Properties

Label 1875.2.a.p.1.8
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 1.8
Root \(2.53767\) 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)
 
\(f(q)\) \(=\) \(q+2.53767 q^{2} -1.00000 q^{3} +4.43979 q^{4} -2.53767 q^{6} +1.04054 q^{7} +6.19138 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.53767 q^{2} -1.00000 q^{3} +4.43979 q^{4} -2.53767 q^{6} +1.04054 q^{7} +6.19138 q^{8} +1.00000 q^{9} +2.97101 q^{11} -4.43979 q^{12} +5.66922 q^{13} +2.64054 q^{14} +6.83213 q^{16} -5.08361 q^{17} +2.53767 q^{18} -5.37156 q^{19} -1.04054 q^{21} +7.53945 q^{22} +3.86039 q^{23} -6.19138 q^{24} +14.3866 q^{26} -1.00000 q^{27} +4.61976 q^{28} +0.679696 q^{29} +0.850111 q^{31} +4.95495 q^{32} -2.97101 q^{33} -12.9006 q^{34} +4.43979 q^{36} -1.61763 q^{37} -13.6313 q^{38} -5.66922 q^{39} +1.16529 q^{41} -2.64054 q^{42} +5.68601 q^{43} +13.1906 q^{44} +9.79640 q^{46} +3.28640 q^{47} -6.83213 q^{48} -5.91729 q^{49} +5.08361 q^{51} +25.1701 q^{52} +12.6861 q^{53} -2.53767 q^{54} +6.44235 q^{56} +5.37156 q^{57} +1.72485 q^{58} +3.21187 q^{59} -5.42093 q^{61} +2.15730 q^{62} +1.04054 q^{63} -1.09021 q^{64} -7.53945 q^{66} -0.929140 q^{67} -22.5702 q^{68} -3.86039 q^{69} -1.41358 q^{71} +6.19138 q^{72} +11.3234 q^{73} -4.10501 q^{74} -23.8486 q^{76} +3.09144 q^{77} -14.3866 q^{78} -1.44707 q^{79} +1.00000 q^{81} +2.95713 q^{82} -11.4756 q^{83} -4.61976 q^{84} +14.4292 q^{86} -0.679696 q^{87} +18.3946 q^{88} +9.07225 q^{89} +5.89903 q^{91} +17.1393 q^{92} -0.850111 q^{93} +8.33982 q^{94} -4.95495 q^{96} -6.02928 q^{97} -15.0161 q^{98} +2.97101 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53767 1.79441 0.897203 0.441618i \(-0.145595\pi\)
0.897203 + 0.441618i \(0.145595\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.43979 2.21989
\(5\) 0 0
\(6\) −2.53767 −1.03600
\(7\) 1.04054 0.393285 0.196643 0.980475i \(-0.436996\pi\)
0.196643 + 0.980475i \(0.436996\pi\)
\(8\) 6.19138 2.18898
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.97101 0.895792 0.447896 0.894086i \(-0.352173\pi\)
0.447896 + 0.894086i \(0.352173\pi\)
\(12\) −4.43979 −1.28166
\(13\) 5.66922 1.57236 0.786180 0.617998i \(-0.212056\pi\)
0.786180 + 0.617998i \(0.212056\pi\)
\(14\) 2.64054 0.705714
\(15\) 0 0
\(16\) 6.83213 1.70803
\(17\) −5.08361 −1.23296 −0.616479 0.787372i \(-0.711441\pi\)
−0.616479 + 0.787372i \(0.711441\pi\)
\(18\) 2.53767 0.598135
\(19\) −5.37156 −1.23232 −0.616160 0.787621i \(-0.711313\pi\)
−0.616160 + 0.787621i \(0.711313\pi\)
\(20\) 0 0
\(21\) −1.04054 −0.227063
\(22\) 7.53945 1.60742
\(23\) 3.86039 0.804946 0.402473 0.915432i \(-0.368151\pi\)
0.402473 + 0.915432i \(0.368151\pi\)
\(24\) −6.19138 −1.26381
\(25\) 0 0
\(26\) 14.3866 2.82145
\(27\) −1.00000 −0.192450
\(28\) 4.61976 0.873052
\(29\) 0.679696 0.126216 0.0631082 0.998007i \(-0.479899\pi\)
0.0631082 + 0.998007i \(0.479899\pi\)
\(30\) 0 0
\(31\) 0.850111 0.152684 0.0763422 0.997082i \(-0.475676\pi\)
0.0763422 + 0.997082i \(0.475676\pi\)
\(32\) 4.95495 0.875921
\(33\) −2.97101 −0.517186
\(34\) −12.9006 −2.21243
\(35\) 0 0
\(36\) 4.43979 0.739964
\(37\) −1.61763 −0.265936 −0.132968 0.991120i \(-0.542451\pi\)
−0.132968 + 0.991120i \(0.542451\pi\)
\(38\) −13.6313 −2.21128
\(39\) −5.66922 −0.907802
\(40\) 0 0
\(41\) 1.16529 0.181988 0.0909939 0.995851i \(-0.470996\pi\)
0.0909939 + 0.995851i \(0.470996\pi\)
\(42\) −2.64054 −0.407444
\(43\) 5.68601 0.867109 0.433554 0.901127i \(-0.357259\pi\)
0.433554 + 0.901127i \(0.357259\pi\)
\(44\) 13.1906 1.98856
\(45\) 0 0
\(46\) 9.79640 1.44440
\(47\) 3.28640 0.479371 0.239686 0.970851i \(-0.422956\pi\)
0.239686 + 0.970851i \(0.422956\pi\)
\(48\) −6.83213 −0.986133
\(49\) −5.91729 −0.845327
\(50\) 0 0
\(51\) 5.08361 0.711848
\(52\) 25.1701 3.49047
\(53\) 12.6861 1.74257 0.871287 0.490773i \(-0.163286\pi\)
0.871287 + 0.490773i \(0.163286\pi\)
\(54\) −2.53767 −0.345334
\(55\) 0 0
\(56\) 6.44235 0.860896
\(57\) 5.37156 0.711481
\(58\) 1.72485 0.226484
\(59\) 3.21187 0.418150 0.209075 0.977900i \(-0.432955\pi\)
0.209075 + 0.977900i \(0.432955\pi\)
\(60\) 0 0
\(61\) −5.42093 −0.694079 −0.347039 0.937851i \(-0.612813\pi\)
−0.347039 + 0.937851i \(0.612813\pi\)
\(62\) 2.15730 0.273978
\(63\) 1.04054 0.131095
\(64\) −1.09021 −0.136276
\(65\) 0 0
\(66\) −7.53945 −0.928042
\(67\) −0.929140 −0.113513 −0.0567563 0.998388i \(-0.518076\pi\)
−0.0567563 + 0.998388i \(0.518076\pi\)
\(68\) −22.5702 −2.73703
\(69\) −3.86039 −0.464736
\(70\) 0 0
\(71\) −1.41358 −0.167761 −0.0838807 0.996476i \(-0.526731\pi\)
−0.0838807 + 0.996476i \(0.526731\pi\)
\(72\) 6.19138 0.729661
\(73\) 11.3234 1.32530 0.662650 0.748929i \(-0.269432\pi\)
0.662650 + 0.748929i \(0.269432\pi\)
\(74\) −4.10501 −0.477198
\(75\) 0 0
\(76\) −23.8486 −2.73562
\(77\) 3.09144 0.352302
\(78\) −14.3866 −1.62897
\(79\) −1.44707 −0.162809 −0.0814043 0.996681i \(-0.525941\pi\)
−0.0814043 + 0.996681i \(0.525941\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.95713 0.326560
\(83\) −11.4756 −1.25961 −0.629806 0.776752i \(-0.716866\pi\)
−0.629806 + 0.776752i \(0.716866\pi\)
\(84\) −4.61976 −0.504057
\(85\) 0 0
\(86\) 14.4292 1.55594
\(87\) −0.679696 −0.0728711
\(88\) 18.3946 1.96087
\(89\) 9.07225 0.961657 0.480828 0.876815i \(-0.340336\pi\)
0.480828 + 0.876815i \(0.340336\pi\)
\(90\) 0 0
\(91\) 5.89903 0.618386
\(92\) 17.1393 1.78689
\(93\) −0.850111 −0.0881524
\(94\) 8.33982 0.860187
\(95\) 0 0
\(96\) −4.95495 −0.505713
\(97\) −6.02928 −0.612181 −0.306091 0.952002i \(-0.599021\pi\)
−0.306091 + 0.952002i \(0.599021\pi\)
\(98\) −15.0161 −1.51686
\(99\) 2.97101 0.298597
\(100\) 0 0
\(101\) −15.3408 −1.52647 −0.763236 0.646120i \(-0.776390\pi\)
−0.763236 + 0.646120i \(0.776390\pi\)
\(102\) 12.9006 1.27734
\(103\) −12.0590 −1.18820 −0.594102 0.804390i \(-0.702493\pi\)
−0.594102 + 0.804390i \(0.702493\pi\)
\(104\) 35.1003 3.44187
\(105\) 0 0
\(106\) 32.1933 3.12689
\(107\) 6.49787 0.628173 0.314086 0.949394i \(-0.398302\pi\)
0.314086 + 0.949394i \(0.398302\pi\)
\(108\) −4.43979 −0.427219
\(109\) −2.31057 −0.221313 −0.110656 0.993859i \(-0.535295\pi\)
−0.110656 + 0.993859i \(0.535295\pi\)
\(110\) 0 0
\(111\) 1.61763 0.153538
\(112\) 7.10907 0.671744
\(113\) −3.87281 −0.364323 −0.182162 0.983269i \(-0.558309\pi\)
−0.182162 + 0.983269i \(0.558309\pi\)
\(114\) 13.6313 1.27669
\(115\) 0 0
\(116\) 3.01771 0.280187
\(117\) 5.66922 0.524120
\(118\) 8.15067 0.750330
\(119\) −5.28968 −0.484904
\(120\) 0 0
\(121\) −2.17312 −0.197556
\(122\) −13.7565 −1.24546
\(123\) −1.16529 −0.105071
\(124\) 3.77431 0.338943
\(125\) 0 0
\(126\) 2.64054 0.235238
\(127\) −11.6938 −1.03765 −0.518827 0.854879i \(-0.673631\pi\)
−0.518827 + 0.854879i \(0.673631\pi\)
\(128\) −12.6765 −1.12045
\(129\) −5.68601 −0.500625
\(130\) 0 0
\(131\) −7.96210 −0.695652 −0.347826 0.937559i \(-0.613080\pi\)
−0.347826 + 0.937559i \(0.613080\pi\)
\(132\) −13.1906 −1.14810
\(133\) −5.58930 −0.484654
\(134\) −2.35785 −0.203688
\(135\) 0 0
\(136\) −31.4746 −2.69892
\(137\) 11.0513 0.944176 0.472088 0.881551i \(-0.343500\pi\)
0.472088 + 0.881551i \(0.343500\pi\)
\(138\) −9.79640 −0.833925
\(139\) −12.2698 −1.04071 −0.520357 0.853949i \(-0.674201\pi\)
−0.520357 + 0.853949i \(0.674201\pi\)
\(140\) 0 0
\(141\) −3.28640 −0.276765
\(142\) −3.58721 −0.301032
\(143\) 16.8433 1.40851
\(144\) 6.83213 0.569344
\(145\) 0 0
\(146\) 28.7350 2.37813
\(147\) 5.91729 0.488050
\(148\) −7.18192 −0.590350
\(149\) −4.62832 −0.379167 −0.189584 0.981865i \(-0.560714\pi\)
−0.189584 + 0.981865i \(0.560714\pi\)
\(150\) 0 0
\(151\) −4.67249 −0.380242 −0.190121 0.981761i \(-0.560888\pi\)
−0.190121 + 0.981761i \(0.560888\pi\)
\(152\) −33.2574 −2.69753
\(153\) −5.08361 −0.410986
\(154\) 7.84506 0.632173
\(155\) 0 0
\(156\) −25.1701 −2.01522
\(157\) −14.9726 −1.19494 −0.597472 0.801890i \(-0.703828\pi\)
−0.597472 + 0.801890i \(0.703828\pi\)
\(158\) −3.67220 −0.292145
\(159\) −12.6861 −1.00608
\(160\) 0 0
\(161\) 4.01687 0.316574
\(162\) 2.53767 0.199378
\(163\) −11.9112 −0.932958 −0.466479 0.884532i \(-0.654478\pi\)
−0.466479 + 0.884532i \(0.654478\pi\)
\(164\) 5.17364 0.403994
\(165\) 0 0
\(166\) −29.1214 −2.26026
\(167\) 10.6081 0.820881 0.410440 0.911887i \(-0.365375\pi\)
0.410440 + 0.911887i \(0.365375\pi\)
\(168\) −6.44235 −0.497038
\(169\) 19.1401 1.47231
\(170\) 0 0
\(171\) −5.37156 −0.410774
\(172\) 25.2447 1.92489
\(173\) 1.25338 0.0952928 0.0476464 0.998864i \(-0.484828\pi\)
0.0476464 + 0.998864i \(0.484828\pi\)
\(174\) −1.72485 −0.130760
\(175\) 0 0
\(176\) 20.2983 1.53004
\(177\) −3.21187 −0.241419
\(178\) 23.0224 1.72560
\(179\) −20.8113 −1.55551 −0.777755 0.628567i \(-0.783642\pi\)
−0.777755 + 0.628567i \(0.783642\pi\)
\(180\) 0 0
\(181\) −6.92706 −0.514884 −0.257442 0.966294i \(-0.582880\pi\)
−0.257442 + 0.966294i \(0.582880\pi\)
\(182\) 14.9698 1.10964
\(183\) 5.42093 0.400726
\(184\) 23.9011 1.76201
\(185\) 0 0
\(186\) −2.15730 −0.158181
\(187\) −15.1034 −1.10447
\(188\) 14.5909 1.06415
\(189\) −1.04054 −0.0756878
\(190\) 0 0
\(191\) 8.16415 0.590737 0.295369 0.955383i \(-0.404558\pi\)
0.295369 + 0.955383i \(0.404558\pi\)
\(192\) 1.09021 0.0786788
\(193\) 13.9629 1.00507 0.502537 0.864556i \(-0.332400\pi\)
0.502537 + 0.864556i \(0.332400\pi\)
\(194\) −15.3004 −1.09850
\(195\) 0 0
\(196\) −26.2715 −1.87653
\(197\) −5.76250 −0.410561 −0.205281 0.978703i \(-0.565811\pi\)
−0.205281 + 0.978703i \(0.565811\pi\)
\(198\) 7.53945 0.535805
\(199\) −26.5748 −1.88384 −0.941919 0.335841i \(-0.890980\pi\)
−0.941919 + 0.335841i \(0.890980\pi\)
\(200\) 0 0
\(201\) 0.929140 0.0655365
\(202\) −38.9301 −2.73911
\(203\) 0.707248 0.0496391
\(204\) 22.5702 1.58023
\(205\) 0 0
\(206\) −30.6017 −2.13212
\(207\) 3.86039 0.268315
\(208\) 38.7329 2.68564
\(209\) −15.9589 −1.10390
\(210\) 0 0
\(211\) −26.4594 −1.82154 −0.910771 0.412912i \(-0.864512\pi\)
−0.910771 + 0.412912i \(0.864512\pi\)
\(212\) 56.3237 3.86833
\(213\) 1.41358 0.0968571
\(214\) 16.4895 1.12720
\(215\) 0 0
\(216\) −6.19138 −0.421270
\(217\) 0.884570 0.0600486
\(218\) −5.86348 −0.397125
\(219\) −11.3234 −0.765163
\(220\) 0 0
\(221\) −28.8201 −1.93865
\(222\) 4.10501 0.275510
\(223\) 27.4456 1.83789 0.918946 0.394383i \(-0.129042\pi\)
0.918946 + 0.394383i \(0.129042\pi\)
\(224\) 5.15581 0.344487
\(225\) 0 0
\(226\) −9.82792 −0.653744
\(227\) −0.130161 −0.00863907 −0.00431954 0.999991i \(-0.501375\pi\)
−0.00431954 + 0.999991i \(0.501375\pi\)
\(228\) 23.8486 1.57941
\(229\) 2.82530 0.186701 0.0933504 0.995633i \(-0.470242\pi\)
0.0933504 + 0.995633i \(0.470242\pi\)
\(230\) 0 0
\(231\) −3.09144 −0.203402
\(232\) 4.20826 0.276286
\(233\) −7.98709 −0.523252 −0.261626 0.965169i \(-0.584259\pi\)
−0.261626 + 0.965169i \(0.584259\pi\)
\(234\) 14.3866 0.940484
\(235\) 0 0
\(236\) 14.2600 0.928247
\(237\) 1.44707 0.0939976
\(238\) −13.4235 −0.870115
\(239\) −19.0619 −1.23301 −0.616506 0.787350i \(-0.711452\pi\)
−0.616506 + 0.787350i \(0.711452\pi\)
\(240\) 0 0
\(241\) 21.1199 1.36045 0.680226 0.733002i \(-0.261882\pi\)
0.680226 + 0.733002i \(0.261882\pi\)
\(242\) −5.51466 −0.354496
\(243\) −1.00000 −0.0641500
\(244\) −24.0678 −1.54078
\(245\) 0 0
\(246\) −2.95713 −0.188540
\(247\) −30.4526 −1.93765
\(248\) 5.26336 0.334224
\(249\) 11.4756 0.727238
\(250\) 0 0
\(251\) 30.2224 1.90762 0.953811 0.300408i \(-0.0971228\pi\)
0.953811 + 0.300408i \(0.0971228\pi\)
\(252\) 4.61976 0.291017
\(253\) 11.4692 0.721065
\(254\) −29.6750 −1.86197
\(255\) 0 0
\(256\) −29.9884 −1.87427
\(257\) −5.10215 −0.318263 −0.159132 0.987257i \(-0.550869\pi\)
−0.159132 + 0.987257i \(0.550869\pi\)
\(258\) −14.4292 −0.898325
\(259\) −1.68320 −0.104589
\(260\) 0 0
\(261\) 0.679696 0.0420721
\(262\) −20.2052 −1.24828
\(263\) −6.41540 −0.395591 −0.197795 0.980243i \(-0.563378\pi\)
−0.197795 + 0.980243i \(0.563378\pi\)
\(264\) −18.3946 −1.13211
\(265\) 0 0
\(266\) −14.1838 −0.869666
\(267\) −9.07225 −0.555213
\(268\) −4.12518 −0.251986
\(269\) −17.4592 −1.06450 −0.532252 0.846586i \(-0.678654\pi\)
−0.532252 + 0.846586i \(0.678654\pi\)
\(270\) 0 0
\(271\) −0.0951857 −0.00578212 −0.00289106 0.999996i \(-0.500920\pi\)
−0.00289106 + 0.999996i \(0.500920\pi\)
\(272\) −34.7319 −2.10593
\(273\) −5.89903 −0.357025
\(274\) 28.0446 1.69424
\(275\) 0 0
\(276\) −17.1393 −1.03166
\(277\) 18.4007 1.10559 0.552796 0.833316i \(-0.313561\pi\)
0.552796 + 0.833316i \(0.313561\pi\)
\(278\) −31.1369 −1.86747
\(279\) 0.850111 0.0508948
\(280\) 0 0
\(281\) 17.9361 1.06998 0.534988 0.844860i \(-0.320316\pi\)
0.534988 + 0.844860i \(0.320316\pi\)
\(282\) −8.33982 −0.496629
\(283\) 22.2399 1.32203 0.661014 0.750374i \(-0.270126\pi\)
0.661014 + 0.750374i \(0.270126\pi\)
\(284\) −6.27601 −0.372413
\(285\) 0 0
\(286\) 42.7428 2.52743
\(287\) 1.21253 0.0715732
\(288\) 4.95495 0.291974
\(289\) 8.84312 0.520184
\(290\) 0 0
\(291\) 6.02928 0.353443
\(292\) 50.2734 2.94203
\(293\) 18.7316 1.09431 0.547155 0.837031i \(-0.315711\pi\)
0.547155 + 0.837031i \(0.315711\pi\)
\(294\) 15.0161 0.875759
\(295\) 0 0
\(296\) −10.0154 −0.582130
\(297\) −2.97101 −0.172395
\(298\) −11.7452 −0.680380
\(299\) 21.8854 1.26566
\(300\) 0 0
\(301\) 5.91650 0.341021
\(302\) −11.8573 −0.682309
\(303\) 15.3408 0.881309
\(304\) −36.6992 −2.10484
\(305\) 0 0
\(306\) −12.9006 −0.737475
\(307\) 7.03850 0.401708 0.200854 0.979621i \(-0.435628\pi\)
0.200854 + 0.979621i \(0.435628\pi\)
\(308\) 13.7253 0.782073
\(309\) 12.0590 0.686010
\(310\) 0 0
\(311\) −29.2790 −1.66026 −0.830130 0.557570i \(-0.811734\pi\)
−0.830130 + 0.557570i \(0.811734\pi\)
\(312\) −35.1003 −1.98716
\(313\) 15.6354 0.883764 0.441882 0.897073i \(-0.354311\pi\)
0.441882 + 0.897073i \(0.354311\pi\)
\(314\) −37.9956 −2.14422
\(315\) 0 0
\(316\) −6.42470 −0.361418
\(317\) 6.24144 0.350554 0.175277 0.984519i \(-0.443918\pi\)
0.175277 + 0.984519i \(0.443918\pi\)
\(318\) −32.1933 −1.80531
\(319\) 2.01938 0.113064
\(320\) 0 0
\(321\) −6.49787 −0.362676
\(322\) 10.1935 0.568062
\(323\) 27.3069 1.51940
\(324\) 4.43979 0.246655
\(325\) 0 0
\(326\) −30.2268 −1.67411
\(327\) 2.31057 0.127775
\(328\) 7.21476 0.398369
\(329\) 3.41962 0.188530
\(330\) 0 0
\(331\) 21.4575 1.17941 0.589705 0.807619i \(-0.299244\pi\)
0.589705 + 0.807619i \(0.299244\pi\)
\(332\) −50.9493 −2.79621
\(333\) −1.61763 −0.0886455
\(334\) 26.9199 1.47299
\(335\) 0 0
\(336\) −7.10907 −0.387832
\(337\) 34.7511 1.89301 0.946507 0.322683i \(-0.104585\pi\)
0.946507 + 0.322683i \(0.104585\pi\)
\(338\) 48.5712 2.64193
\(339\) 3.87281 0.210342
\(340\) 0 0
\(341\) 2.52569 0.136774
\(342\) −13.6313 −0.737094
\(343\) −13.4409 −0.725740
\(344\) 35.2043 1.89809
\(345\) 0 0
\(346\) 3.18067 0.170994
\(347\) 28.7241 1.54199 0.770995 0.636841i \(-0.219759\pi\)
0.770995 + 0.636841i \(0.219759\pi\)
\(348\) −3.01771 −0.161766
\(349\) −12.2834 −0.657515 −0.328758 0.944414i \(-0.606630\pi\)
−0.328758 + 0.944414i \(0.606630\pi\)
\(350\) 0 0
\(351\) −5.66922 −0.302601
\(352\) 14.7212 0.784643
\(353\) −26.9779 −1.43589 −0.717945 0.696100i \(-0.754917\pi\)
−0.717945 + 0.696100i \(0.754917\pi\)
\(354\) −8.15067 −0.433203
\(355\) 0 0
\(356\) 40.2789 2.13478
\(357\) 5.28968 0.279960
\(358\) −52.8123 −2.79122
\(359\) 16.4243 0.866839 0.433419 0.901192i \(-0.357307\pi\)
0.433419 + 0.901192i \(0.357307\pi\)
\(360\) 0 0
\(361\) 9.85366 0.518614
\(362\) −17.5786 −0.923912
\(363\) 2.17312 0.114059
\(364\) 26.1904 1.37275
\(365\) 0 0
\(366\) 13.7565 0.719066
\(367\) −29.8953 −1.56052 −0.780262 0.625453i \(-0.784914\pi\)
−0.780262 + 0.625453i \(0.784914\pi\)
\(368\) 26.3747 1.37487
\(369\) 1.16529 0.0606626
\(370\) 0 0
\(371\) 13.2004 0.685329
\(372\) −3.77431 −0.195689
\(373\) −24.8307 −1.28568 −0.642842 0.765999i \(-0.722245\pi\)
−0.642842 + 0.765999i \(0.722245\pi\)
\(374\) −38.3276 −1.98187
\(375\) 0 0
\(376\) 20.3474 1.04934
\(377\) 3.85335 0.198458
\(378\) −2.64054 −0.135815
\(379\) 19.3966 0.996338 0.498169 0.867080i \(-0.334006\pi\)
0.498169 + 0.867080i \(0.334006\pi\)
\(380\) 0 0
\(381\) 11.6938 0.599090
\(382\) 20.7179 1.06002
\(383\) −11.2508 −0.574891 −0.287446 0.957797i \(-0.592806\pi\)
−0.287446 + 0.957797i \(0.592806\pi\)
\(384\) 12.6765 0.646895
\(385\) 0 0
\(386\) 35.4334 1.80351
\(387\) 5.68601 0.289036
\(388\) −26.7687 −1.35898
\(389\) 14.9017 0.755549 0.377774 0.925898i \(-0.376690\pi\)
0.377774 + 0.925898i \(0.376690\pi\)
\(390\) 0 0
\(391\) −19.6247 −0.992464
\(392\) −36.6362 −1.85041
\(393\) 7.96210 0.401635
\(394\) −14.6233 −0.736713
\(395\) 0 0
\(396\) 13.1906 0.662854
\(397\) 24.0966 1.20937 0.604687 0.796463i \(-0.293298\pi\)
0.604687 + 0.796463i \(0.293298\pi\)
\(398\) −67.4382 −3.38037
\(399\) 5.58930 0.279815
\(400\) 0 0
\(401\) 13.4580 0.672059 0.336030 0.941851i \(-0.390916\pi\)
0.336030 + 0.941851i \(0.390916\pi\)
\(402\) 2.35785 0.117599
\(403\) 4.81947 0.240075
\(404\) −68.1101 −3.38860
\(405\) 0 0
\(406\) 1.79476 0.0890727
\(407\) −4.80598 −0.238224
\(408\) 31.4746 1.55822
\(409\) −35.4737 −1.75406 −0.877030 0.480435i \(-0.840479\pi\)
−0.877030 + 0.480435i \(0.840479\pi\)
\(410\) 0 0
\(411\) −11.0513 −0.545120
\(412\) −53.5392 −2.63769
\(413\) 3.34206 0.164452
\(414\) 9.79640 0.481467
\(415\) 0 0
\(416\) 28.0907 1.37726
\(417\) 12.2698 0.600857
\(418\) −40.4986 −1.98085
\(419\) 15.8120 0.772466 0.386233 0.922401i \(-0.373776\pi\)
0.386233 + 0.922401i \(0.373776\pi\)
\(420\) 0 0
\(421\) 11.9813 0.583935 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(422\) −67.1454 −3.26859
\(423\) 3.28640 0.159790
\(424\) 78.5447 3.81447
\(425\) 0 0
\(426\) 3.58721 0.173801
\(427\) −5.64067 −0.272971
\(428\) 28.8492 1.39448
\(429\) −16.8433 −0.813202
\(430\) 0 0
\(431\) 14.6428 0.705317 0.352659 0.935752i \(-0.385278\pi\)
0.352659 + 0.935752i \(0.385278\pi\)
\(432\) −6.83213 −0.328711
\(433\) −4.27293 −0.205344 −0.102672 0.994715i \(-0.532739\pi\)
−0.102672 + 0.994715i \(0.532739\pi\)
\(434\) 2.24475 0.107751
\(435\) 0 0
\(436\) −10.2584 −0.491291
\(437\) −20.7363 −0.991952
\(438\) −28.7350 −1.37301
\(439\) −20.3073 −0.969214 −0.484607 0.874732i \(-0.661037\pi\)
−0.484607 + 0.874732i \(0.661037\pi\)
\(440\) 0 0
\(441\) −5.91729 −0.281776
\(442\) −73.1361 −3.47873
\(443\) −19.0543 −0.905299 −0.452649 0.891689i \(-0.649521\pi\)
−0.452649 + 0.891689i \(0.649521\pi\)
\(444\) 7.18192 0.340839
\(445\) 0 0
\(446\) 69.6479 3.29793
\(447\) 4.62832 0.218912
\(448\) −1.13440 −0.0535952
\(449\) 29.4793 1.39122 0.695608 0.718421i \(-0.255135\pi\)
0.695608 + 0.718421i \(0.255135\pi\)
\(450\) 0 0
\(451\) 3.46209 0.163023
\(452\) −17.1944 −0.808759
\(453\) 4.67249 0.219533
\(454\) −0.330305 −0.0155020
\(455\) 0 0
\(456\) 33.2574 1.55742
\(457\) −28.3015 −1.32389 −0.661945 0.749553i \(-0.730269\pi\)
−0.661945 + 0.749553i \(0.730269\pi\)
\(458\) 7.16968 0.335017
\(459\) 5.08361 0.237283
\(460\) 0 0
\(461\) −16.5575 −0.771157 −0.385579 0.922675i \(-0.625998\pi\)
−0.385579 + 0.922675i \(0.625998\pi\)
\(462\) −7.84506 −0.364985
\(463\) −9.20933 −0.427994 −0.213997 0.976834i \(-0.568648\pi\)
−0.213997 + 0.976834i \(0.568648\pi\)
\(464\) 4.64377 0.215582
\(465\) 0 0
\(466\) −20.2686 −0.938926
\(467\) −10.6123 −0.491078 −0.245539 0.969387i \(-0.578965\pi\)
−0.245539 + 0.969387i \(0.578965\pi\)
\(468\) 25.1701 1.16349
\(469\) −0.966803 −0.0446428
\(470\) 0 0
\(471\) 14.9726 0.689902
\(472\) 19.8859 0.915323
\(473\) 16.8932 0.776749
\(474\) 3.67220 0.168670
\(475\) 0 0
\(476\) −23.4850 −1.07644
\(477\) 12.6861 0.580858
\(478\) −48.3729 −2.21253
\(479\) 33.6061 1.53550 0.767751 0.640749i \(-0.221376\pi\)
0.767751 + 0.640749i \(0.221376\pi\)
\(480\) 0 0
\(481\) −9.17069 −0.418147
\(482\) 53.5954 2.44120
\(483\) −4.01687 −0.182774
\(484\) −9.64818 −0.438554
\(485\) 0 0
\(486\) −2.53767 −0.115111
\(487\) −29.2486 −1.32538 −0.662690 0.748894i \(-0.730585\pi\)
−0.662690 + 0.748894i \(0.730585\pi\)
\(488\) −33.5630 −1.51933
\(489\) 11.9112 0.538644
\(490\) 0 0
\(491\) 13.9549 0.629778 0.314889 0.949129i \(-0.398033\pi\)
0.314889 + 0.949129i \(0.398033\pi\)
\(492\) −5.17364 −0.233246
\(493\) −3.45531 −0.155619
\(494\) −77.2787 −3.47693
\(495\) 0 0
\(496\) 5.80807 0.260790
\(497\) −1.47088 −0.0659782
\(498\) 29.1214 1.30496
\(499\) −35.7533 −1.60054 −0.800268 0.599642i \(-0.795310\pi\)
−0.800268 + 0.599642i \(0.795310\pi\)
\(500\) 0 0
\(501\) −10.6081 −0.473936
\(502\) 76.6946 3.42305
\(503\) 11.6443 0.519193 0.259596 0.965717i \(-0.416410\pi\)
0.259596 + 0.965717i \(0.416410\pi\)
\(504\) 6.44235 0.286965
\(505\) 0 0
\(506\) 29.1052 1.29388
\(507\) −19.1401 −0.850040
\(508\) −51.9178 −2.30348
\(509\) −8.53362 −0.378246 −0.189123 0.981953i \(-0.560564\pi\)
−0.189123 + 0.981953i \(0.560564\pi\)
\(510\) 0 0
\(511\) 11.7824 0.521222
\(512\) −50.7478 −2.24276
\(513\) 5.37156 0.237160
\(514\) −12.9476 −0.571094
\(515\) 0 0
\(516\) −25.2447 −1.11133
\(517\) 9.76393 0.429417
\(518\) −4.27141 −0.187675
\(519\) −1.25338 −0.0550173
\(520\) 0 0
\(521\) 1.26530 0.0554336 0.0277168 0.999616i \(-0.491176\pi\)
0.0277168 + 0.999616i \(0.491176\pi\)
\(522\) 1.72485 0.0754945
\(523\) −27.2204 −1.19027 −0.595133 0.803627i \(-0.702901\pi\)
−0.595133 + 0.803627i \(0.702901\pi\)
\(524\) −35.3500 −1.54427
\(525\) 0 0
\(526\) −16.2802 −0.709850
\(527\) −4.32163 −0.188253
\(528\) −20.2983 −0.883370
\(529\) −8.09742 −0.352062
\(530\) 0 0
\(531\) 3.21187 0.139383
\(532\) −24.8153 −1.07588
\(533\) 6.60629 0.286150
\(534\) −23.0224 −0.996277
\(535\) 0 0
\(536\) −5.75266 −0.248477
\(537\) 20.8113 0.898074
\(538\) −44.3057 −1.91015
\(539\) −17.5803 −0.757237
\(540\) 0 0
\(541\) 30.3830 1.30627 0.653134 0.757242i \(-0.273454\pi\)
0.653134 + 0.757242i \(0.273454\pi\)
\(542\) −0.241550 −0.0103755
\(543\) 6.92706 0.297269
\(544\) −25.1891 −1.07997
\(545\) 0 0
\(546\) −14.9698 −0.640648
\(547\) −11.3768 −0.486439 −0.243219 0.969971i \(-0.578203\pi\)
−0.243219 + 0.969971i \(0.578203\pi\)
\(548\) 49.0654 2.09597
\(549\) −5.42093 −0.231360
\(550\) 0 0
\(551\) −3.65103 −0.155539
\(552\) −23.9011 −1.01730
\(553\) −1.50573 −0.0640303
\(554\) 46.6950 1.98388
\(555\) 0 0
\(556\) −54.4755 −2.31028
\(557\) 45.7532 1.93862 0.969312 0.245833i \(-0.0790616\pi\)
0.969312 + 0.245833i \(0.0790616\pi\)
\(558\) 2.15730 0.0913259
\(559\) 32.2353 1.36341
\(560\) 0 0
\(561\) 15.1034 0.637668
\(562\) 45.5158 1.91997
\(563\) −8.94289 −0.376898 −0.188449 0.982083i \(-0.560346\pi\)
−0.188449 + 0.982083i \(0.560346\pi\)
\(564\) −14.5909 −0.614389
\(565\) 0 0
\(566\) 56.4377 2.37225
\(567\) 1.04054 0.0436984
\(568\) −8.75203 −0.367227
\(569\) −2.28908 −0.0959632 −0.0479816 0.998848i \(-0.515279\pi\)
−0.0479816 + 0.998848i \(0.515279\pi\)
\(570\) 0 0
\(571\) −30.7595 −1.28725 −0.643623 0.765342i \(-0.722570\pi\)
−0.643623 + 0.765342i \(0.722570\pi\)
\(572\) 74.7806 3.12674
\(573\) −8.16415 −0.341062
\(574\) 3.07700 0.128431
\(575\) 0 0
\(576\) −1.09021 −0.0454252
\(577\) −6.59227 −0.274440 −0.137220 0.990541i \(-0.543817\pi\)
−0.137220 + 0.990541i \(0.543817\pi\)
\(578\) 22.4410 0.933421
\(579\) −13.9629 −0.580280
\(580\) 0 0
\(581\) −11.9408 −0.495387
\(582\) 15.3004 0.634220
\(583\) 37.6906 1.56098
\(584\) 70.1073 2.90106
\(585\) 0 0
\(586\) 47.5346 1.96364
\(587\) −5.53771 −0.228566 −0.114283 0.993448i \(-0.536457\pi\)
−0.114283 + 0.993448i \(0.536457\pi\)
\(588\) 26.2715 1.08342
\(589\) −4.56642 −0.188156
\(590\) 0 0
\(591\) 5.76250 0.237038
\(592\) −11.0518 −0.454228
\(593\) −1.88122 −0.0772524 −0.0386262 0.999254i \(-0.512298\pi\)
−0.0386262 + 0.999254i \(0.512298\pi\)
\(594\) −7.53945 −0.309347
\(595\) 0 0
\(596\) −20.5488 −0.841710
\(597\) 26.5748 1.08763
\(598\) 55.5380 2.27112
\(599\) 17.8272 0.728400 0.364200 0.931321i \(-0.381342\pi\)
0.364200 + 0.931321i \(0.381342\pi\)
\(600\) 0 0
\(601\) 33.0994 1.35015 0.675077 0.737747i \(-0.264110\pi\)
0.675077 + 0.737747i \(0.264110\pi\)
\(602\) 15.0141 0.611931
\(603\) −0.929140 −0.0378375
\(604\) −20.7449 −0.844097
\(605\) 0 0
\(606\) 38.9301 1.58143
\(607\) −23.4603 −0.952226 −0.476113 0.879384i \(-0.657955\pi\)
−0.476113 + 0.879384i \(0.657955\pi\)
\(608\) −26.6158 −1.07941
\(609\) −0.707248 −0.0286591
\(610\) 0 0
\(611\) 18.6314 0.753744
\(612\) −22.5702 −0.912345
\(613\) 47.2281 1.90752 0.953762 0.300564i \(-0.0971750\pi\)
0.953762 + 0.300564i \(0.0971750\pi\)
\(614\) 17.8614 0.720828
\(615\) 0 0
\(616\) 19.1403 0.771184
\(617\) 16.2698 0.654999 0.327499 0.944851i \(-0.393794\pi\)
0.327499 + 0.944851i \(0.393794\pi\)
\(618\) 30.6017 1.23098
\(619\) 35.4599 1.42525 0.712627 0.701543i \(-0.247505\pi\)
0.712627 + 0.701543i \(0.247505\pi\)
\(620\) 0 0
\(621\) −3.86039 −0.154912
\(622\) −74.3005 −2.97918
\(623\) 9.44000 0.378206
\(624\) −38.7329 −1.55056
\(625\) 0 0
\(626\) 39.6775 1.58583
\(627\) 15.9589 0.637339
\(628\) −66.4752 −2.65265
\(629\) 8.22339 0.327888
\(630\) 0 0
\(631\) −29.2364 −1.16388 −0.581941 0.813231i \(-0.697706\pi\)
−0.581941 + 0.813231i \(0.697706\pi\)
\(632\) −8.95939 −0.356385
\(633\) 26.4594 1.05167
\(634\) 15.8387 0.629037
\(635\) 0 0
\(636\) −56.3237 −2.23338
\(637\) −33.5464 −1.32916
\(638\) 5.12453 0.202882
\(639\) −1.41358 −0.0559205
\(640\) 0 0
\(641\) −15.7160 −0.620745 −0.310373 0.950615i \(-0.600454\pi\)
−0.310373 + 0.950615i \(0.600454\pi\)
\(642\) −16.4895 −0.650788
\(643\) 8.08055 0.318666 0.159333 0.987225i \(-0.449066\pi\)
0.159333 + 0.987225i \(0.449066\pi\)
\(644\) 17.8340 0.702760
\(645\) 0 0
\(646\) 69.2961 2.72642
\(647\) −11.0193 −0.433214 −0.216607 0.976259i \(-0.569499\pi\)
−0.216607 + 0.976259i \(0.569499\pi\)
\(648\) 6.19138 0.243220
\(649\) 9.54248 0.374575
\(650\) 0 0
\(651\) −0.884570 −0.0346691
\(652\) −52.8832 −2.07107
\(653\) 31.0861 1.21649 0.608246 0.793748i \(-0.291873\pi\)
0.608246 + 0.793748i \(0.291873\pi\)
\(654\) 5.86348 0.229280
\(655\) 0 0
\(656\) 7.96142 0.310841
\(657\) 11.3234 0.441767
\(658\) 8.67788 0.338299
\(659\) 32.4252 1.26311 0.631553 0.775333i \(-0.282418\pi\)
0.631553 + 0.775333i \(0.282418\pi\)
\(660\) 0 0
\(661\) 15.6076 0.607067 0.303534 0.952821i \(-0.401834\pi\)
0.303534 + 0.952821i \(0.401834\pi\)
\(662\) 54.4521 2.11634
\(663\) 28.8201 1.11928
\(664\) −71.0499 −2.75727
\(665\) 0 0
\(666\) −4.10501 −0.159066
\(667\) 2.62389 0.101597
\(668\) 47.0978 1.82227
\(669\) −27.4456 −1.06111
\(670\) 0 0
\(671\) −16.1056 −0.621750
\(672\) −5.15581 −0.198890
\(673\) −31.0271 −1.19601 −0.598003 0.801494i \(-0.704039\pi\)
−0.598003 + 0.801494i \(0.704039\pi\)
\(674\) 88.1870 3.39684
\(675\) 0 0
\(676\) 84.9778 3.26838
\(677\) 23.0893 0.887394 0.443697 0.896177i \(-0.353667\pi\)
0.443697 + 0.896177i \(0.353667\pi\)
\(678\) 9.82792 0.377439
\(679\) −6.27369 −0.240762
\(680\) 0 0
\(681\) 0.130161 0.00498777
\(682\) 6.40936 0.245427
\(683\) 41.3541 1.58237 0.791185 0.611577i \(-0.209464\pi\)
0.791185 + 0.611577i \(0.209464\pi\)
\(684\) −23.8486 −0.911873
\(685\) 0 0
\(686\) −34.1086 −1.30227
\(687\) −2.82530 −0.107792
\(688\) 38.8476 1.48105
\(689\) 71.9205 2.73995
\(690\) 0 0
\(691\) 4.46909 0.170012 0.0850061 0.996380i \(-0.472909\pi\)
0.0850061 + 0.996380i \(0.472909\pi\)
\(692\) 5.56475 0.211540
\(693\) 3.09144 0.117434
\(694\) 72.8924 2.76696
\(695\) 0 0
\(696\) −4.20826 −0.159514
\(697\) −5.92389 −0.224383
\(698\) −31.1713 −1.17985
\(699\) 7.98709 0.302099
\(700\) 0 0
\(701\) −25.2265 −0.952791 −0.476396 0.879231i \(-0.658057\pi\)
−0.476396 + 0.879231i \(0.658057\pi\)
\(702\) −14.3866 −0.542988
\(703\) 8.68919 0.327719
\(704\) −3.23901 −0.122075
\(705\) 0 0
\(706\) −68.4611 −2.57657
\(707\) −15.9627 −0.600339
\(708\) −14.2600 −0.535924
\(709\) 1.55990 0.0585834 0.0292917 0.999571i \(-0.490675\pi\)
0.0292917 + 0.999571i \(0.490675\pi\)
\(710\) 0 0
\(711\) −1.44707 −0.0542695
\(712\) 56.1698 2.10505
\(713\) 3.28176 0.122903
\(714\) 13.4235 0.502361
\(715\) 0 0
\(716\) −92.3978 −3.45307
\(717\) 19.0619 0.711880
\(718\) 41.6794 1.55546
\(719\) 28.9403 1.07929 0.539645 0.841893i \(-0.318558\pi\)
0.539645 + 0.841893i \(0.318558\pi\)
\(720\) 0 0
\(721\) −12.5478 −0.467304
\(722\) 25.0054 0.930604
\(723\) −21.1199 −0.785458
\(724\) −30.7547 −1.14299
\(725\) 0 0
\(726\) 5.51466 0.204668
\(727\) −36.9595 −1.37075 −0.685375 0.728190i \(-0.740362\pi\)
−0.685375 + 0.728190i \(0.740362\pi\)
\(728\) 36.5231 1.35364
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −28.9055 −1.06911
\(732\) 24.0678 0.889570
\(733\) −4.16161 −0.153713 −0.0768564 0.997042i \(-0.524488\pi\)
−0.0768564 + 0.997042i \(0.524488\pi\)
\(734\) −75.8646 −2.80021
\(735\) 0 0
\(736\) 19.1280 0.705069
\(737\) −2.76048 −0.101684
\(738\) 2.95713 0.108853
\(739\) 18.9577 0.697370 0.348685 0.937240i \(-0.386628\pi\)
0.348685 + 0.937240i \(0.386628\pi\)
\(740\) 0 0
\(741\) 30.4526 1.11870
\(742\) 33.4982 1.22976
\(743\) −11.7060 −0.429452 −0.214726 0.976674i \(-0.568886\pi\)
−0.214726 + 0.976674i \(0.568886\pi\)
\(744\) −5.26336 −0.192964
\(745\) 0 0
\(746\) −63.0122 −2.30704
\(747\) −11.4756 −0.419871
\(748\) −67.0561 −2.45181
\(749\) 6.76126 0.247051
\(750\) 0 0
\(751\) −4.95672 −0.180873 −0.0904367 0.995902i \(-0.528826\pi\)
−0.0904367 + 0.995902i \(0.528826\pi\)
\(752\) 22.4531 0.818782
\(753\) −30.2224 −1.10137
\(754\) 9.77854 0.356113
\(755\) 0 0
\(756\) −4.61976 −0.168019
\(757\) 18.6020 0.676101 0.338051 0.941128i \(-0.390232\pi\)
0.338051 + 0.941128i \(0.390232\pi\)
\(758\) 49.2223 1.78784
\(759\) −11.4692 −0.416307
\(760\) 0 0
\(761\) −11.3585 −0.411747 −0.205873 0.978579i \(-0.566003\pi\)
−0.205873 + 0.978579i \(0.566003\pi\)
\(762\) 29.6750 1.07501
\(763\) −2.40423 −0.0870391
\(764\) 36.2471 1.31137
\(765\) 0 0
\(766\) −28.5510 −1.03159
\(767\) 18.2088 0.657481
\(768\) 29.9884 1.08211
\(769\) 7.27477 0.262335 0.131168 0.991360i \(-0.458127\pi\)
0.131168 + 0.991360i \(0.458127\pi\)
\(770\) 0 0
\(771\) 5.10215 0.183749
\(772\) 61.9925 2.23116
\(773\) 9.32288 0.335321 0.167660 0.985845i \(-0.446379\pi\)
0.167660 + 0.985845i \(0.446379\pi\)
\(774\) 14.4292 0.518648
\(775\) 0 0
\(776\) −37.3296 −1.34005
\(777\) 1.68320 0.0603844
\(778\) 37.8158 1.35576
\(779\) −6.25943 −0.224267
\(780\) 0 0
\(781\) −4.19977 −0.150279
\(782\) −49.8011 −1.78088
\(783\) −0.679696 −0.0242904
\(784\) −40.4277 −1.44385
\(785\) 0 0
\(786\) 20.2052 0.720696
\(787\) −14.8658 −0.529907 −0.264953 0.964261i \(-0.585357\pi\)
−0.264953 + 0.964261i \(0.585357\pi\)
\(788\) −25.5843 −0.911402
\(789\) 6.41540 0.228394
\(790\) 0 0
\(791\) −4.02979 −0.143283
\(792\) 18.3946 0.653625
\(793\) −30.7324 −1.09134
\(794\) 61.1493 2.17011
\(795\) 0 0
\(796\) −117.986 −4.18192
\(797\) 9.57546 0.339180 0.169590 0.985515i \(-0.445756\pi\)
0.169590 + 0.985515i \(0.445756\pi\)
\(798\) 14.1838 0.502102
\(799\) −16.7068 −0.591044
\(800\) 0 0
\(801\) 9.07225 0.320552
\(802\) 34.1520 1.20595
\(803\) 33.6418 1.18719
\(804\) 4.12518 0.145484
\(805\) 0 0
\(806\) 12.2302 0.430792
\(807\) 17.4592 0.614592
\(808\) −94.9810 −3.34142
\(809\) 41.1436 1.44653 0.723266 0.690569i \(-0.242640\pi\)
0.723266 + 0.690569i \(0.242640\pi\)
\(810\) 0 0
\(811\) 43.4398 1.52538 0.762688 0.646766i \(-0.223879\pi\)
0.762688 + 0.646766i \(0.223879\pi\)
\(812\) 3.14003 0.110193
\(813\) 0.0951857 0.00333831
\(814\) −12.1960 −0.427470
\(815\) 0 0
\(816\) 34.7319 1.21586
\(817\) −30.5428 −1.06856
\(818\) −90.0206 −3.14750
\(819\) 5.89903 0.206129
\(820\) 0 0
\(821\) 17.4617 0.609416 0.304708 0.952446i \(-0.401441\pi\)
0.304708 + 0.952446i \(0.401441\pi\)
\(822\) −28.0446 −0.978167
\(823\) 2.22456 0.0775432 0.0387716 0.999248i \(-0.487656\pi\)
0.0387716 + 0.999248i \(0.487656\pi\)
\(824\) −74.6616 −2.60096
\(825\) 0 0
\(826\) 8.48106 0.295094
\(827\) −31.4550 −1.09380 −0.546898 0.837199i \(-0.684192\pi\)
−0.546898 + 0.837199i \(0.684192\pi\)
\(828\) 17.1393 0.595632
\(829\) 53.2947 1.85100 0.925500 0.378748i \(-0.123645\pi\)
0.925500 + 0.378748i \(0.123645\pi\)
\(830\) 0 0
\(831\) −18.4007 −0.638314
\(832\) −6.18061 −0.214274
\(833\) 30.0812 1.04225
\(834\) 31.1369 1.07818
\(835\) 0 0
\(836\) −70.8543 −2.45055
\(837\) −0.850111 −0.0293841
\(838\) 40.1256 1.38612
\(839\) −8.37331 −0.289079 −0.144539 0.989499i \(-0.546170\pi\)
−0.144539 + 0.989499i \(0.546170\pi\)
\(840\) 0 0
\(841\) −28.5380 −0.984069
\(842\) 30.4047 1.04782
\(843\) −17.9361 −0.617750
\(844\) −117.474 −4.04363
\(845\) 0 0
\(846\) 8.33982 0.286729
\(847\) −2.26121 −0.0776960
\(848\) 86.6733 2.97637
\(849\) −22.2399 −0.763273
\(850\) 0 0
\(851\) −6.24467 −0.214064
\(852\) 6.27601 0.215013
\(853\) 1.22370 0.0418989 0.0209494 0.999781i \(-0.493331\pi\)
0.0209494 + 0.999781i \(0.493331\pi\)
\(854\) −14.3142 −0.489821
\(855\) 0 0
\(856\) 40.2308 1.37506
\(857\) 14.3684 0.490816 0.245408 0.969420i \(-0.421078\pi\)
0.245408 + 0.969420i \(0.421078\pi\)
\(858\) −42.7428 −1.45921
\(859\) −10.3090 −0.351738 −0.175869 0.984414i \(-0.556274\pi\)
−0.175869 + 0.984414i \(0.556274\pi\)
\(860\) 0 0
\(861\) −1.21253 −0.0413228
\(862\) 37.1586 1.26563
\(863\) −30.2724 −1.03048 −0.515242 0.857045i \(-0.672298\pi\)
−0.515242 + 0.857045i \(0.672298\pi\)
\(864\) −4.95495 −0.168571
\(865\) 0 0
\(866\) −10.8433 −0.368470
\(867\) −8.84312 −0.300328
\(868\) 3.92730 0.133301
\(869\) −4.29927 −0.145843
\(870\) 0 0
\(871\) −5.26750 −0.178482
\(872\) −14.3056 −0.484450
\(873\) −6.02928 −0.204060
\(874\) −52.6220 −1.77996
\(875\) 0 0
\(876\) −50.2734 −1.69858
\(877\) −30.6345 −1.03445 −0.517227 0.855848i \(-0.673036\pi\)
−0.517227 + 0.855848i \(0.673036\pi\)
\(878\) −51.5333 −1.73916
\(879\) −18.7316 −0.631800
\(880\) 0 0
\(881\) 15.2211 0.512813 0.256406 0.966569i \(-0.417461\pi\)
0.256406 + 0.966569i \(0.417461\pi\)
\(882\) −15.0161 −0.505620
\(883\) −40.1985 −1.35279 −0.676393 0.736541i \(-0.736458\pi\)
−0.676393 + 0.736541i \(0.736458\pi\)
\(884\) −127.955 −4.30360
\(885\) 0 0
\(886\) −48.3537 −1.62447
\(887\) 31.8432 1.06919 0.534594 0.845109i \(-0.320464\pi\)
0.534594 + 0.845109i \(0.320464\pi\)
\(888\) 10.0154 0.336093
\(889\) −12.1678 −0.408094
\(890\) 0 0
\(891\) 2.97101 0.0995325
\(892\) 121.853 4.07993
\(893\) −17.6531 −0.590739
\(894\) 11.7452 0.392817
\(895\) 0 0
\(896\) −13.1903 −0.440658
\(897\) −21.8854 −0.730732
\(898\) 74.8089 2.49641
\(899\) 0.577817 0.0192713
\(900\) 0 0
\(901\) −64.4914 −2.14852
\(902\) 8.78565 0.292530
\(903\) −5.91650 −0.196889
\(904\) −23.9780 −0.797498
\(905\) 0 0
\(906\) 11.8573 0.393931
\(907\) 28.8507 0.957970 0.478985 0.877823i \(-0.341005\pi\)
0.478985 + 0.877823i \(0.341005\pi\)
\(908\) −0.577886 −0.0191778
\(909\) −15.3408 −0.508824
\(910\) 0 0
\(911\) 48.3760 1.60277 0.801385 0.598149i \(-0.204097\pi\)
0.801385 + 0.598149i \(0.204097\pi\)
\(912\) 36.6992 1.21523
\(913\) −34.0941 −1.12835
\(914\) −71.8200 −2.37560
\(915\) 0 0
\(916\) 12.5437 0.414456
\(917\) −8.28485 −0.273590
\(918\) 12.9006 0.425782
\(919\) −8.38022 −0.276438 −0.138219 0.990402i \(-0.544138\pi\)
−0.138219 + 0.990402i \(0.544138\pi\)
\(920\) 0 0
\(921\) −7.03850 −0.231926
\(922\) −42.0174 −1.38377
\(923\) −8.01392 −0.263781
\(924\) −13.7253 −0.451530
\(925\) 0 0
\(926\) −23.3703 −0.767995
\(927\) −12.0590 −0.396068
\(928\) 3.36786 0.110556
\(929\) 14.3126 0.469580 0.234790 0.972046i \(-0.424560\pi\)
0.234790 + 0.972046i \(0.424560\pi\)
\(930\) 0 0
\(931\) 31.7851 1.04171
\(932\) −35.4610 −1.16156
\(933\) 29.2790 0.958552
\(934\) −26.9305 −0.881194
\(935\) 0 0
\(936\) 35.1003 1.14729
\(937\) 21.9244 0.716240 0.358120 0.933675i \(-0.383418\pi\)
0.358120 + 0.933675i \(0.383418\pi\)
\(938\) −2.45343 −0.0801074
\(939\) −15.6354 −0.510241
\(940\) 0 0
\(941\) 22.3934 0.730005 0.365003 0.931007i \(-0.381068\pi\)
0.365003 + 0.931007i \(0.381068\pi\)
\(942\) 37.9956 1.23796
\(943\) 4.49847 0.146490
\(944\) 21.9439 0.714213
\(945\) 0 0
\(946\) 42.8694 1.39380
\(947\) −43.5638 −1.41563 −0.707816 0.706397i \(-0.750319\pi\)
−0.707816 + 0.706397i \(0.750319\pi\)
\(948\) 6.42470 0.208665
\(949\) 64.1947 2.08385
\(950\) 0 0
\(951\) −6.24144 −0.202393
\(952\) −32.7504 −1.06145
\(953\) −2.37169 −0.0768266 −0.0384133 0.999262i \(-0.512230\pi\)
−0.0384133 + 0.999262i \(0.512230\pi\)
\(954\) 32.1933 1.04230
\(955\) 0 0
\(956\) −84.6308 −2.73716
\(957\) −2.01938 −0.0652774
\(958\) 85.2813 2.75531
\(959\) 11.4993 0.371331
\(960\) 0 0
\(961\) −30.2773 −0.976687
\(962\) −23.2722 −0.750326
\(963\) 6.49787 0.209391
\(964\) 93.7678 3.02006
\(965\) 0 0
\(966\) −10.1935 −0.327971
\(967\) 19.7857 0.636266 0.318133 0.948046i \(-0.396944\pi\)
0.318133 + 0.948046i \(0.396944\pi\)
\(968\) −13.4546 −0.432447
\(969\) −27.3069 −0.877225
\(970\) 0 0
\(971\) 49.3838 1.58480 0.792401 0.610000i \(-0.208831\pi\)
0.792401 + 0.610000i \(0.208831\pi\)
\(972\) −4.43979 −0.142406
\(973\) −12.7672 −0.409298
\(974\) −74.2234 −2.37827
\(975\) 0 0
\(976\) −37.0365 −1.18551
\(977\) 47.7652 1.52814 0.764072 0.645130i \(-0.223197\pi\)
0.764072 + 0.645130i \(0.223197\pi\)
\(978\) 30.2268 0.966545
\(979\) 26.9537 0.861445
\(980\) 0 0
\(981\) −2.31057 −0.0737709
\(982\) 35.4131 1.13008
\(983\) −51.9282 −1.65625 −0.828127 0.560541i \(-0.810593\pi\)
−0.828127 + 0.560541i \(0.810593\pi\)
\(984\) −7.21476 −0.229998
\(985\) 0 0
\(986\) −8.76846 −0.279245
\(987\) −3.41962 −0.108848
\(988\) −135.203 −4.30138
\(989\) 21.9502 0.697976
\(990\) 0 0
\(991\) −16.3790 −0.520297 −0.260148 0.965569i \(-0.583771\pi\)
−0.260148 + 0.965569i \(0.583771\pi\)
\(992\) 4.21226 0.133739
\(993\) −21.4575 −0.680933
\(994\) −3.73262 −0.118392
\(995\) 0 0
\(996\) 50.9493 1.61439
\(997\) 22.2515 0.704713 0.352357 0.935866i \(-0.385380\pi\)
0.352357 + 0.935866i \(0.385380\pi\)
\(998\) −90.7302 −2.87201
\(999\) 1.61763 0.0511795
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.1.8 8
3.2 odd 2 5625.2.a.t.1.1 8
5.2 odd 4 1875.2.b.h.1249.16 16
5.3 odd 4 1875.2.b.h.1249.1 16
5.4 even 2 1875.2.a.m.1.1 8
15.14 odd 2 5625.2.a.bd.1.8 8
25.3 odd 20 375.2.i.c.49.1 16
25.4 even 10 375.2.g.e.76.4 16
25.6 even 5 375.2.g.d.301.1 16
25.8 odd 20 75.2.i.a.64.4 yes 16
25.17 odd 20 375.2.i.c.199.1 16
25.19 even 10 375.2.g.e.301.4 16
25.21 even 5 375.2.g.d.76.1 16
25.22 odd 20 75.2.i.a.34.4 16
75.8 even 20 225.2.m.b.64.1 16
75.47 even 20 225.2.m.b.109.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.4 16 25.22 odd 20
75.2.i.a.64.4 yes 16 25.8 odd 20
225.2.m.b.64.1 16 75.8 even 20
225.2.m.b.109.1 16 75.47 even 20
375.2.g.d.76.1 16 25.21 even 5
375.2.g.d.301.1 16 25.6 even 5
375.2.g.e.76.4 16 25.4 even 10
375.2.g.e.301.4 16 25.19 even 10
375.2.i.c.49.1 16 25.3 odd 20
375.2.i.c.199.1 16 25.17 odd 20
1875.2.a.m.1.1 8 5.4 even 2
1875.2.a.p.1.8 8 1.1 even 1 trivial
1875.2.b.h.1249.1 16 5.3 odd 4
1875.2.b.h.1249.16 16 5.2 odd 4
5625.2.a.t.1.1 8 3.2 odd 2
5625.2.a.bd.1.8 8 15.14 odd 2