Properties

Label 1875.2.a.p.1.7
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.5444000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 1.7
Root \(2.35083\) 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)
 
\(f(q)\) \(=\) \(q+2.35083 q^{2} -1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} +3.48189 q^{7} +3.58831 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.35083 q^{2} -1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} +3.48189 q^{7} +3.58831 q^{8} +1.00000 q^{9} +2.93111 q^{11} -3.52640 q^{12} -1.87575 q^{13} +8.18532 q^{14} +1.38270 q^{16} +6.78566 q^{17} +2.35083 q^{18} -2.94950 q^{19} -3.48189 q^{21} +6.89053 q^{22} -5.49019 q^{23} -3.58831 q^{24} -4.40956 q^{26} -1.00000 q^{27} +12.2785 q^{28} +2.55593 q^{29} +0.418084 q^{31} -3.92613 q^{32} -2.93111 q^{33} +15.9519 q^{34} +3.52640 q^{36} +5.23959 q^{37} -6.93377 q^{38} +1.87575 q^{39} +1.67869 q^{41} -8.18532 q^{42} +10.9233 q^{43} +10.3363 q^{44} -12.9065 q^{46} -7.49178 q^{47} -1.38270 q^{48} +5.12353 q^{49} -6.78566 q^{51} -6.61463 q^{52} +3.70953 q^{53} -2.35083 q^{54} +12.4941 q^{56} +2.94950 q^{57} +6.00857 q^{58} -7.10854 q^{59} -6.43710 q^{61} +0.982844 q^{62} +3.48189 q^{63} -11.9951 q^{64} -6.89053 q^{66} +10.0415 q^{67} +23.9290 q^{68} +5.49019 q^{69} +0.728602 q^{71} +3.58831 q^{72} -3.59269 q^{73} +12.3174 q^{74} -10.4011 q^{76} +10.2058 q^{77} +4.40956 q^{78} +3.07265 q^{79} +1.00000 q^{81} +3.94632 q^{82} +10.1152 q^{83} -12.2785 q^{84} +25.6788 q^{86} -2.55593 q^{87} +10.5177 q^{88} +0.287512 q^{89} -6.53114 q^{91} -19.3606 q^{92} -0.418084 q^{93} -17.6119 q^{94} +3.92613 q^{96} -10.4090 q^{97} +12.0446 q^{98} +2.93111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35083 1.66229 0.831144 0.556058i \(-0.187687\pi\)
0.831144 + 0.556058i \(0.187687\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.52640 1.76320
\(5\) 0 0
\(6\) −2.35083 −0.959722
\(7\) 3.48189 1.31603 0.658015 0.753005i \(-0.271396\pi\)
0.658015 + 0.753005i \(0.271396\pi\)
\(8\) 3.58831 1.26866
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.93111 0.883762 0.441881 0.897074i \(-0.354311\pi\)
0.441881 + 0.897074i \(0.354311\pi\)
\(12\) −3.52640 −1.01798
\(13\) −1.87575 −0.520239 −0.260119 0.965576i \(-0.583762\pi\)
−0.260119 + 0.965576i \(0.583762\pi\)
\(14\) 8.18532 2.18762
\(15\) 0 0
\(16\) 1.38270 0.345674
\(17\) 6.78566 1.64576 0.822882 0.568212i \(-0.192365\pi\)
0.822882 + 0.568212i \(0.192365\pi\)
\(18\) 2.35083 0.554096
\(19\) −2.94950 −0.676662 −0.338331 0.941027i \(-0.609862\pi\)
−0.338331 + 0.941027i \(0.609862\pi\)
\(20\) 0 0
\(21\) −3.48189 −0.759810
\(22\) 6.89053 1.46907
\(23\) −5.49019 −1.14478 −0.572392 0.819980i \(-0.693985\pi\)
−0.572392 + 0.819980i \(0.693985\pi\)
\(24\) −3.58831 −0.732460
\(25\) 0 0
\(26\) −4.40956 −0.864786
\(27\) −1.00000 −0.192450
\(28\) 12.2785 2.32042
\(29\) 2.55593 0.474625 0.237313 0.971433i \(-0.423733\pi\)
0.237313 + 0.971433i \(0.423733\pi\)
\(30\) 0 0
\(31\) 0.418084 0.0750901 0.0375451 0.999295i \(-0.488046\pi\)
0.0375451 + 0.999295i \(0.488046\pi\)
\(32\) −3.92613 −0.694048
\(33\) −2.93111 −0.510240
\(34\) 15.9519 2.73573
\(35\) 0 0
\(36\) 3.52640 0.587733
\(37\) 5.23959 0.861383 0.430691 0.902499i \(-0.358270\pi\)
0.430691 + 0.902499i \(0.358270\pi\)
\(38\) −6.93377 −1.12481
\(39\) 1.87575 0.300360
\(40\) 0 0
\(41\) 1.67869 0.262167 0.131084 0.991371i \(-0.458154\pi\)
0.131084 + 0.991371i \(0.458154\pi\)
\(42\) −8.18532 −1.26302
\(43\) 10.9233 1.66578 0.832892 0.553436i \(-0.186684\pi\)
0.832892 + 0.553436i \(0.186684\pi\)
\(44\) 10.3363 1.55825
\(45\) 0 0
\(46\) −12.9065 −1.90296
\(47\) −7.49178 −1.09279 −0.546394 0.837528i \(-0.684000\pi\)
−0.546394 + 0.837528i \(0.684000\pi\)
\(48\) −1.38270 −0.199575
\(49\) 5.12353 0.731933
\(50\) 0 0
\(51\) −6.78566 −0.950183
\(52\) −6.61463 −0.917285
\(53\) 3.70953 0.509543 0.254771 0.967001i \(-0.418000\pi\)
0.254771 + 0.967001i \(0.418000\pi\)
\(54\) −2.35083 −0.319907
\(55\) 0 0
\(56\) 12.4941 1.66959
\(57\) 2.94950 0.390671
\(58\) 6.00857 0.788964
\(59\) −7.10854 −0.925453 −0.462727 0.886501i \(-0.653129\pi\)
−0.462727 + 0.886501i \(0.653129\pi\)
\(60\) 0 0
\(61\) −6.43710 −0.824186 −0.412093 0.911142i \(-0.635202\pi\)
−0.412093 + 0.911142i \(0.635202\pi\)
\(62\) 0.982844 0.124821
\(63\) 3.48189 0.438676
\(64\) −11.9951 −1.49938
\(65\) 0 0
\(66\) −6.89053 −0.848166
\(67\) 10.0415 1.22677 0.613383 0.789786i \(-0.289808\pi\)
0.613383 + 0.789786i \(0.289808\pi\)
\(68\) 23.9290 2.90181
\(69\) 5.49019 0.660942
\(70\) 0 0
\(71\) 0.728602 0.0864691 0.0432346 0.999065i \(-0.486234\pi\)
0.0432346 + 0.999065i \(0.486234\pi\)
\(72\) 3.58831 0.422886
\(73\) −3.59269 −0.420492 −0.210246 0.977648i \(-0.567427\pi\)
−0.210246 + 0.977648i \(0.567427\pi\)
\(74\) 12.3174 1.43187
\(75\) 0 0
\(76\) −10.4011 −1.19309
\(77\) 10.2058 1.16306
\(78\) 4.40956 0.499285
\(79\) 3.07265 0.345700 0.172850 0.984948i \(-0.444702\pi\)
0.172850 + 0.984948i \(0.444702\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.94632 0.435798
\(83\) 10.1152 1.11029 0.555146 0.831753i \(-0.312662\pi\)
0.555146 + 0.831753i \(0.312662\pi\)
\(84\) −12.2785 −1.33970
\(85\) 0 0
\(86\) 25.6788 2.76901
\(87\) −2.55593 −0.274025
\(88\) 10.5177 1.12119
\(89\) 0.287512 0.0304762 0.0152381 0.999884i \(-0.495149\pi\)
0.0152381 + 0.999884i \(0.495149\pi\)
\(90\) 0 0
\(91\) −6.53114 −0.684649
\(92\) −19.3606 −2.01848
\(93\) −0.418084 −0.0433533
\(94\) −17.6119 −1.81653
\(95\) 0 0
\(96\) 3.92613 0.400709
\(97\) −10.4090 −1.05688 −0.528439 0.848971i \(-0.677222\pi\)
−0.528439 + 0.848971i \(0.677222\pi\)
\(98\) 12.0446 1.21668
\(99\) 2.93111 0.294587
\(100\) 0 0
\(101\) 7.65744 0.761943 0.380972 0.924587i \(-0.375590\pi\)
0.380972 + 0.924587i \(0.375590\pi\)
\(102\) −15.9519 −1.57948
\(103\) −2.98602 −0.294221 −0.147111 0.989120i \(-0.546997\pi\)
−0.147111 + 0.989120i \(0.546997\pi\)
\(104\) −6.73076 −0.660005
\(105\) 0 0
\(106\) 8.72047 0.847007
\(107\) −7.07213 −0.683689 −0.341844 0.939757i \(-0.611052\pi\)
−0.341844 + 0.939757i \(0.611052\pi\)
\(108\) −3.52640 −0.339328
\(109\) −13.3022 −1.27412 −0.637058 0.770816i \(-0.719849\pi\)
−0.637058 + 0.770816i \(0.719849\pi\)
\(110\) 0 0
\(111\) −5.23959 −0.497320
\(112\) 4.81440 0.454918
\(113\) 10.0233 0.942911 0.471456 0.881890i \(-0.343729\pi\)
0.471456 + 0.881890i \(0.343729\pi\)
\(114\) 6.93377 0.649407
\(115\) 0 0
\(116\) 9.01325 0.836859
\(117\) −1.87575 −0.173413
\(118\) −16.7110 −1.53837
\(119\) 23.6269 2.16587
\(120\) 0 0
\(121\) −2.40861 −0.218965
\(122\) −15.1325 −1.37003
\(123\) −1.67869 −0.151362
\(124\) 1.47433 0.132399
\(125\) 0 0
\(126\) 8.18532 0.729206
\(127\) 10.5730 0.938205 0.469103 0.883144i \(-0.344577\pi\)
0.469103 + 0.883144i \(0.344577\pi\)
\(128\) −20.3461 −1.79836
\(129\) −10.9233 −0.961741
\(130\) 0 0
\(131\) −9.02608 −0.788612 −0.394306 0.918979i \(-0.629015\pi\)
−0.394306 + 0.918979i \(0.629015\pi\)
\(132\) −10.3363 −0.899656
\(133\) −10.2698 −0.890507
\(134\) 23.6059 2.03924
\(135\) 0 0
\(136\) 24.3490 2.08791
\(137\) −19.6646 −1.68006 −0.840029 0.542541i \(-0.817462\pi\)
−0.840029 + 0.542541i \(0.817462\pi\)
\(138\) 12.9065 1.09868
\(139\) −11.6520 −0.988310 −0.494155 0.869374i \(-0.664522\pi\)
−0.494155 + 0.869374i \(0.664522\pi\)
\(140\) 0 0
\(141\) 7.49178 0.630922
\(142\) 1.71282 0.143737
\(143\) −5.49801 −0.459767
\(144\) 1.38270 0.115225
\(145\) 0 0
\(146\) −8.44580 −0.698979
\(147\) −5.12353 −0.422582
\(148\) 18.4769 1.51879
\(149\) −7.33020 −0.600513 −0.300257 0.953858i \(-0.597072\pi\)
−0.300257 + 0.953858i \(0.597072\pi\)
\(150\) 0 0
\(151\) −16.7358 −1.36194 −0.680968 0.732313i \(-0.738441\pi\)
−0.680968 + 0.732313i \(0.738441\pi\)
\(152\) −10.5837 −0.858453
\(153\) 6.78566 0.548588
\(154\) 23.9921 1.93333
\(155\) 0 0
\(156\) 6.61463 0.529595
\(157\) 7.88635 0.629399 0.314700 0.949191i \(-0.398096\pi\)
0.314700 + 0.949191i \(0.398096\pi\)
\(158\) 7.22328 0.574653
\(159\) −3.70953 −0.294185
\(160\) 0 0
\(161\) −19.1162 −1.50657
\(162\) 2.35083 0.184699
\(163\) −9.93992 −0.778555 −0.389277 0.921121i \(-0.627275\pi\)
−0.389277 + 0.921121i \(0.627275\pi\)
\(164\) 5.91973 0.462254
\(165\) 0 0
\(166\) 23.7792 1.84563
\(167\) 5.57767 0.431613 0.215806 0.976436i \(-0.430762\pi\)
0.215806 + 0.976436i \(0.430762\pi\)
\(168\) −12.4941 −0.963939
\(169\) −9.48157 −0.729352
\(170\) 0 0
\(171\) −2.94950 −0.225554
\(172\) 38.5198 2.93711
\(173\) 16.5682 1.25966 0.629828 0.776734i \(-0.283125\pi\)
0.629828 + 0.776734i \(0.283125\pi\)
\(174\) −6.00857 −0.455508
\(175\) 0 0
\(176\) 4.05284 0.305494
\(177\) 7.10854 0.534311
\(178\) 0.675892 0.0506602
\(179\) −5.36021 −0.400641 −0.200321 0.979730i \(-0.564198\pi\)
−0.200321 + 0.979730i \(0.564198\pi\)
\(180\) 0 0
\(181\) −0.327745 −0.0243611 −0.0121805 0.999926i \(-0.503877\pi\)
−0.0121805 + 0.999926i \(0.503877\pi\)
\(182\) −15.3536 −1.13808
\(183\) 6.43710 0.475844
\(184\) −19.7005 −1.45234
\(185\) 0 0
\(186\) −0.982844 −0.0720656
\(187\) 19.8895 1.45446
\(188\) −26.4190 −1.92680
\(189\) −3.48189 −0.253270
\(190\) 0 0
\(191\) −3.46992 −0.251074 −0.125537 0.992089i \(-0.540065\pi\)
−0.125537 + 0.992089i \(0.540065\pi\)
\(192\) 11.9951 0.865668
\(193\) −24.3134 −1.75012 −0.875058 0.484018i \(-0.839177\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(194\) −24.4699 −1.75684
\(195\) 0 0
\(196\) 18.0676 1.29055
\(197\) 14.9561 1.06558 0.532789 0.846248i \(-0.321144\pi\)
0.532789 + 0.846248i \(0.321144\pi\)
\(198\) 6.89053 0.489689
\(199\) −11.3251 −0.802817 −0.401408 0.915899i \(-0.631479\pi\)
−0.401408 + 0.915899i \(0.631479\pi\)
\(200\) 0 0
\(201\) −10.0415 −0.708274
\(202\) 18.0013 1.26657
\(203\) 8.89948 0.624621
\(204\) −23.9290 −1.67536
\(205\) 0 0
\(206\) −7.01963 −0.489081
\(207\) −5.49019 −0.381595
\(208\) −2.59359 −0.179833
\(209\) −8.64530 −0.598008
\(210\) 0 0
\(211\) 7.54283 0.519270 0.259635 0.965707i \(-0.416398\pi\)
0.259635 + 0.965707i \(0.416398\pi\)
\(212\) 13.0813 0.898426
\(213\) −0.728602 −0.0499230
\(214\) −16.6254 −1.13649
\(215\) 0 0
\(216\) −3.58831 −0.244153
\(217\) 1.45572 0.0988208
\(218\) −31.2711 −2.11795
\(219\) 3.59269 0.242771
\(220\) 0 0
\(221\) −12.7282 −0.856190
\(222\) −12.3174 −0.826688
\(223\) −7.79539 −0.522018 −0.261009 0.965336i \(-0.584055\pi\)
−0.261009 + 0.965336i \(0.584055\pi\)
\(224\) −13.6703 −0.913387
\(225\) 0 0
\(226\) 23.5630 1.56739
\(227\) −12.8319 −0.851686 −0.425843 0.904797i \(-0.640022\pi\)
−0.425843 + 0.904797i \(0.640022\pi\)
\(228\) 10.4011 0.688831
\(229\) −14.5033 −0.958405 −0.479203 0.877704i \(-0.659074\pi\)
−0.479203 + 0.877704i \(0.659074\pi\)
\(230\) 0 0
\(231\) −10.2058 −0.671491
\(232\) 9.17148 0.602137
\(233\) 11.3970 0.746646 0.373323 0.927702i \(-0.378218\pi\)
0.373323 + 0.927702i \(0.378218\pi\)
\(234\) −4.40956 −0.288262
\(235\) 0 0
\(236\) −25.0676 −1.63176
\(237\) −3.07265 −0.199590
\(238\) 55.5428 3.60031
\(239\) 6.90072 0.446371 0.223185 0.974776i \(-0.428354\pi\)
0.223185 + 0.974776i \(0.428354\pi\)
\(240\) 0 0
\(241\) −28.4467 −1.83242 −0.916208 0.400704i \(-0.868766\pi\)
−0.916208 + 0.400704i \(0.868766\pi\)
\(242\) −5.66224 −0.363982
\(243\) −1.00000 −0.0641500
\(244\) −22.6998 −1.45320
\(245\) 0 0
\(246\) −3.94632 −0.251608
\(247\) 5.53252 0.352026
\(248\) 1.50021 0.0952637
\(249\) −10.1152 −0.641028
\(250\) 0 0
\(251\) −12.3258 −0.777999 −0.389000 0.921238i \(-0.627179\pi\)
−0.389000 + 0.921238i \(0.627179\pi\)
\(252\) 12.2785 0.773474
\(253\) −16.0923 −1.01172
\(254\) 24.8554 1.55957
\(255\) 0 0
\(256\) −23.8400 −1.49000
\(257\) 20.8274 1.29918 0.649590 0.760285i \(-0.274941\pi\)
0.649590 + 0.760285i \(0.274941\pi\)
\(258\) −25.6788 −1.59869
\(259\) 18.2437 1.13361
\(260\) 0 0
\(261\) 2.55593 0.158208
\(262\) −21.2188 −1.31090
\(263\) 2.23517 0.137826 0.0689131 0.997623i \(-0.478047\pi\)
0.0689131 + 0.997623i \(0.478047\pi\)
\(264\) −10.5177 −0.647320
\(265\) 0 0
\(266\) −24.1426 −1.48028
\(267\) −0.287512 −0.0175955
\(268\) 35.4104 2.16303
\(269\) −19.3036 −1.17696 −0.588482 0.808511i \(-0.700274\pi\)
−0.588482 + 0.808511i \(0.700274\pi\)
\(270\) 0 0
\(271\) −11.9411 −0.725372 −0.362686 0.931911i \(-0.618140\pi\)
−0.362686 + 0.931911i \(0.618140\pi\)
\(272\) 9.38252 0.568899
\(273\) 6.53114 0.395282
\(274\) −46.2281 −2.79274
\(275\) 0 0
\(276\) 19.3606 1.16537
\(277\) 26.1466 1.57100 0.785499 0.618863i \(-0.212406\pi\)
0.785499 + 0.618863i \(0.212406\pi\)
\(278\) −27.3919 −1.64285
\(279\) 0.418084 0.0250300
\(280\) 0 0
\(281\) 10.1219 0.603820 0.301910 0.953336i \(-0.402376\pi\)
0.301910 + 0.953336i \(0.402376\pi\)
\(282\) 17.6119 1.04877
\(283\) −31.8638 −1.89410 −0.947052 0.321081i \(-0.895954\pi\)
−0.947052 + 0.321081i \(0.895954\pi\)
\(284\) 2.56934 0.152462
\(285\) 0 0
\(286\) −12.9249 −0.764265
\(287\) 5.84501 0.345020
\(288\) −3.92613 −0.231349
\(289\) 29.0452 1.70854
\(290\) 0 0
\(291\) 10.4090 0.610189
\(292\) −12.6693 −0.741412
\(293\) 16.6235 0.971153 0.485576 0.874194i \(-0.338610\pi\)
0.485576 + 0.874194i \(0.338610\pi\)
\(294\) −12.0446 −0.702453
\(295\) 0 0
\(296\) 18.8012 1.09280
\(297\) −2.93111 −0.170080
\(298\) −17.2320 −0.998225
\(299\) 10.2982 0.595561
\(300\) 0 0
\(301\) 38.0336 2.19222
\(302\) −39.3429 −2.26393
\(303\) −7.65744 −0.439908
\(304\) −4.07827 −0.233905
\(305\) 0 0
\(306\) 15.9519 0.911911
\(307\) −14.1643 −0.808402 −0.404201 0.914670i \(-0.632450\pi\)
−0.404201 + 0.914670i \(0.632450\pi\)
\(308\) 35.9897 2.05070
\(309\) 2.98602 0.169869
\(310\) 0 0
\(311\) 15.0500 0.853406 0.426703 0.904392i \(-0.359675\pi\)
0.426703 + 0.904392i \(0.359675\pi\)
\(312\) 6.73076 0.381054
\(313\) −3.57476 −0.202057 −0.101029 0.994884i \(-0.532213\pi\)
−0.101029 + 0.994884i \(0.532213\pi\)
\(314\) 18.5395 1.04624
\(315\) 0 0
\(316\) 10.8354 0.609539
\(317\) 12.7820 0.717907 0.358954 0.933355i \(-0.383134\pi\)
0.358954 + 0.933355i \(0.383134\pi\)
\(318\) −8.72047 −0.489020
\(319\) 7.49172 0.419456
\(320\) 0 0
\(321\) 7.07213 0.394728
\(322\) −44.9390 −2.50435
\(323\) −20.0143 −1.11363
\(324\) 3.52640 0.195911
\(325\) 0 0
\(326\) −23.3671 −1.29418
\(327\) 13.3022 0.735612
\(328\) 6.02366 0.332601
\(329\) −26.0855 −1.43814
\(330\) 0 0
\(331\) 4.70504 0.258612 0.129306 0.991605i \(-0.458725\pi\)
0.129306 + 0.991605i \(0.458725\pi\)
\(332\) 35.6704 1.95767
\(333\) 5.23959 0.287128
\(334\) 13.1121 0.717465
\(335\) 0 0
\(336\) −4.81440 −0.262647
\(337\) −17.4048 −0.948098 −0.474049 0.880499i \(-0.657208\pi\)
−0.474049 + 0.880499i \(0.657208\pi\)
\(338\) −22.2896 −1.21239
\(339\) −10.0233 −0.544390
\(340\) 0 0
\(341\) 1.22545 0.0663618
\(342\) −6.93377 −0.374936
\(343\) −6.53364 −0.352783
\(344\) 39.1961 2.11331
\(345\) 0 0
\(346\) 38.9490 2.09391
\(347\) 14.0106 0.752130 0.376065 0.926593i \(-0.377277\pi\)
0.376065 + 0.926593i \(0.377277\pi\)
\(348\) −9.01325 −0.483161
\(349\) 35.9459 1.92414 0.962069 0.272806i \(-0.0879518\pi\)
0.962069 + 0.272806i \(0.0879518\pi\)
\(350\) 0 0
\(351\) 1.87575 0.100120
\(352\) −11.5079 −0.613373
\(353\) 8.80198 0.468482 0.234241 0.972179i \(-0.424740\pi\)
0.234241 + 0.972179i \(0.424740\pi\)
\(354\) 16.7110 0.888178
\(355\) 0 0
\(356\) 1.01388 0.0537357
\(357\) −23.6269 −1.25047
\(358\) −12.6009 −0.665981
\(359\) 6.02962 0.318231 0.159116 0.987260i \(-0.449136\pi\)
0.159116 + 0.987260i \(0.449136\pi\)
\(360\) 0 0
\(361\) −10.3004 −0.542129
\(362\) −0.770472 −0.0404951
\(363\) 2.40861 0.126419
\(364\) −23.0314 −1.20717
\(365\) 0 0
\(366\) 15.1325 0.790989
\(367\) 28.7662 1.50158 0.750792 0.660539i \(-0.229672\pi\)
0.750792 + 0.660539i \(0.229672\pi\)
\(368\) −7.59128 −0.395723
\(369\) 1.67869 0.0873891
\(370\) 0 0
\(371\) 12.9162 0.670574
\(372\) −1.47433 −0.0764405
\(373\) 33.6000 1.73974 0.869872 0.493278i \(-0.164201\pi\)
0.869872 + 0.493278i \(0.164201\pi\)
\(374\) 46.7568 2.41774
\(375\) 0 0
\(376\) −26.8828 −1.38637
\(377\) −4.79429 −0.246918
\(378\) −8.18532 −0.421008
\(379\) 4.63403 0.238034 0.119017 0.992892i \(-0.462026\pi\)
0.119017 + 0.992892i \(0.462026\pi\)
\(380\) 0 0
\(381\) −10.5730 −0.541673
\(382\) −8.15718 −0.417358
\(383\) −12.9058 −0.659453 −0.329727 0.944076i \(-0.606957\pi\)
−0.329727 + 0.944076i \(0.606957\pi\)
\(384\) 20.3461 1.03828
\(385\) 0 0
\(386\) −57.1567 −2.90920
\(387\) 10.9233 0.555261
\(388\) −36.7064 −1.86349
\(389\) −17.5246 −0.888532 −0.444266 0.895895i \(-0.646535\pi\)
−0.444266 + 0.895895i \(0.646535\pi\)
\(390\) 0 0
\(391\) −37.2546 −1.88405
\(392\) 18.3848 0.928573
\(393\) 9.02608 0.455305
\(394\) 35.1593 1.77130
\(395\) 0 0
\(396\) 10.3363 0.519416
\(397\) −27.0176 −1.35597 −0.677987 0.735074i \(-0.737147\pi\)
−0.677987 + 0.735074i \(0.737147\pi\)
\(398\) −26.6234 −1.33451
\(399\) 10.2698 0.514134
\(400\) 0 0
\(401\) −15.9792 −0.797965 −0.398983 0.916958i \(-0.630637\pi\)
−0.398983 + 0.916958i \(0.630637\pi\)
\(402\) −23.6059 −1.17735
\(403\) −0.784220 −0.0390648
\(404\) 27.0032 1.34346
\(405\) 0 0
\(406\) 20.9212 1.03830
\(407\) 15.3578 0.761258
\(408\) −24.3490 −1.20546
\(409\) −30.9962 −1.53266 −0.766331 0.642446i \(-0.777920\pi\)
−0.766331 + 0.642446i \(0.777920\pi\)
\(410\) 0 0
\(411\) 19.6646 0.969982
\(412\) −10.5299 −0.518771
\(413\) −24.7511 −1.21792
\(414\) −12.9065 −0.634321
\(415\) 0 0
\(416\) 7.36442 0.361070
\(417\) 11.6520 0.570601
\(418\) −20.3236 −0.994061
\(419\) 25.9153 1.26605 0.633024 0.774132i \(-0.281814\pi\)
0.633024 + 0.774132i \(0.281814\pi\)
\(420\) 0 0
\(421\) 3.92646 0.191364 0.0956821 0.995412i \(-0.469497\pi\)
0.0956821 + 0.995412i \(0.469497\pi\)
\(422\) 17.7319 0.863176
\(423\) −7.49178 −0.364263
\(424\) 13.3109 0.646436
\(425\) 0 0
\(426\) −1.71282 −0.0829863
\(427\) −22.4132 −1.08465
\(428\) −24.9392 −1.20548
\(429\) 5.49801 0.265447
\(430\) 0 0
\(431\) −35.4632 −1.70820 −0.854101 0.520108i \(-0.825892\pi\)
−0.854101 + 0.520108i \(0.825892\pi\)
\(432\) −1.38270 −0.0665251
\(433\) 10.0959 0.485178 0.242589 0.970129i \(-0.422003\pi\)
0.242589 + 0.970129i \(0.422003\pi\)
\(434\) 3.42215 0.164269
\(435\) 0 0
\(436\) −46.9088 −2.24652
\(437\) 16.1933 0.774632
\(438\) 8.44580 0.403556
\(439\) 7.10560 0.339132 0.169566 0.985519i \(-0.445763\pi\)
0.169566 + 0.985519i \(0.445763\pi\)
\(440\) 0 0
\(441\) 5.12353 0.243978
\(442\) −29.9218 −1.42323
\(443\) 20.6841 0.982733 0.491366 0.870953i \(-0.336498\pi\)
0.491366 + 0.870953i \(0.336498\pi\)
\(444\) −18.4769 −0.876874
\(445\) 0 0
\(446\) −18.3256 −0.867744
\(447\) 7.33020 0.346706
\(448\) −41.7654 −1.97323
\(449\) 19.4940 0.919980 0.459990 0.887924i \(-0.347853\pi\)
0.459990 + 0.887924i \(0.347853\pi\)
\(450\) 0 0
\(451\) 4.92042 0.231694
\(452\) 35.3461 1.66254
\(453\) 16.7358 0.786314
\(454\) −30.1657 −1.41575
\(455\) 0 0
\(456\) 10.5837 0.495628
\(457\) −4.34194 −0.203107 −0.101554 0.994830i \(-0.532381\pi\)
−0.101554 + 0.994830i \(0.532381\pi\)
\(458\) −34.0948 −1.59315
\(459\) −6.78566 −0.316728
\(460\) 0 0
\(461\) 11.7897 0.549102 0.274551 0.961573i \(-0.411471\pi\)
0.274551 + 0.961573i \(0.411471\pi\)
\(462\) −23.9921 −1.11621
\(463\) −9.07870 −0.421923 −0.210962 0.977494i \(-0.567659\pi\)
−0.210962 + 0.977494i \(0.567659\pi\)
\(464\) 3.53409 0.164066
\(465\) 0 0
\(466\) 26.7925 1.24114
\(467\) −34.9181 −1.61582 −0.807910 0.589307i \(-0.799401\pi\)
−0.807910 + 0.589307i \(0.799401\pi\)
\(468\) −6.61463 −0.305762
\(469\) 34.9634 1.61446
\(470\) 0 0
\(471\) −7.88635 −0.363384
\(472\) −25.5076 −1.17408
\(473\) 32.0173 1.47216
\(474\) −7.22328 −0.331776
\(475\) 0 0
\(476\) 83.3179 3.81887
\(477\) 3.70953 0.169848
\(478\) 16.2224 0.741996
\(479\) −8.64649 −0.395068 −0.197534 0.980296i \(-0.563293\pi\)
−0.197534 + 0.980296i \(0.563293\pi\)
\(480\) 0 0
\(481\) −9.82814 −0.448125
\(482\) −66.8734 −3.04600
\(483\) 19.1162 0.869819
\(484\) −8.49373 −0.386079
\(485\) 0 0
\(486\) −2.35083 −0.106636
\(487\) 2.84462 0.128902 0.0644510 0.997921i \(-0.479470\pi\)
0.0644510 + 0.997921i \(0.479470\pi\)
\(488\) −23.0983 −1.04561
\(489\) 9.93992 0.449499
\(490\) 0 0
\(491\) 36.8041 1.66095 0.830473 0.557059i \(-0.188070\pi\)
0.830473 + 0.557059i \(0.188070\pi\)
\(492\) −5.91973 −0.266882
\(493\) 17.3437 0.781121
\(494\) 13.0060 0.585168
\(495\) 0 0
\(496\) 0.578084 0.0259567
\(497\) 2.53691 0.113796
\(498\) −23.7792 −1.06557
\(499\) 5.85775 0.262229 0.131114 0.991367i \(-0.458144\pi\)
0.131114 + 0.991367i \(0.458144\pi\)
\(500\) 0 0
\(501\) −5.57767 −0.249192
\(502\) −28.9759 −1.29326
\(503\) −30.2874 −1.35045 −0.675223 0.737613i \(-0.735953\pi\)
−0.675223 + 0.737613i \(0.735953\pi\)
\(504\) 12.4941 0.556530
\(505\) 0 0
\(506\) −37.8304 −1.68177
\(507\) 9.48157 0.421091
\(508\) 37.2848 1.65424
\(509\) 15.1430 0.671201 0.335601 0.942004i \(-0.391061\pi\)
0.335601 + 0.942004i \(0.391061\pi\)
\(510\) 0 0
\(511\) −12.5093 −0.553380
\(512\) −15.3517 −0.678457
\(513\) 2.94950 0.130224
\(514\) 48.9618 2.15961
\(515\) 0 0
\(516\) −38.5198 −1.69574
\(517\) −21.9592 −0.965765
\(518\) 42.8877 1.88438
\(519\) −16.5682 −0.727263
\(520\) 0 0
\(521\) 39.0832 1.71227 0.856134 0.516754i \(-0.172860\pi\)
0.856134 + 0.516754i \(0.172860\pi\)
\(522\) 6.00857 0.262988
\(523\) −38.7976 −1.69650 −0.848251 0.529594i \(-0.822344\pi\)
−0.848251 + 0.529594i \(0.822344\pi\)
\(524\) −31.8296 −1.39048
\(525\) 0 0
\(526\) 5.25449 0.229107
\(527\) 2.83698 0.123581
\(528\) −4.05284 −0.176377
\(529\) 7.14224 0.310532
\(530\) 0 0
\(531\) −7.10854 −0.308484
\(532\) −36.2155 −1.57014
\(533\) −3.14880 −0.136390
\(534\) −0.675892 −0.0292487
\(535\) 0 0
\(536\) 36.0320 1.55635
\(537\) 5.36021 0.231310
\(538\) −45.3796 −1.95645
\(539\) 15.0176 0.646855
\(540\) 0 0
\(541\) 14.2280 0.611710 0.305855 0.952078i \(-0.401058\pi\)
0.305855 + 0.952078i \(0.401058\pi\)
\(542\) −28.0716 −1.20578
\(543\) 0.327745 0.0140649
\(544\) −26.6414 −1.14224
\(545\) 0 0
\(546\) 15.3536 0.657073
\(547\) 39.3229 1.68133 0.840663 0.541559i \(-0.182166\pi\)
0.840663 + 0.541559i \(0.182166\pi\)
\(548\) −69.3452 −2.96228
\(549\) −6.43710 −0.274729
\(550\) 0 0
\(551\) −7.53873 −0.321161
\(552\) 19.7005 0.838509
\(553\) 10.6986 0.454952
\(554\) 61.4662 2.61145
\(555\) 0 0
\(556\) −41.0896 −1.74259
\(557\) 35.3849 1.49931 0.749654 0.661830i \(-0.230220\pi\)
0.749654 + 0.661830i \(0.230220\pi\)
\(558\) 0.982844 0.0416071
\(559\) −20.4893 −0.866605
\(560\) 0 0
\(561\) −19.8895 −0.839735
\(562\) 23.7948 1.00372
\(563\) 15.0386 0.633800 0.316900 0.948459i \(-0.397358\pi\)
0.316900 + 0.948459i \(0.397358\pi\)
\(564\) 26.4190 1.11244
\(565\) 0 0
\(566\) −74.9063 −3.14855
\(567\) 3.48189 0.146225
\(568\) 2.61445 0.109700
\(569\) −2.72409 −0.114200 −0.0570998 0.998368i \(-0.518185\pi\)
−0.0570998 + 0.998368i \(0.518185\pi\)
\(570\) 0 0
\(571\) 12.6236 0.528282 0.264141 0.964484i \(-0.414912\pi\)
0.264141 + 0.964484i \(0.414912\pi\)
\(572\) −19.3882 −0.810661
\(573\) 3.46992 0.144958
\(574\) 13.7406 0.573522
\(575\) 0 0
\(576\) −11.9951 −0.499794
\(577\) −23.5844 −0.981831 −0.490915 0.871207i \(-0.663338\pi\)
−0.490915 + 0.871207i \(0.663338\pi\)
\(578\) 68.2803 2.84009
\(579\) 24.3134 1.01043
\(580\) 0 0
\(581\) 35.2201 1.46118
\(582\) 24.4699 1.01431
\(583\) 10.8730 0.450315
\(584\) −12.8917 −0.533461
\(585\) 0 0
\(586\) 39.0789 1.61434
\(587\) −22.6772 −0.935988 −0.467994 0.883732i \(-0.655023\pi\)
−0.467994 + 0.883732i \(0.655023\pi\)
\(588\) −18.0676 −0.745097
\(589\) −1.23314 −0.0508106
\(590\) 0 0
\(591\) −14.9561 −0.615212
\(592\) 7.24477 0.297758
\(593\) −8.74287 −0.359027 −0.179513 0.983756i \(-0.557452\pi\)
−0.179513 + 0.983756i \(0.557452\pi\)
\(594\) −6.89053 −0.282722
\(595\) 0 0
\(596\) −25.8492 −1.05882
\(597\) 11.3251 0.463506
\(598\) 24.2094 0.989994
\(599\) 16.3209 0.666854 0.333427 0.942776i \(-0.391795\pi\)
0.333427 + 0.942776i \(0.391795\pi\)
\(600\) 0 0
\(601\) 36.2713 1.47954 0.739768 0.672862i \(-0.234935\pi\)
0.739768 + 0.672862i \(0.234935\pi\)
\(602\) 89.4105 3.64410
\(603\) 10.0415 0.408922
\(604\) −59.0170 −2.40137
\(605\) 0 0
\(606\) −18.0013 −0.731254
\(607\) −16.6820 −0.677102 −0.338551 0.940948i \(-0.609937\pi\)
−0.338551 + 0.940948i \(0.609937\pi\)
\(608\) 11.5801 0.469636
\(609\) −8.89948 −0.360625
\(610\) 0 0
\(611\) 14.0527 0.568511
\(612\) 23.9290 0.967271
\(613\) 24.7967 1.00153 0.500765 0.865583i \(-0.333052\pi\)
0.500765 + 0.865583i \(0.333052\pi\)
\(614\) −33.2980 −1.34380
\(615\) 0 0
\(616\) 36.6215 1.47552
\(617\) 27.9669 1.12591 0.562953 0.826489i \(-0.309665\pi\)
0.562953 + 0.826489i \(0.309665\pi\)
\(618\) 7.01963 0.282371
\(619\) 27.2686 1.09602 0.548009 0.836473i \(-0.315386\pi\)
0.548009 + 0.836473i \(0.315386\pi\)
\(620\) 0 0
\(621\) 5.49019 0.220314
\(622\) 35.3799 1.41861
\(623\) 1.00108 0.0401076
\(624\) 2.59359 0.103827
\(625\) 0 0
\(626\) −8.40365 −0.335878
\(627\) 8.64530 0.345260
\(628\) 27.8104 1.10976
\(629\) 35.5541 1.41763
\(630\) 0 0
\(631\) 42.2603 1.68235 0.841177 0.540759i \(-0.181863\pi\)
0.841177 + 0.540759i \(0.181863\pi\)
\(632\) 11.0256 0.438575
\(633\) −7.54283 −0.299801
\(634\) 30.0482 1.19337
\(635\) 0 0
\(636\) −13.0813 −0.518707
\(637\) −9.61045 −0.380780
\(638\) 17.6118 0.697256
\(639\) 0.728602 0.0288230
\(640\) 0 0
\(641\) 45.8456 1.81079 0.905396 0.424569i \(-0.139574\pi\)
0.905396 + 0.424569i \(0.139574\pi\)
\(642\) 16.6254 0.656151
\(643\) −46.6710 −1.84052 −0.920261 0.391304i \(-0.872024\pi\)
−0.920261 + 0.391304i \(0.872024\pi\)
\(644\) −67.4115 −2.65639
\(645\) 0 0
\(646\) −47.0502 −1.85117
\(647\) −12.1264 −0.476740 −0.238370 0.971174i \(-0.576613\pi\)
−0.238370 + 0.971174i \(0.576613\pi\)
\(648\) 3.58831 0.140962
\(649\) −20.8359 −0.817880
\(650\) 0 0
\(651\) −1.45572 −0.0570542
\(652\) −35.0521 −1.37275
\(653\) −1.72017 −0.0673154 −0.0336577 0.999433i \(-0.510716\pi\)
−0.0336577 + 0.999433i \(0.510716\pi\)
\(654\) 31.2711 1.22280
\(655\) 0 0
\(656\) 2.32112 0.0906246
\(657\) −3.59269 −0.140164
\(658\) −61.3226 −2.39060
\(659\) 5.47947 0.213450 0.106725 0.994289i \(-0.465964\pi\)
0.106725 + 0.994289i \(0.465964\pi\)
\(660\) 0 0
\(661\) 0.797448 0.0310171 0.0155086 0.999880i \(-0.495063\pi\)
0.0155086 + 0.999880i \(0.495063\pi\)
\(662\) 11.0607 0.429888
\(663\) 12.7282 0.494322
\(664\) 36.2966 1.40858
\(665\) 0 0
\(666\) 12.3174 0.477289
\(667\) −14.0326 −0.543344
\(668\) 19.6691 0.761020
\(669\) 7.79539 0.301387
\(670\) 0 0
\(671\) −18.8678 −0.728384
\(672\) 13.6703 0.527344
\(673\) −51.3280 −1.97855 −0.989274 0.146074i \(-0.953336\pi\)
−0.989274 + 0.146074i \(0.953336\pi\)
\(674\) −40.9156 −1.57601
\(675\) 0 0
\(676\) −33.4358 −1.28599
\(677\) 15.0929 0.580069 0.290034 0.957016i \(-0.406333\pi\)
0.290034 + 0.957016i \(0.406333\pi\)
\(678\) −23.5630 −0.904933
\(679\) −36.2431 −1.39088
\(680\) 0 0
\(681\) 12.8319 0.491721
\(682\) 2.88082 0.110312
\(683\) −10.8602 −0.415555 −0.207777 0.978176i \(-0.566623\pi\)
−0.207777 + 0.978176i \(0.566623\pi\)
\(684\) −10.4011 −0.397697
\(685\) 0 0
\(686\) −15.3595 −0.586428
\(687\) 14.5033 0.553336
\(688\) 15.1036 0.575819
\(689\) −6.95814 −0.265084
\(690\) 0 0
\(691\) 8.44552 0.321283 0.160641 0.987013i \(-0.448644\pi\)
0.160641 + 0.987013i \(0.448644\pi\)
\(692\) 58.4261 2.22103
\(693\) 10.2058 0.387686
\(694\) 32.9366 1.25026
\(695\) 0 0
\(696\) −9.17148 −0.347644
\(697\) 11.3910 0.431466
\(698\) 84.5026 3.19847
\(699\) −11.3970 −0.431076
\(700\) 0 0
\(701\) 48.9399 1.84843 0.924216 0.381869i \(-0.124719\pi\)
0.924216 + 0.381869i \(0.124719\pi\)
\(702\) 4.40956 0.166428
\(703\) −15.4542 −0.582865
\(704\) −35.1588 −1.32510
\(705\) 0 0
\(706\) 20.6920 0.778752
\(707\) 26.6623 1.00274
\(708\) 25.0676 0.942097
\(709\) 24.7834 0.930762 0.465381 0.885111i \(-0.345917\pi\)
0.465381 + 0.885111i \(0.345917\pi\)
\(710\) 0 0
\(711\) 3.07265 0.115233
\(712\) 1.03168 0.0386639
\(713\) −2.29536 −0.0859620
\(714\) −55.5428 −2.07864
\(715\) 0 0
\(716\) −18.9023 −0.706410
\(717\) −6.90072 −0.257712
\(718\) 14.1746 0.528992
\(719\) −18.2171 −0.679382 −0.339691 0.940537i \(-0.610322\pi\)
−0.339691 + 0.940537i \(0.610322\pi\)
\(720\) 0 0
\(721\) −10.3970 −0.387204
\(722\) −24.2146 −0.901174
\(723\) 28.4467 1.05795
\(724\) −1.15576 −0.0429534
\(725\) 0 0
\(726\) 5.66224 0.210145
\(727\) 16.6699 0.618253 0.309127 0.951021i \(-0.399963\pi\)
0.309127 + 0.951021i \(0.399963\pi\)
\(728\) −23.4357 −0.868586
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 74.1216 2.74149
\(732\) 22.6998 0.839008
\(733\) 20.0644 0.741094 0.370547 0.928814i \(-0.379170\pi\)
0.370547 + 0.928814i \(0.379170\pi\)
\(734\) 67.6245 2.49606
\(735\) 0 0
\(736\) 21.5552 0.794535
\(737\) 29.4328 1.08417
\(738\) 3.94632 0.145266
\(739\) −34.0571 −1.25281 −0.626406 0.779497i \(-0.715475\pi\)
−0.626406 + 0.779497i \(0.715475\pi\)
\(740\) 0 0
\(741\) −5.53252 −0.203242
\(742\) 30.3637 1.11469
\(743\) −38.5355 −1.41373 −0.706865 0.707348i \(-0.749891\pi\)
−0.706865 + 0.707348i \(0.749891\pi\)
\(744\) −1.50021 −0.0550005
\(745\) 0 0
\(746\) 78.9880 2.89195
\(747\) 10.1152 0.370097
\(748\) 70.1383 2.56451
\(749\) −24.6244 −0.899754
\(750\) 0 0
\(751\) −36.0351 −1.31494 −0.657470 0.753481i \(-0.728373\pi\)
−0.657470 + 0.753481i \(0.728373\pi\)
\(752\) −10.3589 −0.377749
\(753\) 12.3258 0.449178
\(754\) −11.2706 −0.410449
\(755\) 0 0
\(756\) −12.2785 −0.446566
\(757\) 35.0131 1.27257 0.636287 0.771453i \(-0.280470\pi\)
0.636287 + 0.771453i \(0.280470\pi\)
\(758\) 10.8938 0.395681
\(759\) 16.0923 0.584115
\(760\) 0 0
\(761\) 31.3577 1.13672 0.568358 0.822781i \(-0.307579\pi\)
0.568358 + 0.822781i \(0.307579\pi\)
\(762\) −24.8554 −0.900416
\(763\) −46.3166 −1.67678
\(764\) −12.2363 −0.442694
\(765\) 0 0
\(766\) −30.3392 −1.09620
\(767\) 13.3338 0.481456
\(768\) 23.8400 0.860253
\(769\) 2.39815 0.0864793 0.0432397 0.999065i \(-0.486232\pi\)
0.0432397 + 0.999065i \(0.486232\pi\)
\(770\) 0 0
\(771\) −20.8274 −0.750082
\(772\) −85.7388 −3.08581
\(773\) −10.9226 −0.392858 −0.196429 0.980518i \(-0.562935\pi\)
−0.196429 + 0.980518i \(0.562935\pi\)
\(774\) 25.6788 0.923004
\(775\) 0 0
\(776\) −37.3508 −1.34082
\(777\) −18.2437 −0.654487
\(778\) −41.1973 −1.47700
\(779\) −4.95130 −0.177399
\(780\) 0 0
\(781\) 2.13561 0.0764181
\(782\) −87.5792 −3.13183
\(783\) −2.55593 −0.0913417
\(784\) 7.08430 0.253011
\(785\) 0 0
\(786\) 21.2188 0.756848
\(787\) 11.5724 0.412511 0.206256 0.978498i \(-0.433872\pi\)
0.206256 + 0.978498i \(0.433872\pi\)
\(788\) 52.7412 1.87883
\(789\) −2.23517 −0.0795740
\(790\) 0 0
\(791\) 34.8999 1.24090
\(792\) 10.5177 0.373731
\(793\) 12.0744 0.428773
\(794\) −63.5137 −2.25402
\(795\) 0 0
\(796\) −39.9369 −1.41553
\(797\) 21.8923 0.775467 0.387733 0.921772i \(-0.373258\pi\)
0.387733 + 0.921772i \(0.373258\pi\)
\(798\) 24.1426 0.854639
\(799\) −50.8367 −1.79847
\(800\) 0 0
\(801\) 0.287512 0.0101587
\(802\) −37.5645 −1.32645
\(803\) −10.5306 −0.371615
\(804\) −35.4104 −1.24883
\(805\) 0 0
\(806\) −1.84357 −0.0649369
\(807\) 19.3036 0.679520
\(808\) 27.4772 0.966646
\(809\) −21.5498 −0.757652 −0.378826 0.925468i \(-0.623672\pi\)
−0.378826 + 0.925468i \(0.623672\pi\)
\(810\) 0 0
\(811\) −43.9615 −1.54370 −0.771848 0.635807i \(-0.780667\pi\)
−0.771848 + 0.635807i \(0.780667\pi\)
\(812\) 31.3831 1.10133
\(813\) 11.9411 0.418794
\(814\) 36.1036 1.26543
\(815\) 0 0
\(816\) −9.38252 −0.328454
\(817\) −32.2182 −1.12717
\(818\) −72.8667 −2.54772
\(819\) −6.53114 −0.228216
\(820\) 0 0
\(821\) 33.5061 1.16937 0.584686 0.811260i \(-0.301218\pi\)
0.584686 + 0.811260i \(0.301218\pi\)
\(822\) 46.2281 1.61239
\(823\) 55.9860 1.95155 0.975774 0.218781i \(-0.0702080\pi\)
0.975774 + 0.218781i \(0.0702080\pi\)
\(824\) −10.7148 −0.373266
\(825\) 0 0
\(826\) −58.1857 −2.02454
\(827\) −40.8100 −1.41910 −0.709552 0.704653i \(-0.751103\pi\)
−0.709552 + 0.704653i \(0.751103\pi\)
\(828\) −19.3606 −0.672828
\(829\) 0.122914 0.00426896 0.00213448 0.999998i \(-0.499321\pi\)
0.00213448 + 0.999998i \(0.499321\pi\)
\(830\) 0 0
\(831\) −26.1466 −0.907016
\(832\) 22.4997 0.780036
\(833\) 34.7666 1.20459
\(834\) 27.3919 0.948503
\(835\) 0 0
\(836\) −30.4868 −1.05441
\(837\) −0.418084 −0.0144511
\(838\) 60.9226 2.10454
\(839\) 14.8083 0.511241 0.255620 0.966777i \(-0.417720\pi\)
0.255620 + 0.966777i \(0.417720\pi\)
\(840\) 0 0
\(841\) −22.4672 −0.774731
\(842\) 9.23045 0.318102
\(843\) −10.1219 −0.348616
\(844\) 26.5990 0.915577
\(845\) 0 0
\(846\) −17.6119 −0.605509
\(847\) −8.38651 −0.288164
\(848\) 5.12916 0.176136
\(849\) 31.8638 1.09356
\(850\) 0 0
\(851\) −28.7664 −0.986098
\(852\) −2.56934 −0.0880242
\(853\) 7.26474 0.248740 0.124370 0.992236i \(-0.460309\pi\)
0.124370 + 0.992236i \(0.460309\pi\)
\(854\) −52.6897 −1.80301
\(855\) 0 0
\(856\) −25.3770 −0.867367
\(857\) −26.8175 −0.916068 −0.458034 0.888935i \(-0.651446\pi\)
−0.458034 + 0.888935i \(0.651446\pi\)
\(858\) 12.9249 0.441249
\(859\) −20.6038 −0.702991 −0.351496 0.936190i \(-0.614327\pi\)
−0.351496 + 0.936190i \(0.614327\pi\)
\(860\) 0 0
\(861\) −5.84501 −0.199197
\(862\) −83.3679 −2.83952
\(863\) 13.7488 0.468015 0.234008 0.972235i \(-0.424816\pi\)
0.234008 + 0.972235i \(0.424816\pi\)
\(864\) 3.92613 0.133570
\(865\) 0 0
\(866\) 23.7338 0.806506
\(867\) −29.0452 −0.986427
\(868\) 5.13346 0.174241
\(869\) 9.00627 0.305517
\(870\) 0 0
\(871\) −18.8353 −0.638211
\(872\) −47.7323 −1.61642
\(873\) −10.4090 −0.352293
\(874\) 38.0678 1.28766
\(875\) 0 0
\(876\) 12.6693 0.428055
\(877\) 22.0430 0.744339 0.372170 0.928165i \(-0.378614\pi\)
0.372170 + 0.928165i \(0.378614\pi\)
\(878\) 16.7041 0.563735
\(879\) −16.6235 −0.560695
\(880\) 0 0
\(881\) −9.03929 −0.304541 −0.152271 0.988339i \(-0.548659\pi\)
−0.152271 + 0.988339i \(0.548659\pi\)
\(882\) 12.0446 0.405561
\(883\) −1.60120 −0.0538848 −0.0269424 0.999637i \(-0.508577\pi\)
−0.0269424 + 0.999637i \(0.508577\pi\)
\(884\) −44.8847 −1.50963
\(885\) 0 0
\(886\) 48.6249 1.63358
\(887\) 17.4253 0.585083 0.292541 0.956253i \(-0.405499\pi\)
0.292541 + 0.956253i \(0.405499\pi\)
\(888\) −18.8012 −0.630929
\(889\) 36.8141 1.23471
\(890\) 0 0
\(891\) 2.93111 0.0981958
\(892\) −27.4897 −0.920423
\(893\) 22.0970 0.739448
\(894\) 17.2320 0.576326
\(895\) 0 0
\(896\) −70.8427 −2.36669
\(897\) −10.2982 −0.343847
\(898\) 45.8272 1.52927
\(899\) 1.06860 0.0356397
\(900\) 0 0
\(901\) 25.1716 0.838588
\(902\) 11.5671 0.385141
\(903\) −38.0336 −1.26568
\(904\) 35.9666 1.19623
\(905\) 0 0
\(906\) 39.3429 1.30708
\(907\) −20.8690 −0.692942 −0.346471 0.938061i \(-0.612620\pi\)
−0.346471 + 0.938061i \(0.612620\pi\)
\(908\) −45.2506 −1.50169
\(909\) 7.65744 0.253981
\(910\) 0 0
\(911\) 49.1748 1.62923 0.814617 0.580000i \(-0.196947\pi\)
0.814617 + 0.580000i \(0.196947\pi\)
\(912\) 4.07827 0.135045
\(913\) 29.6489 0.981234
\(914\) −10.2072 −0.337623
\(915\) 0 0
\(916\) −51.1445 −1.68986
\(917\) −31.4278 −1.03784
\(918\) −15.9519 −0.526492
\(919\) −11.1959 −0.369318 −0.184659 0.982803i \(-0.559118\pi\)
−0.184659 + 0.982803i \(0.559118\pi\)
\(920\) 0 0
\(921\) 14.1643 0.466731
\(922\) 27.7156 0.912765
\(923\) −1.36667 −0.0449846
\(924\) −35.9897 −1.18397
\(925\) 0 0
\(926\) −21.3425 −0.701357
\(927\) −2.98602 −0.0980738
\(928\) −10.0349 −0.329413
\(929\) 48.2290 1.58234 0.791171 0.611596i \(-0.209472\pi\)
0.791171 + 0.611596i \(0.209472\pi\)
\(930\) 0 0
\(931\) −15.1119 −0.495271
\(932\) 40.1906 1.31649
\(933\) −15.0500 −0.492714
\(934\) −82.0866 −2.68596
\(935\) 0 0
\(936\) −6.73076 −0.220002
\(937\) 50.5397 1.65106 0.825529 0.564359i \(-0.190877\pi\)
0.825529 + 0.564359i \(0.190877\pi\)
\(938\) 82.1930 2.68370
\(939\) 3.57476 0.116658
\(940\) 0 0
\(941\) −38.1644 −1.24412 −0.622061 0.782969i \(-0.713705\pi\)
−0.622061 + 0.782969i \(0.713705\pi\)
\(942\) −18.5395 −0.604049
\(943\) −9.21634 −0.300125
\(944\) −9.82896 −0.319906
\(945\) 0 0
\(946\) 75.2672 2.44715
\(947\) 24.5190 0.796759 0.398380 0.917221i \(-0.369573\pi\)
0.398380 + 0.917221i \(0.369573\pi\)
\(948\) −10.8354 −0.351917
\(949\) 6.73897 0.218756
\(950\) 0 0
\(951\) −12.7820 −0.414484
\(952\) 84.7806 2.74775
\(953\) 13.0019 0.421173 0.210586 0.977575i \(-0.432463\pi\)
0.210586 + 0.977575i \(0.432463\pi\)
\(954\) 8.72047 0.282336
\(955\) 0 0
\(956\) 24.3347 0.787041
\(957\) −7.49172 −0.242173
\(958\) −20.3264 −0.656717
\(959\) −68.4698 −2.21101
\(960\) 0 0
\(961\) −30.8252 −0.994361
\(962\) −23.1043 −0.744912
\(963\) −7.07213 −0.227896
\(964\) −100.315 −3.23091
\(965\) 0 0
\(966\) 44.9390 1.44589
\(967\) −43.3212 −1.39312 −0.696559 0.717500i \(-0.745286\pi\)
−0.696559 + 0.717500i \(0.745286\pi\)
\(968\) −8.64284 −0.277791
\(969\) 20.0143 0.642952
\(970\) 0 0
\(971\) −29.5324 −0.947741 −0.473870 0.880595i \(-0.657143\pi\)
−0.473870 + 0.880595i \(0.657143\pi\)
\(972\) −3.52640 −0.113109
\(973\) −40.5709 −1.30064
\(974\) 6.68721 0.214272
\(975\) 0 0
\(976\) −8.90056 −0.284900
\(977\) −60.6153 −1.93926 −0.969628 0.244583i \(-0.921349\pi\)
−0.969628 + 0.244583i \(0.921349\pi\)
\(978\) 23.3671 0.747196
\(979\) 0.842729 0.0269337
\(980\) 0 0
\(981\) −13.3022 −0.424706
\(982\) 86.5202 2.76097
\(983\) −40.2796 −1.28472 −0.642359 0.766404i \(-0.722044\pi\)
−0.642359 + 0.766404i \(0.722044\pi\)
\(984\) −6.02366 −0.192027
\(985\) 0 0
\(986\) 40.7721 1.29845
\(987\) 26.0855 0.830311
\(988\) 19.5099 0.620692
\(989\) −59.9709 −1.90696
\(990\) 0 0
\(991\) 38.1495 1.21186 0.605930 0.795518i \(-0.292801\pi\)
0.605930 + 0.795518i \(0.292801\pi\)
\(992\) −1.64145 −0.0521161
\(993\) −4.70504 −0.149310
\(994\) 5.96384 0.189162
\(995\) 0 0
\(996\) −35.6704 −1.13026
\(997\) 26.5777 0.841724 0.420862 0.907125i \(-0.361728\pi\)
0.420862 + 0.907125i \(0.361728\pi\)
\(998\) 13.7706 0.435900
\(999\) −5.23959 −0.165773
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.p.1.7 8
3.2 odd 2 5625.2.a.t.1.2 8
5.2 odd 4 1875.2.b.h.1249.15 16
5.3 odd 4 1875.2.b.h.1249.2 16
5.4 even 2 1875.2.a.m.1.2 8
15.14 odd 2 5625.2.a.bd.1.7 8
25.2 odd 20 375.2.i.c.274.4 16
25.9 even 10 375.2.g.e.151.1 16
25.11 even 5 375.2.g.d.226.4 16
25.12 odd 20 75.2.i.a.19.1 yes 16
25.13 odd 20 375.2.i.c.349.4 16
25.14 even 10 375.2.g.e.226.1 16
25.16 even 5 375.2.g.d.151.4 16
25.23 odd 20 75.2.i.a.4.1 16
75.23 even 20 225.2.m.b.154.4 16
75.62 even 20 225.2.m.b.19.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.4.1 16 25.23 odd 20
75.2.i.a.19.1 yes 16 25.12 odd 20
225.2.m.b.19.4 16 75.62 even 20
225.2.m.b.154.4 16 75.23 even 20
375.2.g.d.151.4 16 25.16 even 5
375.2.g.d.226.4 16 25.11 even 5
375.2.g.e.151.1 16 25.9 even 10
375.2.g.e.226.1 16 25.14 even 10
375.2.i.c.274.4 16 25.2 odd 20
375.2.i.c.349.4 16 25.13 odd 20
1875.2.a.m.1.2 8 5.4 even 2
1875.2.a.p.1.7 8 1.1 even 1 trivial
1875.2.b.h.1249.2 16 5.3 odd 4
1875.2.b.h.1249.15 16 5.2 odd 4
5625.2.a.t.1.2 8 3.2 odd 2
5625.2.a.bd.1.7 8 15.14 odd 2