −19.0166 −0.947287
\(404\) −70.8142 −3.52314
\(405\) 0 0
\(406\) −26.9992 −1.33995
\(407\) −4.94300 −0.245015
\(408\) −34.0200 −1.68424
\(409\) 6.66478 0.329552 0.164776 0.986331i \(-0.447310\pi\)
0.164776 + 0.986331i \(0.447310\pi\)
\(410\) 0 0
\(411\) 20.5100 1.01168
\(412\) 7.14664 0.352090
\(413\) 2.25387 0.110906
\(414\) 10.9173 0.536556
\(415\) 0 0
\(416\) −45.8223 −2.24662
\(417\) −18.3184 −0.897054
\(418\) 3.41351 0.166960
\(419\) 21.3540 1.04321 0.521606 0.853186i \(-0.325333\pi\)
0.521606 + 0.853186i \(0.325333\pi\)
\(420\) 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
\(423\) −1.91782 −0.0932476
\(424\) 22.0526 1.07097
\(425\) 0 0
\(426\) −16.0814 −0.779147
\(427\) −12.2835 −0.594438
\(428\) 57.4823 2.77851
\(429\) −2.88825 −0.139446
\(430\) 0 0
\(431\) −30.4129 −1.46494 −0.732469 0.680800i \(-0.761632\pi\)
−0.732469 + 0.680800i \(0.761632\pi\)
\(432\) −12.8101 −0.616324
\(433\) 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
\(438\) −8.86899 −0.423777
\(439\) 12.8045 0.611125 0.305563 0.952172i \(-0.401155\pi\)
0.305563 + 0.952172i \(0.401155\pi\)
\(440\) 0 0
\(441\) −4.15389 −0.197804
\(442\) 28.2838 1.34532
\(443\) 14.3147 0.680110 0.340055 0.940405i \(-0.389554\pi\)
0.340055 + 0.940405i \(0.389554\pi\)
\(444\) 23.9182 1.13511
\(445\) 0 0
\(446\) −35.2303 −1.66820
\(447\) −0.705938 −0.0333897
\(448\) 34.3652 1.62360
\(449\) −11.9663 −0.564724 −0.282362 0.959308i \(-0.591118\pi\)
−0.282362 + 0.959308i \(0.591118\pi\)
\(450\) 0 0
\(451\) −12.1490 −0.572076
\(452\) 31.5923 1.48598
\(453\) −7.04538 −0.331021
\(454\) 70.6173 3.31424
\(455\) 0 0
\(456\) −10.1891 −0.477147
\(457\) −8.22154 −0.384587 −0.192294 0.981337i \(-0.561593\pi\)
−0.192294 + 0.981337i \(0.561593\pi\)
\(458\) −5.36346 −0.250618
\(459\) 3.93167 0.183515
\(460\) 0 0
\(461\) 1.27216 0.0592505 0.0296253 0.999561i \(-0.490569\pi\)
0.0296253 + 0.999561i \(0.490569\pi\)
\(462\) 4.89045 0.227524
\(463\) 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 \(-0.391609\pi\)
0.333979 + 0.942581i \(0.391609\pi\)
\(468\) 13.9757 0.646026
\(469\) 20.9953 0.969474
\(470\) 0 0
\(471\) 5.12880 0.236323
\(472\) −11.5600 −0.532094
\(473\) 2.78968 0.128270
\(474\) −10.7590 −0.494178
\(475\) 0 0
\(476\) −34.6250 −1.58703
\(477\) −2.54861 −0.116693
\(478\) 18.8354 0.861511
\(479\) −16.8618 −0.770436 −0.385218 0.922826i \(-0.625874\pi\)
−0.385218 + 0.922826i \(0.625874\pi\)
\(480\) 0 0
\(481\) −12.2667 −0.559314
\(482\) −6.22196 −0.283403
\(483\) 6.85436 0.311884
\(484\) −51.3466 −2.33394
\(485\) 0 0
\(486\) 2.68704 0.121887
\(487\) 39.8235 1.80457 0.902287 0.431135i \(-0.141887\pi\)
0.902287 + 0.431135i \(0.141887\pi\)
\(488\) 63.0016 2.85195
\(489\) 24.2613 1.09713
\(490\) 0 0
\(491\) 14.0255 0.632961 0.316480 0.948599i \(-0.397499\pi\)
0.316480 + 0.948599i \(0.397499\pi\)
\(492\) 58.7869 2.65032
\(493\) −23.4168 −1.05464
\(494\) 8.47108 0.381132
\(495\) 0 0
\(496\) −90.9911 −4.08562
\(497\) −10.0966 −0.452895
\(498\) 23.8470 1.06861
\(499\) 15.2315 0.681857 0.340929 0.940089i \(-0.389259\pi\)
0.340929 + 0.940089i \(0.389259\pi\)
\(500\) 0 0
\(501\) −4.54890 −0.203230
\(502\) 16.6028 0.741020
\(503\) 23.7330 1.05820 0.529101 0.848559i \(-0.322529\pi\)
0.529101 + 0.848559i \(0.322529\pi\)
\(504\) −14.5976 −0.650231
\(505\) 0 0
\(506\) 11.7778 0.523586
\(507\) 5.83243 0.259027
\(508\) −105.637 −4.68688
\(509\) −12.5097 −0.554482 −0.277241 0.960800i \(-0.589420\pi\)
−0.277241 + 0.960800i \(0.589420\pi\)
\(510\) 0 0
\(511\) −5.56834 −0.246329
\(512\) 2.43464 0.107597
\(513\) 1.17755 0.0519899
\(514\) −35.5331 −1.56730
\(515\) 0 0
\(516\) −13.4988 −0.594249
\(517\) −2.06898 −0.0909936
\(518\) 20.7703 0.912595
\(519\) 2.67729 0.117520
\(520\) 0 0
\(521\) −27.9638 −1.22512 −0.612559 0.790425i \(-0.709860\pi\)
−0.612559 + 0.790425i \(0.709860\pi\)
\(522\) −16.0039 −0.700471
\(523\) 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
\(528\) −13.8197 −0.601426
\(529\) −6.49247 −0.282281
\(530\) 0 0
\(531\) 1.33599 0.0579769
\(532\) −10.3703 −0.449609
\(533\) −30.1495 −1.30592
\(534\) 41.4976 1.79578
\(535\) 0 0
\(536\) −107.685 −4.65127
\(537\) 15.6266 0.674340
\(538\) −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
\(543\) −0.202843 −0.00870482
\(544\) 67.2927 2.88515
\(545\) 0 0
\(546\) 12.1363 0.519386
\(547\) −5.62681 −0.240585 −0.120293 0.992738i \(-0.538383\pi\)
−0.120293 + 0.992738i \(0.538383\pi\)
\(548\) −107.066 −4.57365
\(549\) −7.28106 −0.310748
\(550\) 0 0
\(551\) −7.01341 −0.298781
\(552\) −35.1559 −1.49633
\(553\) −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 \(-0.855557\pi\)
−0.898796 + 0.438366i \(0.855557\pi\)
\(558\) 19.0863 0.807989
\(559\) 6.92298 0.292811
\(560\) 0 0
\(561\) 4.24155 0.179079
\(562\) −46.1261 −1.94571
\(563\) 5.78597 0.243850 0.121925 0.992539i \(-0.461093\pi\)
0.121925 + 0.992539i \(0.461093\pi\)
\(564\) 10.0114 0.421556
\(565\) 0 0
\(566\) 25.8495 1.08653
\(567\) 1.68704 0.0708491
\(568\) 51.7853 2.17286
\(569\) −22.0056 −0.922522 −0.461261 0.887265i \(-0.652603\pi\)
−0.461261 + 0.887265i \(0.652603\pi\)
\(570\) 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
\(573\) 7.91975 0.330852
\(574\) 51.0499 2.13078
\(575\) 0 0
\(576\) 20.3701 0.848754
\(577\) 40.2350 1.67501 0.837503 0.546433i \(-0.184015\pi\)
0.837503 + 0.546433i \(0.184015\pi\)
\(578\) 4.14337 0.172341
\(579\) 16.4269 0.682677
\(580\) 0 0
\(581\) 14.9722 0.621151
\(582\) 28.9346 1.19938
\(583\) −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 \(-0.518555\pi\)
−0.0582587 + 0.998302i \(0.518555\pi\)
\(588\) 21.6841 0.894238
\(589\) 8.36423 0.344642
\(590\) 0 0
\(591\) 17.1369 0.704916
\(592\) −58.6939 −2.41231
\(593\) −24.6805 −1.01351 −0.506754 0.862091i \(-0.669155\pi\)
−0.506754 + 0.862091i \(0.669155\pi\)
\(594\) 2.89883 0.118940
\(595\) 0 0
\(596\) 3.68513 0.150949
\(597\) 2.01848 0.0826108
\(598\) 29.2282 1.19523
\(599\) −7.71547 −0.315245 −0.157623 0.987499i \(-0.550383\pi\)
−0.157623 + 0.987499i \(0.550383\pi\)
\(600\) 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
\(603\) 12.4451 0.506802
\(604\) 36.7783 1.49649
\(605\) 0 0
\(606\) −36.4509 −1.48072
\(607\) −24.6423 −1.00020 −0.500099 0.865968i \(-0.666703\pi\)
−0.500099 + 0.865968i \(0.666703\pi\)
\(608\) 20.1543 0.817367
\(609\) −10.0479 −0.407163
\(610\) 0 0
\(611\) −5.13445 −0.207718
\(612\) −20.5241 −0.829637
\(613\) −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 \(-0.396035\pi\)
0.320840 + 0.947133i \(0.396035\pi\)
\(618\) 3.67866 0.147977
\(619\) −47.1700 −1.89592 −0.947961 0.318385i \(-0.896859\pi\)
−0.947961 + 0.318385i \(0.896859\pi\)
\(620\) 0 0
\(621\) 4.06295 0.163040
\(622\) −49.8808 −2.00004
\(623\) 26.0540 1.04383
\(624\) −34.2955 −1.37292
\(625\) 0 0
\(626\) 26.0630 1.04169
\(627\) 1.27036 0.0507332
\(628\) −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
\(633\) 23.8213 0.946812
\(634\) −20.0896 −0.797859
\(635\) 0 0
\(636\) 13.3042 0.527547
\(637\) −11.1209 −0.440627
\(638\) −17.2653 −0.683539
\(639\) −5.98480 −0.236755
\(640\) 0 0
\(641\) 18.0042 0.711122 0.355561 0.934653i \(-0.384290\pi\)
0.355561 + 0.934653i \(0.384290\pi\)
\(642\) 29.5884 1.16776
\(643\) 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 \(-0.377985\pi\)
0.374001 + 0.927428i \(0.377985\pi\)
\(648\) −8.65280 −0.339914
\(649\) 1.44129 0.0565755
\(650\) 0 0
\(651\) 11.9832 0.469660
\(652\) −126.649 −4.95995
\(653\) −2.31971 −0.0907774 −0.0453887 0.998969i \(-0.514453\pi\)
−0.0453887 + 0.998969i \(0.514453\pi\)
\(654\) 9.26733 0.362381
\(655\) 0 0
\(656\) −144.260 −5.63239
\(657\) −3.30065 −0.128771
\(658\) 8.69378 0.338919
\(659\) 7.19180 0.280153 0.140076 0.990141i \(-0.455265\pi\)
0.140076 + 0.990141i \(0.455265\pi\)
\(660\) 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
\(663\) 10.5260 0.408796
\(664\) −76.7920 −2.98011
\(665\) 0 0
\(666\) 12.3117 0.477067
\(667\) −24.1987 −0.936977
\(668\) 23.7461 0.918765
\(669\) −13.1112 −0.506907
\(670\) 0 0
\(671\) −7.85494 −0.303237
\(672\) 28.8746 1.11386
\(673\) −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 \(-0.671743\pi\)
−0.513746 + 0.857942i \(0.671743\pi\)
\(678\) 16.2618 0.624532
\(679\) 18.1664 0.697163
\(680\) 0 0
\(681\) 26.2807 1.00708
\(682\) 20.5907 0.788457
\(683\) 28.4950 1.09033 0.545165 0.838329i \(-0.316467\pi\)
0.545165 + 0.838329i \(0.316467\pi\)
\(684\) −6.14702 −0.235037
\(685\) 0 0
\(686\) 50.5623 1.93048
\(687\) −1.99605 −0.0761540
\(688\) 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
\(693\) 1.82001 0.0691365
\(694\) 16.1263 0.612145
\(695\) 0 0
\(696\) 51.5357 1.95345
\(697\) 44.2763 1.67708
\(698\) 61.0451 2.31059
\(699\) 0.525548 0.0198781
\(700\) 0 0
\(701\) 0.973305 0.0367612 0.0183806 0.999831i \(-0.494149\pi\)
0.0183806 + 0.999831i \(0.494149\pi\)
\(702\) 7.19384 0.271514
\(703\) 5.39536 0.203490
\(704\) 21.9756 0.828237
\(705\) 0 0
\(706\) 13.0454 0.490971
\(707\) −22.8854 −0.860696
\(708\) −6.97412 −0.262103
\(709\) −11.3242 −0.425291 −0.212645 0.977129i \(-0.568208\pi\)
−0.212645 + 0.977129i \(0.568208\pi\)
\(710\) 0 0
\(711\) −4.00404 −0.150163
\(712\) −133.630 −5.00801
\(713\) 28.8595 1.08080
\(714\) −17.8229 −0.667004
\(715\) 0 0
\(716\) −81.5742 −3.04857
\(717\) 7.00971 0.261783
\(718\) −77.6312 −2.89717
\(719\) −30.7320 −1.14611 −0.573056 0.819516i \(-0.694242\pi\)
−0.573056 + 0.819516i \(0.694242\pi\)
\(720\) 0 0
\(721\) 2.30962 0.0860149
\(722\) 47.3279 1.76136
\(723\) −2.31554 −0.0861159
\(724\) 1.05888 0.0393529
\(725\) 0 0
\(726\) −26.4302 −0.980915
\(727\) −6.56836 −0.243607 −0.121803 0.992554i \(-0.538868\pi\)
−0.121803 + 0.992554i \(0.538868\pi\)
\(728\) −39.0813 −1.44845
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.1668 −0.376032
\(732\) 38.0086 1.40484
\(733\) −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
\(738\) 30.2600 1.11389
\(739\) −15.3072 −0.563083 −0.281542 0.959549i \(-0.590846\pi\)
−0.281542 + 0.959549i \(0.590846\pi\)
\(740\) 0 0
\(741\) 3.15257 0.115812
\(742\) 11.5532 0.424132
\(743\) 16.5455 0.606995 0.303498 0.952832i \(-0.401846\pi\)
0.303498 + 0.952832i \(0.401846\pi\)
\(744\) −61.4617 −2.25330
\(745\) 0 0
\(746\) 88.7774 3.25037
\(747\) 8.87482 0.324713
\(748\) −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
\(753\) 6.17885 0.225170
\(754\) −42.8461 −1.56036
\(755\) 0 0
\(756\) −8.80669 −0.320296
\(757\) −26.5282 −0.964184 −0.482092 0.876121i \(-0.660123\pi\)
−0.482092 + 0.876121i \(0.660123\pi\)
\(758\) 40.7239 1.47916
\(759\) 4.38318 0.159099
\(760\) 0 0
\(761\) −28.4241 −1.03037 −0.515187 0.857078i \(-0.672277\pi\)
−0.515187 + 0.857078i \(0.672277\pi\)
\(762\) −54.3755 −1.96982
\(763\) 5.81843 0.210641
\(764\) −41.3426 −1.49572
\(765\) 0 0
\(766\) 21.5181 0.777481
\(767\) 3.57675 0.129149
\(768\) −14.3555 −0.518009
\(769\) −8.95952 −0.323089 −0.161544 0.986865i \(-0.551647\pi\)
−0.161544 + 0.986865i \(0.551647\pi\)
\(770\) 0 0
\(771\) −13.2239 −0.476246
\(772\) −85.7515 −3.08626
\(773\) 23.1042 0.831001 0.415500 0.909593i \(-0.363607\pi\)
0.415500 + 0.909593i \(0.363607\pi\)
\(774\) −6.94834 −0.249753
\(775\) 0 0
\(776\) −93.1752 −3.34479
\(777\) 7.72980 0.277305
\(778\) −2.88757 −0.103524
\(779\) 13.2609 0.475120
\(780\) 0 0
\(781\) −6.45651 −0.231032
\(782\) −42.9232 −1.53493
\(783\) −5.95595 −0.212848
\(784\) −53.2115 −1.90041
\(785\) 0 0
\(786\) −8.19578 −0.292334
\(787\) −2.03390 −0.0725009 −0.0362504 0.999343i \(-0.511541\pi\)
−0.0362504 + 0.999343i \(0.511541\pi\)
\(788\) −89.4578 −3.18680
\(789\) 6.97643 0.248367
\(790\) 0 0
\(791\) 10.2099 0.363022
\(792\) −9.33480 −0.331698
\(793\) −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 \(-0.569281\pi\)
−0.215937 + 0.976407i \(0.569281\pi\)
\(798\) −5.33800 −0.188963
\(799\) 7.54024 0.266754
\(800\) 0 0
\(801\) 15.4436 0.545673
\(802\) 5.35572 0.189117
\(803\) −3.56080 −0.125658
\(804\) −64.9656 −2.29116
\(805\) 0 0
\(806\) 51.0985 1.79987
\(807\) −23.7199 −0.834979
\(808\) 117.379 4.12938
\(809\) 8.83355 0.310571 0.155286 0.987870i \(-0.450370\pi\)
0.155286 + 0.987870i \(0.450370\pi\)
\(810\) 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
\(813\) −28.9910 −1.01676
\(814\) 13.2820 0.465535
\(815\) 0 0
\(816\) 50.3649 1.76312
\(817\) −3.04498 −0.106530
\(818\) −17.9085 −0.626158
\(819\) 4.51660 0.157823
\(820\) 0 0
\(821\) −29.4930 −1.02931 −0.514656 0.857397i \(-0.672080\pi\)
−0.514656 + 0.857397i \(0.672080\pi\)
\(822\) −55.1113 −1.92223
\(823\) −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 \(-0.274167\pi\)
0.651435 + 0.758704i \(0.274167\pi\)
\(828\) −21.2094 −0.737077
\(829\) −9.55614 −0.331899 −0.165949 0.986134i \(-0.553069\pi\)
−0.165949 + 0.986134i \(0.553069\pi\)
\(830\) 0 0
\(831\) 8.58667 0.297868
\(832\) 54.5355 1.89068
\(833\) 16.3317 0.565860
\(834\) 49.2222 1.70443
\(835\) 0 0
\(836\) −6.63152 −0.229356
\(837\) 7.10310 0.245519
\(838\) −57.3792 −1.98213
\(839\) 54.7310 1.88952 0.944761 0.327759i \(-0.106293\pi\)
0.944761 + 0.327759i \(0.106293\pi\)
\(840\) 0 0
\(841\) 6.47334 0.223219
\(842\) −60.0197 −2.06842
\(843\) −17.1661 −0.591233
\(844\) −124.352 −4.28037
\(845\) 0 0
\(846\) 5.15327 0.177173
\(847\) −16.5940 −0.570177
\(848\) −32.6478 −1.12113
\(849\) 9.62005 0.330159
\(850\) 0 0
\(851\) 18.6159 0.638144
\(852\) 31.2419 1.07033
\(853\) −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 \(-0.547305\pi\)
−0.148065 + 0.988978i \(0.547305\pi\)
\(858\) 7.76084 0.264951
\(859\) −30.4078 −1.03750 −0.518750 0.854926i \(-0.673603\pi\)
−0.518750 + 0.854926i \(0.673603\pi\)
\(860\) 0 0
\(861\) 18.9985 0.647468
\(862\) 81.7208 2.78342
\(863\) −18.0425 −0.614175 −0.307088 0.951681i \(-0.599354\pi\)
−0.307088 + 0.951681i \(0.599354\pi\)
\(864\) 17.1155 0.582283
\(865\) 0 0
\(866\) −1.54945 −0.0526526
\(867\) 1.54198 0.0523684
\(868\) −62.5548 −2.12325
\(869\) −4.31963 −0.146533
\(870\) 0 0
\(871\) 33.3183 1.12895
\(872\) −29.8426 −1.01060
\(873\) 10.7682 0.364449
\(874\) −12.8556 −0.434848
\(875\) 0 0
\(876\) 17.2301 0.582149
\(877\) 22.3057 0.753211 0.376605 0.926374i \(-0.377091\pi\)
0.376605 + 0.926374i \(0.377091\pi\)
\(878\) −34.4062 −1.16115
\(879\) −16.8202 −0.567332
\(880\) 0 0
\(881\) 7.75956 0.261426 0.130713 0.991420i \(-0.458273\pi\)
0.130713 + 0.991420i \(0.458273\pi\)
\(882\) 11.1617 0.375833
\(883\) −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 \(-0.453866\pi\)
0.144428 + 0.989515i \(0.453866\pi\)
\(888\) −39.6460 −1.33043
\(889\) −34.1393 −1.14499
\(890\) 0 0
\(891\) 1.07882 0.0361418
\(892\) 68.4429 2.29164
\(893\) 2.25832 0.0755719
\(894\) 1.89688 0.0634413
\(895\) 0 0
\(896\) −34.5915 −1.15562
\(897\) 10.8775 0.363187
\(898\) 32.1539 1.07299
\(899\) −42.3057 −1.41097
\(900\) 0 0
\(901\) 10.0203 0.333824
\(902\) 32.6450 1.08696
\(903\) −4.36247 −0.145174
\(904\) −52.3663 −1.74168
\(905\) 0 0
\(906\) 18.9312 0.628948
\(907\) 6.26125 0.207902 0.103951 0.994582i \(-0.466852\pi\)
0.103951 + 0.994582i \(0.466852\pi\)
\(908\) −137.190 −4.55282
\(909\) −13.5654 −0.449937
\(910\) 0 0
\(911\) 0.905575 0.0300030 0.0150015 0.999887i \(-0.495225\pi\)
0.0150015 + 0.999887i \(0.495225\pi\)
\(912\) 15.0844 0.499495
\(913\) 9.57431 0.316863
\(914\) 22.0916 0.730726
\(915\) 0 0
\(916\) 10.4198 0.344279
\(917\) −5.14567 −0.169925
\(918\) −10.5646 −0.348682
\(919\) 43.9933 1.45120 0.725602 0.688114i \(-0.241561\pi\)
0.725602 + 0.688114i \(0.241561\pi\)
\(920\) 0 0
\(921\) −11.3212 −0.373046
\(922\) −3.41836 −0.112578
\(923\) −16.0227 −0.527394
\(924\) −9.50081 −0.312554
\(925\) 0 0
\(926\) −87.8029 −2.88538
\(927\) 1.36904 0.0449651
\(928\) −101.939 −3.34632
\(929\) −36.7447 −1.20555 −0.602777 0.797910i \(-0.705939\pi\)
−0.602777 + 0.797910i \(0.705939\pi\)
\(930\) 0 0
\(931\) 4.89140 0.160309
\(932\) −2.74347 −0.0898652
\(933\) −18.5635 −0.607740
\(934\) −38.7866 −1.26914
\(935\) 0 0
\(936\) −23.1656 −0.757190
\(937\) 24.6490 0.805248 0.402624 0.915366i \(-0.368098\pi\)
0.402624 + 0.915366i \(0.368098\pi\)
\(938\) −56.4153 −1.84203
\(939\) 9.69951 0.316532
\(940\) 0 0
\(941\) −59.1479 −1.92817 −0.964083 0.265602i \(-0.914429\pi\)
−0.964083 + 0.265602i \(0.914429\pi\)
\(942\) −13.7813 −0.449019
\(943\) 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 \(-0.662636\pi\)
−0.488995 + 0.872287i \(0.662636\pi\)
\(948\) 20.9019 0.678861
\(949\) −8.83661 −0.286849
\(950\) 0 0
\(951\) −7.47647 −0.242441
\(952\) 57.3931 1.86012
\(953\) −20.6393 −0.668572 −0.334286 0.942472i \(-0.608495\pi\)
−0.334286 + 0.942472i \(0.608495\pi\)
\(954\) 6.84822 0.221719
\(955\) 0 0
\(956\) −36.5921 −1.18347
\(957\) −6.42538 −0.207703
\(958\) 45.3084 1.46385
\(959\) −34.6013 −1.11733
\(960\) 0 0
\(961\) 19.4540 0.627549
\(962\) 32.9612 1.06271
\(963\) 11.0115 0.354842
\(964\) 12.0876 0.389315
\(965\) 0 0
\(966\) −18.4180 −0.592588
\(967\) −22.6016 −0.726817 −0.363408 0.931630i \(-0.618387\pi\)
−0.363408 + 0.931630i \(0.618387\pi\)
\(968\) 85.1103 2.73555
\(969\) −4.62972 −0.148728
\(970\) 0 0
\(971\) 24.1381 0.774628 0.387314 0.921948i \(-0.373403\pi\)
0.387314 + 0.921948i \(0.373403\pi\)
\(972\) −5.22020 −0.167438
\(973\) 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 \(-0.379794\pi\)
0.368727 + 0.929538i \(0.379794\pi\)
\(978\) −65.1911 −2.08458
\(979\) 16.6608 0.532483
\(980\) 0 0
\(981\) 3.44890 0.110115
\(982\) −37.6870 −1.20264
\(983\) 29.8713 0.952745 0.476373 0.879243i \(-0.341951\pi\)
0.476373 + 0.879243i \(0.341951\pi\)
\(984\) −97.4430 −3.10637
\(985\) 0 0
\(986\) 62.9220 2.00384
\(987\) 3.23544 0.102985
\(988\) −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
\(993\) 25.2570 0.801507
\(994\) 27.1300 0.860513
\(995\) 0 0
\(996\) −46.3283 −1.46797
\(997\) 12.4934 0.395669 0.197835 0.980235i \(-0.436609\pi\)
0.197835 + 0.980235i \(0.436609\pi\)
\(998\) −40.9278 −1.29555
\(999\) 4.58187 0.144964
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 1875.2.a.j.1.1 6
3.2 odd 2 5625.2.a.p.1.6 6
5.2 odd 4 1875.2.b.f.1249.1 12
5.3 odd 4 1875.2.b.f.1249.12 12
5.4 even 2 1875.2.a.k.1.6 6
15.14 odd 2 5625.2.a.q.1.1 6
25.2 odd 20 375.2.i.d.274.1 24
25.9 even 10 375.2.g.c.151.3 12
25.11 even 5 75.2.g.c.46.1 yes 12
25.12 odd 20 375.2.i.d.349.6 24
25.13 odd 20 375.2.i.d.349.1 24
25.14 even 10 375.2.g.c.226.3 12
25.16 even 5 75.2.g.c.31.1 12
25.23 odd 20 375.2.i.d.274.6 24
75.11 odd 10 225.2.h.d.46.3 12
75.41 odd 10 225.2.h.d.181.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.31.1 12 25.16 even 5
75.2.g.c.46.1 yes 12 25.11 even 5
225.2.h.d.46.3 12 75.11 odd 10
225.2.h.d.181.3 12 75.41 odd 10
375.2.g.c.151.3 12 25.9 even 10
375.2.g.c.226.3 12 25.14 even 10
375.2.i.d.274.1 24 25.2 odd 20
375.2.i.d.274.6 24 25.23 odd 20
375.2.i.d.349.1 24 25.13 odd 20
375.2.i.d.349.6 24 25.12 odd 20
1875.2.a.j.1.1 6 1.1 even 1 trivial
1875.2.a.k.1.6 6 5.4 even 2
1875.2.b.f.1249.1 12 5.2 odd 4
1875.2.b.f.1249.12 12 5.3 odd 4
5625.2.a.p.1.6 6 3.2 odd 2
5625.2.a.q.1.1 6 15.14 odd 2