Properties

Label 1425.2.a.q.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
Analytic rank $1$
Dimension $2$
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] = [1425,2,Mod(1,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.61803 q^{12} +6.47214 q^{13} +1.38197 q^{14} +1.85410 q^{16} +1.23607 q^{17} -0.618034 q^{18} -1.00000 q^{19} +2.23607 q^{21} +2.47214 q^{22} +3.23607 q^{23} -2.23607 q^{24} -4.00000 q^{26} -1.00000 q^{27} +3.61803 q^{28} +7.47214 q^{29} -10.4721 q^{31} -5.61803 q^{32} +4.00000 q^{33} -0.763932 q^{34} -1.61803 q^{36} +2.76393 q^{37} +0.618034 q^{38} -6.47214 q^{39} -5.00000 q^{41} -1.38197 q^{42} +4.00000 q^{43} +6.47214 q^{44} -2.00000 q^{46} +0.472136 q^{47} -1.85410 q^{48} -2.00000 q^{49} -1.23607 q^{51} -10.4721 q^{52} -5.00000 q^{53} +0.618034 q^{54} -5.00000 q^{56} +1.00000 q^{57} -4.61803 q^{58} -14.7082 q^{59} +7.94427 q^{61} +6.47214 q^{62} -2.23607 q^{63} -0.236068 q^{64} -2.47214 q^{66} -10.9443 q^{67} -2.00000 q^{68} -3.23607 q^{69} +9.18034 q^{71} +2.23607 q^{72} +3.47214 q^{73} -1.70820 q^{74} +1.61803 q^{76} +8.94427 q^{77} +4.00000 q^{78} -2.29180 q^{79} +1.00000 q^{81} +3.09017 q^{82} -6.94427 q^{83} -3.61803 q^{84} -2.47214 q^{86} -7.47214 q^{87} -8.94427 q^{88} -9.94427 q^{89} -14.4721 q^{91} -5.23607 q^{92} +10.4721 q^{93} -0.291796 q^{94} +5.61803 q^{96} -1.70820 q^{97} +1.23607 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9} - 8 q^{11} + q^{12} + 4 q^{13} + 5 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 2 q^{19} - 4 q^{22} + 2 q^{23} - 8 q^{26} - 2 q^{27} + 5 q^{28} + 6 q^{29} - 12 q^{31} - 9 q^{32} + 8 q^{33} - 6 q^{34} - q^{36} + 10 q^{37} - q^{38} - 4 q^{39} - 10 q^{41} - 5 q^{42} + 8 q^{43} + 4 q^{44} - 4 q^{46} - 8 q^{47} + 3 q^{48} - 4 q^{49} + 2 q^{51} - 12 q^{52} - 10 q^{53} - q^{54} - 10 q^{56} + 2 q^{57} - 7 q^{58} - 16 q^{59} - 2 q^{61} + 4 q^{62} + 4 q^{64} + 4 q^{66} - 4 q^{67} - 4 q^{68} - 2 q^{69} - 4 q^{71} - 2 q^{73} + 10 q^{74} + q^{76} + 8 q^{78} - 18 q^{79} + 2 q^{81} - 5 q^{82} + 4 q^{83} - 5 q^{84} + 4 q^{86} - 6 q^{87} - 2 q^{89} - 20 q^{91} - 6 q^{92} + 12 q^{93} - 14 q^{94} + 9 q^{96} + 10 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) −2.23607 −0.845154 −0.422577 0.906327i \(-0.638874\pi\)
−0.422577 + 0.906327i \(0.638874\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.61803 0.467086
\(13\) 6.47214 1.79505 0.897524 0.440966i \(-0.145364\pi\)
0.897524 + 0.440966i \(0.145364\pi\)
\(14\) 1.38197 0.369346
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 1.23607 0.299791 0.149895 0.988702i \(-0.452106\pi\)
0.149895 + 0.988702i \(0.452106\pi\)
\(18\) −0.618034 −0.145672
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.23607 0.487950
\(22\) 2.47214 0.527061
\(23\) 3.23607 0.674767 0.337383 0.941367i \(-0.390458\pi\)
0.337383 + 0.941367i \(0.390458\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 3.61803 0.683744
\(29\) 7.47214 1.38754 0.693770 0.720196i \(-0.255948\pi\)
0.693770 + 0.720196i \(0.255948\pi\)
\(30\) 0 0
\(31\) −10.4721 −1.88085 −0.940426 0.340000i \(-0.889573\pi\)
−0.940426 + 0.340000i \(0.889573\pi\)
\(32\) −5.61803 −0.993137
\(33\) 4.00000 0.696311
\(34\) −0.763932 −0.131013
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) 2.76393 0.454388 0.227194 0.973850i \(-0.427045\pi\)
0.227194 + 0.973850i \(0.427045\pi\)
\(38\) 0.618034 0.100258
\(39\) −6.47214 −1.03637
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) −1.38197 −0.213242
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.47214 0.975711
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 0.472136 0.0688681 0.0344341 0.999407i \(-0.489037\pi\)
0.0344341 + 0.999407i \(0.489037\pi\)
\(48\) −1.85410 −0.267617
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −1.23607 −0.173084
\(52\) −10.4721 −1.45222
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) −5.00000 −0.668153
\(57\) 1.00000 0.132453
\(58\) −4.61803 −0.606378
\(59\) −14.7082 −1.91485 −0.957423 0.288690i \(-0.906780\pi\)
−0.957423 + 0.288690i \(0.906780\pi\)
\(60\) 0 0
\(61\) 7.94427 1.01716 0.508580 0.861015i \(-0.330171\pi\)
0.508580 + 0.861015i \(0.330171\pi\)
\(62\) 6.47214 0.821962
\(63\) −2.23607 −0.281718
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) −2.47214 −0.304299
\(67\) −10.9443 −1.33706 −0.668528 0.743687i \(-0.733075\pi\)
−0.668528 + 0.743687i \(0.733075\pi\)
\(68\) −2.00000 −0.242536
\(69\) −3.23607 −0.389577
\(70\) 0 0
\(71\) 9.18034 1.08951 0.544753 0.838597i \(-0.316623\pi\)
0.544753 + 0.838597i \(0.316623\pi\)
\(72\) 2.23607 0.263523
\(73\) 3.47214 0.406383 0.203191 0.979139i \(-0.434869\pi\)
0.203191 + 0.979139i \(0.434869\pi\)
\(74\) −1.70820 −0.198575
\(75\) 0 0
\(76\) 1.61803 0.185601
\(77\) 8.94427 1.01929
\(78\) 4.00000 0.452911
\(79\) −2.29180 −0.257847 −0.128924 0.991655i \(-0.541152\pi\)
−0.128924 + 0.991655i \(0.541152\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.09017 0.341252
\(83\) −6.94427 −0.762233 −0.381116 0.924527i \(-0.624460\pi\)
−0.381116 + 0.924527i \(0.624460\pi\)
\(84\) −3.61803 −0.394760
\(85\) 0 0
\(86\) −2.47214 −0.266577
\(87\) −7.47214 −0.801097
\(88\) −8.94427 −0.953463
\(89\) −9.94427 −1.05409 −0.527045 0.849837i \(-0.676700\pi\)
−0.527045 + 0.849837i \(0.676700\pi\)
\(90\) 0 0
\(91\) −14.4721 −1.51709
\(92\) −5.23607 −0.545898
\(93\) 10.4721 1.08591
\(94\) −0.291796 −0.0300965
\(95\) 0 0
\(96\) 5.61803 0.573388
\(97\) −1.70820 −0.173442 −0.0867209 0.996233i \(-0.527639\pi\)
−0.0867209 + 0.996233i \(0.527639\pi\)
\(98\) 1.23607 0.124862
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −14.1803 −1.41100 −0.705498 0.708712i \(-0.749277\pi\)
−0.705498 + 0.708712i \(0.749277\pi\)
\(102\) 0.763932 0.0756405
\(103\) 16.9443 1.66957 0.834784 0.550577i \(-0.185592\pi\)
0.834784 + 0.550577i \(0.185592\pi\)
\(104\) 14.4721 1.41911
\(105\) 0 0
\(106\) 3.09017 0.300144
\(107\) −16.2361 −1.56960 −0.784800 0.619749i \(-0.787234\pi\)
−0.784800 + 0.619749i \(0.787234\pi\)
\(108\) 1.61803 0.155695
\(109\) −7.70820 −0.738312 −0.369156 0.929367i \(-0.620353\pi\)
−0.369156 + 0.929367i \(0.620353\pi\)
\(110\) 0 0
\(111\) −2.76393 −0.262341
\(112\) −4.14590 −0.391751
\(113\) −12.4721 −1.17328 −0.586640 0.809848i \(-0.699550\pi\)
−0.586640 + 0.809848i \(0.699550\pi\)
\(114\) −0.618034 −0.0578842
\(115\) 0 0
\(116\) −12.0902 −1.12254
\(117\) 6.47214 0.598349
\(118\) 9.09017 0.836818
\(119\) −2.76393 −0.253369
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −4.90983 −0.444515
\(123\) 5.00000 0.450835
\(124\) 16.9443 1.52164
\(125\) 0 0
\(126\) 1.38197 0.123115
\(127\) 10.7639 0.955145 0.477572 0.878592i \(-0.341517\pi\)
0.477572 + 0.878592i \(0.341517\pi\)
\(128\) 11.3820 1.00603
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −6.94427 −0.606724 −0.303362 0.952875i \(-0.598109\pi\)
−0.303362 + 0.952875i \(0.598109\pi\)
\(132\) −6.47214 −0.563327
\(133\) 2.23607 0.193892
\(134\) 6.76393 0.584315
\(135\) 0 0
\(136\) 2.76393 0.237005
\(137\) 4.29180 0.366673 0.183336 0.983050i \(-0.441310\pi\)
0.183336 + 0.983050i \(0.441310\pi\)
\(138\) 2.00000 0.170251
\(139\) −15.7639 −1.33708 −0.668540 0.743677i \(-0.733080\pi\)
−0.668540 + 0.743677i \(0.733080\pi\)
\(140\) 0 0
\(141\) −0.472136 −0.0397610
\(142\) −5.67376 −0.476132
\(143\) −25.8885 −2.16491
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) −2.14590 −0.177596
\(147\) 2.00000 0.164957
\(148\) −4.47214 −0.367607
\(149\) −19.4164 −1.59065 −0.795327 0.606181i \(-0.792701\pi\)
−0.795327 + 0.606181i \(0.792701\pi\)
\(150\) 0 0
\(151\) 11.7082 0.952800 0.476400 0.879229i \(-0.341941\pi\)
0.476400 + 0.879229i \(0.341941\pi\)
\(152\) −2.23607 −0.181369
\(153\) 1.23607 0.0999302
\(154\) −5.52786 −0.445448
\(155\) 0 0
\(156\) 10.4721 0.838442
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 1.41641 0.112683
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) −7.23607 −0.570282
\(162\) −0.618034 −0.0485573
\(163\) −11.1803 −0.875712 −0.437856 0.899045i \(-0.644262\pi\)
−0.437856 + 0.899045i \(0.644262\pi\)
\(164\) 8.09017 0.631736
\(165\) 0 0
\(166\) 4.29180 0.333108
\(167\) −24.5967 −1.90335 −0.951677 0.307102i \(-0.900641\pi\)
−0.951677 + 0.307102i \(0.900641\pi\)
\(168\) 5.00000 0.385758
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −6.47214 −0.493496
\(173\) −13.4721 −1.02427 −0.512134 0.858906i \(-0.671145\pi\)
−0.512134 + 0.858906i \(0.671145\pi\)
\(174\) 4.61803 0.350092
\(175\) 0 0
\(176\) −7.41641 −0.559033
\(177\) 14.7082 1.10554
\(178\) 6.14590 0.460655
\(179\) 22.1246 1.65367 0.826836 0.562444i \(-0.190139\pi\)
0.826836 + 0.562444i \(0.190139\pi\)
\(180\) 0 0
\(181\) 10.4721 0.778388 0.389194 0.921156i \(-0.372754\pi\)
0.389194 + 0.921156i \(0.372754\pi\)
\(182\) 8.94427 0.662994
\(183\) −7.94427 −0.587257
\(184\) 7.23607 0.533450
\(185\) 0 0
\(186\) −6.47214 −0.474560
\(187\) −4.94427 −0.361561
\(188\) −0.763932 −0.0557155
\(189\) 2.23607 0.162650
\(190\) 0 0
\(191\) 7.41641 0.536632 0.268316 0.963331i \(-0.413533\pi\)
0.268316 + 0.963331i \(0.413533\pi\)
\(192\) 0.236068 0.0170367
\(193\) −3.23607 −0.232937 −0.116469 0.993194i \(-0.537157\pi\)
−0.116469 + 0.993194i \(0.537157\pi\)
\(194\) 1.05573 0.0757969
\(195\) 0 0
\(196\) 3.23607 0.231148
\(197\) −22.6525 −1.61392 −0.806961 0.590605i \(-0.798889\pi\)
−0.806961 + 0.590605i \(0.798889\pi\)
\(198\) 2.47214 0.175687
\(199\) 1.76393 0.125042 0.0625209 0.998044i \(-0.480086\pi\)
0.0625209 + 0.998044i \(0.480086\pi\)
\(200\) 0 0
\(201\) 10.9443 0.771949
\(202\) 8.76393 0.616628
\(203\) −16.7082 −1.17269
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −10.4721 −0.729628
\(207\) 3.23607 0.224922
\(208\) 12.0000 0.832050
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 5.81966 0.400642 0.200321 0.979730i \(-0.435802\pi\)
0.200321 + 0.979730i \(0.435802\pi\)
\(212\) 8.09017 0.555635
\(213\) −9.18034 −0.629027
\(214\) 10.0344 0.685940
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) 23.4164 1.58961
\(218\) 4.76393 0.322654
\(219\) −3.47214 −0.234625
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 1.70820 0.114647
\(223\) 9.23607 0.618493 0.309246 0.950982i \(-0.399923\pi\)
0.309246 + 0.950982i \(0.399923\pi\)
\(224\) 12.5623 0.839354
\(225\) 0 0
\(226\) 7.70820 0.512742
\(227\) 28.1246 1.86670 0.933348 0.358973i \(-0.116873\pi\)
0.933348 + 0.358973i \(0.116873\pi\)
\(228\) −1.61803 −0.107157
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) −8.94427 −0.588490
\(232\) 16.7082 1.09695
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 23.7984 1.54914
\(237\) 2.29180 0.148868
\(238\) 1.70820 0.110726
\(239\) 0.291796 0.0188747 0.00943736 0.999955i \(-0.496996\pi\)
0.00943736 + 0.999955i \(0.496996\pi\)
\(240\) 0 0
\(241\) −16.1803 −1.04227 −0.521134 0.853475i \(-0.674491\pi\)
−0.521134 + 0.853475i \(0.674491\pi\)
\(242\) −3.09017 −0.198644
\(243\) −1.00000 −0.0641500
\(244\) −12.8541 −0.822900
\(245\) 0 0
\(246\) −3.09017 −0.197022
\(247\) −6.47214 −0.411812
\(248\) −23.4164 −1.48694
\(249\) 6.94427 0.440075
\(250\) 0 0
\(251\) 3.70820 0.234060 0.117030 0.993128i \(-0.462663\pi\)
0.117030 + 0.993128i \(0.462663\pi\)
\(252\) 3.61803 0.227915
\(253\) −12.9443 −0.813799
\(254\) −6.65248 −0.417413
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −13.0000 −0.810918 −0.405459 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(258\) 2.47214 0.153908
\(259\) −6.18034 −0.384028
\(260\) 0 0
\(261\) 7.47214 0.462514
\(262\) 4.29180 0.265148
\(263\) −10.2918 −0.634619 −0.317310 0.948322i \(-0.602779\pi\)
−0.317310 + 0.948322i \(0.602779\pi\)
\(264\) 8.94427 0.550482
\(265\) 0 0
\(266\) −1.38197 −0.0847338
\(267\) 9.94427 0.608580
\(268\) 17.7082 1.08170
\(269\) 20.4721 1.24821 0.624104 0.781341i \(-0.285464\pi\)
0.624104 + 0.781341i \(0.285464\pi\)
\(270\) 0 0
\(271\) 3.18034 0.193192 0.0965959 0.995324i \(-0.469205\pi\)
0.0965959 + 0.995324i \(0.469205\pi\)
\(272\) 2.29180 0.138961
\(273\) 14.4721 0.875894
\(274\) −2.65248 −0.160242
\(275\) 0 0
\(276\) 5.23607 0.315174
\(277\) 5.58359 0.335486 0.167743 0.985831i \(-0.446352\pi\)
0.167743 + 0.985831i \(0.446352\pi\)
\(278\) 9.74265 0.584325
\(279\) −10.4721 −0.626950
\(280\) 0 0
\(281\) 7.88854 0.470591 0.235296 0.971924i \(-0.424394\pi\)
0.235296 + 0.971924i \(0.424394\pi\)
\(282\) 0.291796 0.0173762
\(283\) 9.88854 0.587813 0.293906 0.955834i \(-0.405045\pi\)
0.293906 + 0.955834i \(0.405045\pi\)
\(284\) −14.8541 −0.881429
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) 11.1803 0.659955
\(288\) −5.61803 −0.331046
\(289\) −15.4721 −0.910126
\(290\) 0 0
\(291\) 1.70820 0.100137
\(292\) −5.61803 −0.328771
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −1.23607 −0.0720889
\(295\) 0 0
\(296\) 6.18034 0.359225
\(297\) 4.00000 0.232104
\(298\) 12.0000 0.695141
\(299\) 20.9443 1.21124
\(300\) 0 0
\(301\) −8.94427 −0.515539
\(302\) −7.23607 −0.416389
\(303\) 14.1803 0.814639
\(304\) −1.85410 −0.106340
\(305\) 0 0
\(306\) −0.763932 −0.0436711
\(307\) −3.05573 −0.174400 −0.0871998 0.996191i \(-0.527792\pi\)
−0.0871998 + 0.996191i \(0.527792\pi\)
\(308\) −14.4721 −0.824626
\(309\) −16.9443 −0.963926
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −14.4721 −0.819323
\(313\) −30.9443 −1.74907 −0.874537 0.484959i \(-0.838834\pi\)
−0.874537 + 0.484959i \(0.838834\pi\)
\(314\) −4.32624 −0.244144
\(315\) 0 0
\(316\) 3.70820 0.208603
\(317\) −7.94427 −0.446195 −0.223097 0.974796i \(-0.571617\pi\)
−0.223097 + 0.974796i \(0.571617\pi\)
\(318\) −3.09017 −0.173288
\(319\) −29.8885 −1.67344
\(320\) 0 0
\(321\) 16.2361 0.906209
\(322\) 4.47214 0.249222
\(323\) −1.23607 −0.0687767
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 6.90983 0.382700
\(327\) 7.70820 0.426265
\(328\) −11.1803 −0.617331
\(329\) −1.05573 −0.0582042
\(330\) 0 0
\(331\) −26.3607 −1.44891 −0.724457 0.689320i \(-0.757909\pi\)
−0.724457 + 0.689320i \(0.757909\pi\)
\(332\) 11.2361 0.616659
\(333\) 2.76393 0.151463
\(334\) 15.2016 0.831796
\(335\) 0 0
\(336\) 4.14590 0.226177
\(337\) −8.47214 −0.461507 −0.230753 0.973012i \(-0.574119\pi\)
−0.230753 + 0.973012i \(0.574119\pi\)
\(338\) −17.8541 −0.971135
\(339\) 12.4721 0.677393
\(340\) 0 0
\(341\) 41.8885 2.26839
\(342\) 0.618034 0.0334195
\(343\) 20.1246 1.08663
\(344\) 8.94427 0.482243
\(345\) 0 0
\(346\) 8.32624 0.447621
\(347\) −1.70820 −0.0917012 −0.0458506 0.998948i \(-0.514600\pi\)
−0.0458506 + 0.998948i \(0.514600\pi\)
\(348\) 12.0902 0.648101
\(349\) −17.8328 −0.954569 −0.477284 0.878749i \(-0.658379\pi\)
−0.477284 + 0.878749i \(0.658379\pi\)
\(350\) 0 0
\(351\) −6.47214 −0.345457
\(352\) 22.4721 1.19777
\(353\) −26.4721 −1.40897 −0.704485 0.709719i \(-0.748822\pi\)
−0.704485 + 0.709719i \(0.748822\pi\)
\(354\) −9.09017 −0.483137
\(355\) 0 0
\(356\) 16.0902 0.852777
\(357\) 2.76393 0.146283
\(358\) −13.6738 −0.722681
\(359\) −25.2361 −1.33191 −0.665954 0.745992i \(-0.731975\pi\)
−0.665954 + 0.745992i \(0.731975\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −6.47214 −0.340168
\(363\) −5.00000 −0.262432
\(364\) 23.4164 1.22735
\(365\) 0 0
\(366\) 4.90983 0.256641
\(367\) 33.3050 1.73850 0.869252 0.494369i \(-0.164601\pi\)
0.869252 + 0.494369i \(0.164601\pi\)
\(368\) 6.00000 0.312772
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 11.1803 0.580454
\(372\) −16.9443 −0.878520
\(373\) 12.1803 0.630674 0.315337 0.948980i \(-0.397882\pi\)
0.315337 + 0.948980i \(0.397882\pi\)
\(374\) 3.05573 0.158008
\(375\) 0 0
\(376\) 1.05573 0.0544450
\(377\) 48.3607 2.49070
\(378\) −1.38197 −0.0710807
\(379\) 6.76393 0.347440 0.173720 0.984795i \(-0.444421\pi\)
0.173720 + 0.984795i \(0.444421\pi\)
\(380\) 0 0
\(381\) −10.7639 −0.551453
\(382\) −4.58359 −0.234517
\(383\) 16.1246 0.823929 0.411965 0.911200i \(-0.364843\pi\)
0.411965 + 0.911200i \(0.364843\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) 2.76393 0.140317
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −4.47214 −0.225877
\(393\) 6.94427 0.350292
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 6.47214 0.325237
\(397\) 22.3607 1.12225 0.561125 0.827731i \(-0.310369\pi\)
0.561125 + 0.827731i \(0.310369\pi\)
\(398\) −1.09017 −0.0546453
\(399\) −2.23607 −0.111943
\(400\) 0 0
\(401\) −20.8328 −1.04034 −0.520171 0.854062i \(-0.674132\pi\)
−0.520171 + 0.854062i \(0.674132\pi\)
\(402\) −6.76393 −0.337354
\(403\) −67.7771 −3.37622
\(404\) 22.9443 1.14152
\(405\) 0 0
\(406\) 10.3262 0.512483
\(407\) −11.0557 −0.548012
\(408\) −2.76393 −0.136835
\(409\) 8.76393 0.433349 0.216674 0.976244i \(-0.430479\pi\)
0.216674 + 0.976244i \(0.430479\pi\)
\(410\) 0 0
\(411\) −4.29180 −0.211699
\(412\) −27.4164 −1.35071
\(413\) 32.8885 1.61834
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) −36.3607 −1.78273
\(417\) 15.7639 0.771963
\(418\) −2.47214 −0.120916
\(419\) 39.3050 1.92017 0.960086 0.279704i \(-0.0902363\pi\)
0.960086 + 0.279704i \(0.0902363\pi\)
\(420\) 0 0
\(421\) 11.5279 0.561834 0.280917 0.959732i \(-0.409361\pi\)
0.280917 + 0.959732i \(0.409361\pi\)
\(422\) −3.59675 −0.175087
\(423\) 0.472136 0.0229560
\(424\) −11.1803 −0.542965
\(425\) 0 0
\(426\) 5.67376 0.274895
\(427\) −17.7639 −0.859657
\(428\) 26.2705 1.26983
\(429\) 25.8885 1.24991
\(430\) 0 0
\(431\) −5.18034 −0.249528 −0.124764 0.992186i \(-0.539817\pi\)
−0.124764 + 0.992186i \(0.539817\pi\)
\(432\) −1.85410 −0.0892055
\(433\) −19.5279 −0.938449 −0.469225 0.883079i \(-0.655467\pi\)
−0.469225 + 0.883079i \(0.655467\pi\)
\(434\) −14.4721 −0.694685
\(435\) 0 0
\(436\) 12.4721 0.597307
\(437\) −3.23607 −0.154802
\(438\) 2.14590 0.102535
\(439\) −16.7639 −0.800099 −0.400049 0.916494i \(-0.631007\pi\)
−0.400049 + 0.916494i \(0.631007\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −4.94427 −0.235175
\(443\) 13.4164 0.637433 0.318716 0.947850i \(-0.396748\pi\)
0.318716 + 0.947850i \(0.396748\pi\)
\(444\) 4.47214 0.212238
\(445\) 0 0
\(446\) −5.70820 −0.270291
\(447\) 19.4164 0.918365
\(448\) 0.527864 0.0249392
\(449\) −25.4721 −1.20210 −0.601052 0.799210i \(-0.705252\pi\)
−0.601052 + 0.799210i \(0.705252\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 20.1803 0.949203
\(453\) −11.7082 −0.550099
\(454\) −17.3820 −0.815776
\(455\) 0 0
\(456\) 2.23607 0.104713
\(457\) 8.88854 0.415789 0.207894 0.978151i \(-0.433339\pi\)
0.207894 + 0.978151i \(0.433339\pi\)
\(458\) 2.76393 0.129150
\(459\) −1.23607 −0.0576947
\(460\) 0 0
\(461\) −1.81966 −0.0847500 −0.0423750 0.999102i \(-0.513492\pi\)
−0.0423750 + 0.999102i \(0.513492\pi\)
\(462\) 5.52786 0.257180
\(463\) 8.58359 0.398913 0.199457 0.979907i \(-0.436082\pi\)
0.199457 + 0.979907i \(0.436082\pi\)
\(464\) 13.8541 0.643161
\(465\) 0 0
\(466\) −1.81966 −0.0842941
\(467\) −26.1803 −1.21148 −0.605741 0.795662i \(-0.707123\pi\)
−0.605741 + 0.795662i \(0.707123\pi\)
\(468\) −10.4721 −0.484075
\(469\) 24.4721 1.13002
\(470\) 0 0
\(471\) −7.00000 −0.322543
\(472\) −32.8885 −1.51382
\(473\) −16.0000 −0.735681
\(474\) −1.41641 −0.0650578
\(475\) 0 0
\(476\) 4.47214 0.204980
\(477\) −5.00000 −0.228934
\(478\) −0.180340 −0.00824855
\(479\) −38.7639 −1.77117 −0.885585 0.464478i \(-0.846242\pi\)
−0.885585 + 0.464478i \(0.846242\pi\)
\(480\) 0 0
\(481\) 17.8885 0.815647
\(482\) 10.0000 0.455488
\(483\) 7.23607 0.329252
\(484\) −8.09017 −0.367735
\(485\) 0 0
\(486\) 0.618034 0.0280346
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 17.7639 0.804135
\(489\) 11.1803 0.505592
\(490\) 0 0
\(491\) −8.29180 −0.374204 −0.187102 0.982341i \(-0.559909\pi\)
−0.187102 + 0.982341i \(0.559909\pi\)
\(492\) −8.09017 −0.364733
\(493\) 9.23607 0.415972
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −19.4164 −0.871822
\(497\) −20.5279 −0.920801
\(498\) −4.29180 −0.192320
\(499\) −31.1803 −1.39582 −0.697912 0.716184i \(-0.745887\pi\)
−0.697912 + 0.716184i \(0.745887\pi\)
\(500\) 0 0
\(501\) 24.5967 1.09890
\(502\) −2.29180 −0.102288
\(503\) −9.52786 −0.424826 −0.212413 0.977180i \(-0.568132\pi\)
−0.212413 + 0.977180i \(0.568132\pi\)
\(504\) −5.00000 −0.222718
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) −28.8885 −1.28299
\(508\) −17.4164 −0.772728
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 0 0
\(511\) −7.76393 −0.343456
\(512\) −18.7082 −0.826794
\(513\) 1.00000 0.0441511
\(514\) 8.03444 0.354384
\(515\) 0 0
\(516\) 6.47214 0.284920
\(517\) −1.88854 −0.0830581
\(518\) 3.81966 0.167826
\(519\) 13.4721 0.591361
\(520\) 0 0
\(521\) −35.8328 −1.56986 −0.784932 0.619582i \(-0.787302\pi\)
−0.784932 + 0.619582i \(0.787302\pi\)
\(522\) −4.61803 −0.202126
\(523\) 41.1246 1.79825 0.899127 0.437688i \(-0.144203\pi\)
0.899127 + 0.437688i \(0.144203\pi\)
\(524\) 11.2361 0.490850
\(525\) 0 0
\(526\) 6.36068 0.277339
\(527\) −12.9443 −0.563861
\(528\) 7.41641 0.322758
\(529\) −12.5279 −0.544690
\(530\) 0 0
\(531\) −14.7082 −0.638282
\(532\) −3.61803 −0.156862
\(533\) −32.3607 −1.40170
\(534\) −6.14590 −0.265959
\(535\) 0 0
\(536\) −24.4721 −1.05704
\(537\) −22.1246 −0.954747
\(538\) −12.6525 −0.545487
\(539\) 8.00000 0.344584
\(540\) 0 0
\(541\) 24.8328 1.06765 0.533823 0.845596i \(-0.320755\pi\)
0.533823 + 0.845596i \(0.320755\pi\)
\(542\) −1.96556 −0.0844280
\(543\) −10.4721 −0.449402
\(544\) −6.94427 −0.297733
\(545\) 0 0
\(546\) −8.94427 −0.382780
\(547\) −7.70820 −0.329579 −0.164790 0.986329i \(-0.552694\pi\)
−0.164790 + 0.986329i \(0.552694\pi\)
\(548\) −6.94427 −0.296645
\(549\) 7.94427 0.339053
\(550\) 0 0
\(551\) −7.47214 −0.318324
\(552\) −7.23607 −0.307988
\(553\) 5.12461 0.217921
\(554\) −3.45085 −0.146613
\(555\) 0 0
\(556\) 25.5066 1.08172
\(557\) −33.8885 −1.43590 −0.717952 0.696093i \(-0.754920\pi\)
−0.717952 + 0.696093i \(0.754920\pi\)
\(558\) 6.47214 0.273987
\(559\) 25.8885 1.09497
\(560\) 0 0
\(561\) 4.94427 0.208747
\(562\) −4.87539 −0.205656
\(563\) 23.6525 0.996833 0.498417 0.866938i \(-0.333915\pi\)
0.498417 + 0.866938i \(0.333915\pi\)
\(564\) 0.763932 0.0321673
\(565\) 0 0
\(566\) −6.11146 −0.256884
\(567\) −2.23607 −0.0939060
\(568\) 20.5279 0.861330
\(569\) −20.8885 −0.875693 −0.437847 0.899050i \(-0.644259\pi\)
−0.437847 + 0.899050i \(0.644259\pi\)
\(570\) 0 0
\(571\) −15.6525 −0.655036 −0.327518 0.944845i \(-0.606212\pi\)
−0.327518 + 0.944845i \(0.606212\pi\)
\(572\) 41.8885 1.75145
\(573\) −7.41641 −0.309825
\(574\) −6.90983 −0.288411
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) −1.41641 −0.0589658 −0.0294829 0.999565i \(-0.509386\pi\)
−0.0294829 + 0.999565i \(0.509386\pi\)
\(578\) 9.56231 0.397739
\(579\) 3.23607 0.134486
\(580\) 0 0
\(581\) 15.5279 0.644204
\(582\) −1.05573 −0.0437613
\(583\) 20.0000 0.828315
\(584\) 7.76393 0.321274
\(585\) 0 0
\(586\) −1.23607 −0.0510615
\(587\) 12.6525 0.522224 0.261112 0.965309i \(-0.415911\pi\)
0.261112 + 0.965309i \(0.415911\pi\)
\(588\) −3.23607 −0.133453
\(589\) 10.4721 0.431497
\(590\) 0 0
\(591\) 22.6525 0.931798
\(592\) 5.12461 0.210620
\(593\) −13.8885 −0.570334 −0.285167 0.958478i \(-0.592049\pi\)
−0.285167 + 0.958478i \(0.592049\pi\)
\(594\) −2.47214 −0.101433
\(595\) 0 0
\(596\) 31.4164 1.28687
\(597\) −1.76393 −0.0721929
\(598\) −12.9443 −0.529331
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −35.7771 −1.45938 −0.729689 0.683779i \(-0.760335\pi\)
−0.729689 + 0.683779i \(0.760335\pi\)
\(602\) 5.52786 0.225299
\(603\) −10.9443 −0.445685
\(604\) −18.9443 −0.770831
\(605\) 0 0
\(606\) −8.76393 −0.356010
\(607\) −19.1246 −0.776244 −0.388122 0.921608i \(-0.626876\pi\)
−0.388122 + 0.921608i \(0.626876\pi\)
\(608\) 5.61803 0.227841
\(609\) 16.7082 0.677051
\(610\) 0 0
\(611\) 3.05573 0.123622
\(612\) −2.00000 −0.0808452
\(613\) 23.0000 0.928961 0.464481 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) 1.88854 0.0762154
\(615\) 0 0
\(616\) 20.0000 0.805823
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 10.4721 0.421251
\(619\) 30.7082 1.23427 0.617133 0.786858i \(-0.288294\pi\)
0.617133 + 0.786858i \(0.288294\pi\)
\(620\) 0 0
\(621\) −3.23607 −0.129859
\(622\) −2.47214 −0.0991236
\(623\) 22.2361 0.890869
\(624\) −12.0000 −0.480384
\(625\) 0 0
\(626\) 19.1246 0.764373
\(627\) −4.00000 −0.159745
\(628\) −11.3262 −0.451966
\(629\) 3.41641 0.136221
\(630\) 0 0
\(631\) −43.4164 −1.72838 −0.864190 0.503166i \(-0.832169\pi\)
−0.864190 + 0.503166i \(0.832169\pi\)
\(632\) −5.12461 −0.203846
\(633\) −5.81966 −0.231311
\(634\) 4.90983 0.194994
\(635\) 0 0
\(636\) −8.09017 −0.320796
\(637\) −12.9443 −0.512871
\(638\) 18.4721 0.731319
\(639\) 9.18034 0.363169
\(640\) 0 0
\(641\) 6.94427 0.274282 0.137141 0.990552i \(-0.456209\pi\)
0.137141 + 0.990552i \(0.456209\pi\)
\(642\) −10.0344 −0.396028
\(643\) −17.1803 −0.677526 −0.338763 0.940872i \(-0.610009\pi\)
−0.338763 + 0.940872i \(0.610009\pi\)
\(644\) 11.7082 0.461368
\(645\) 0 0
\(646\) 0.763932 0.0300565
\(647\) 17.3050 0.680328 0.340164 0.940366i \(-0.389517\pi\)
0.340164 + 0.940366i \(0.389517\pi\)
\(648\) 2.23607 0.0878410
\(649\) 58.8328 2.30939
\(650\) 0 0
\(651\) −23.4164 −0.917761
\(652\) 18.0902 0.708466
\(653\) 25.8885 1.01310 0.506549 0.862211i \(-0.330921\pi\)
0.506549 + 0.862211i \(0.330921\pi\)
\(654\) −4.76393 −0.186284
\(655\) 0 0
\(656\) −9.27051 −0.361953
\(657\) 3.47214 0.135461
\(658\) 0.652476 0.0254362
\(659\) 24.9443 0.971691 0.485845 0.874045i \(-0.338512\pi\)
0.485845 + 0.874045i \(0.338512\pi\)
\(660\) 0 0
\(661\) −23.1246 −0.899443 −0.449722 0.893169i \(-0.648477\pi\)
−0.449722 + 0.893169i \(0.648477\pi\)
\(662\) 16.2918 0.633199
\(663\) −8.00000 −0.310694
\(664\) −15.5279 −0.602598
\(665\) 0 0
\(666\) −1.70820 −0.0661916
\(667\) 24.1803 0.936266
\(668\) 39.7984 1.53985
\(669\) −9.23607 −0.357087
\(670\) 0 0
\(671\) −31.7771 −1.22674
\(672\) −12.5623 −0.484601
\(673\) −34.2492 −1.32021 −0.660105 0.751173i \(-0.729488\pi\)
−0.660105 + 0.751173i \(0.729488\pi\)
\(674\) 5.23607 0.201686
\(675\) 0 0
\(676\) −46.7426 −1.79779
\(677\) 5.83282 0.224173 0.112087 0.993698i \(-0.464247\pi\)
0.112087 + 0.993698i \(0.464247\pi\)
\(678\) −7.70820 −0.296032
\(679\) 3.81966 0.146585
\(680\) 0 0
\(681\) −28.1246 −1.07774
\(682\) −25.8885 −0.991324
\(683\) 8.34752 0.319409 0.159705 0.987165i \(-0.448946\pi\)
0.159705 + 0.987165i \(0.448946\pi\)
\(684\) 1.61803 0.0618671
\(685\) 0 0
\(686\) −12.4377 −0.474873
\(687\) 4.47214 0.170623
\(688\) 7.41641 0.282748
\(689\) −32.3607 −1.23284
\(690\) 0 0
\(691\) −7.05573 −0.268413 −0.134206 0.990953i \(-0.542848\pi\)
−0.134206 + 0.990953i \(0.542848\pi\)
\(692\) 21.7984 0.828650
\(693\) 8.94427 0.339765
\(694\) 1.05573 0.0400749
\(695\) 0 0
\(696\) −16.7082 −0.633323
\(697\) −6.18034 −0.234097
\(698\) 11.0213 0.417162
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) 17.3475 0.655207 0.327603 0.944815i \(-0.393759\pi\)
0.327603 + 0.944815i \(0.393759\pi\)
\(702\) 4.00000 0.150970
\(703\) −2.76393 −0.104244
\(704\) 0.944272 0.0355886
\(705\) 0 0
\(706\) 16.3607 0.615742
\(707\) 31.7082 1.19251
\(708\) −23.7984 −0.894398
\(709\) −40.4164 −1.51787 −0.758935 0.651166i \(-0.774280\pi\)
−0.758935 + 0.651166i \(0.774280\pi\)
\(710\) 0 0
\(711\) −2.29180 −0.0859491
\(712\) −22.2361 −0.833332
\(713\) −33.8885 −1.26914
\(714\) −1.70820 −0.0639279
\(715\) 0 0
\(716\) −35.7984 −1.33785
\(717\) −0.291796 −0.0108973
\(718\) 15.5967 0.582065
\(719\) −37.5279 −1.39955 −0.699777 0.714362i \(-0.746717\pi\)
−0.699777 + 0.714362i \(0.746717\pi\)
\(720\) 0 0
\(721\) −37.8885 −1.41104
\(722\) −0.618034 −0.0230008
\(723\) 16.1803 0.601753
\(724\) −16.9443 −0.629729
\(725\) 0 0
\(726\) 3.09017 0.114687
\(727\) −15.7639 −0.584652 −0.292326 0.956319i \(-0.594429\pi\)
−0.292326 + 0.956319i \(0.594429\pi\)
\(728\) −32.3607 −1.19937
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.94427 0.182871
\(732\) 12.8541 0.475101
\(733\) −12.4164 −0.458610 −0.229305 0.973355i \(-0.573645\pi\)
−0.229305 + 0.973355i \(0.573645\pi\)
\(734\) −20.5836 −0.759754
\(735\) 0 0
\(736\) −18.1803 −0.670136
\(737\) 43.7771 1.61255
\(738\) 3.09017 0.113751
\(739\) −4.59675 −0.169094 −0.0845470 0.996419i \(-0.526944\pi\)
−0.0845470 + 0.996419i \(0.526944\pi\)
\(740\) 0 0
\(741\) 6.47214 0.237760
\(742\) −6.90983 −0.253668
\(743\) 42.1246 1.54540 0.772701 0.634770i \(-0.218905\pi\)
0.772701 + 0.634770i \(0.218905\pi\)
\(744\) 23.4164 0.858487
\(745\) 0 0
\(746\) −7.52786 −0.275615
\(747\) −6.94427 −0.254078
\(748\) 8.00000 0.292509
\(749\) 36.3050 1.32655
\(750\) 0 0
\(751\) −20.4721 −0.747039 −0.373519 0.927622i \(-0.621849\pi\)
−0.373519 + 0.927622i \(0.621849\pi\)
\(752\) 0.875388 0.0319221
\(753\) −3.70820 −0.135134
\(754\) −29.8885 −1.08848
\(755\) 0 0
\(756\) −3.61803 −0.131587
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −4.18034 −0.151837
\(759\) 12.9443 0.469847
\(760\) 0 0
\(761\) −25.1246 −0.910766 −0.455383 0.890296i \(-0.650498\pi\)
−0.455383 + 0.890296i \(0.650498\pi\)
\(762\) 6.65248 0.240994
\(763\) 17.2361 0.623988
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −9.96556 −0.360070
\(767\) −95.1935 −3.43724
\(768\) 6.56231 0.236797
\(769\) −5.58359 −0.201349 −0.100675 0.994919i \(-0.532100\pi\)
−0.100675 + 0.994919i \(0.532100\pi\)
\(770\) 0 0
\(771\) 13.0000 0.468184
\(772\) 5.23607 0.188450
\(773\) −3.94427 −0.141866 −0.0709328 0.997481i \(-0.522598\pi\)
−0.0709328 + 0.997481i \(0.522598\pi\)
\(774\) −2.47214 −0.0888591
\(775\) 0 0
\(776\) −3.81966 −0.137118
\(777\) 6.18034 0.221718
\(778\) −4.94427 −0.177261
\(779\) 5.00000 0.179144
\(780\) 0 0
\(781\) −36.7214 −1.31399
\(782\) −2.47214 −0.0884034
\(783\) −7.47214 −0.267032
\(784\) −3.70820 −0.132436
\(785\) 0 0
\(786\) −4.29180 −0.153083
\(787\) −19.3475 −0.689665 −0.344832 0.938664i \(-0.612064\pi\)
−0.344832 + 0.938664i \(0.612064\pi\)
\(788\) 36.6525 1.30569
\(789\) 10.2918 0.366398
\(790\) 0 0
\(791\) 27.8885 0.991602
\(792\) −8.94427 −0.317821
\(793\) 51.4164 1.82585
\(794\) −13.8197 −0.490441
\(795\) 0 0
\(796\) −2.85410 −0.101161
\(797\) −31.3607 −1.11085 −0.555426 0.831566i \(-0.687445\pi\)
−0.555426 + 0.831566i \(0.687445\pi\)
\(798\) 1.38197 0.0489211
\(799\) 0.583592 0.0206460
\(800\) 0 0
\(801\) −9.94427 −0.351364
\(802\) 12.8754 0.454646
\(803\) −13.8885 −0.490116
\(804\) −17.7082 −0.624520
\(805\) 0 0
\(806\) 41.8885 1.47546
\(807\) −20.4721 −0.720653
\(808\) −31.7082 −1.11549
\(809\) −16.8328 −0.591810 −0.295905 0.955217i \(-0.595621\pi\)
−0.295905 + 0.955217i \(0.595621\pi\)
\(810\) 0 0
\(811\) 46.7214 1.64061 0.820304 0.571927i \(-0.193804\pi\)
0.820304 + 0.571927i \(0.193804\pi\)
\(812\) 27.0344 0.948723
\(813\) −3.18034 −0.111539
\(814\) 6.83282 0.239490
\(815\) 0 0
\(816\) −2.29180 −0.0802289
\(817\) −4.00000 −0.139942
\(818\) −5.41641 −0.189380
\(819\) −14.4721 −0.505697
\(820\) 0 0
\(821\) 5.12461 0.178850 0.0894251 0.995994i \(-0.471497\pi\)
0.0894251 + 0.995994i \(0.471497\pi\)
\(822\) 2.65248 0.0925157
\(823\) −33.1803 −1.15659 −0.578297 0.815826i \(-0.696282\pi\)
−0.578297 + 0.815826i \(0.696282\pi\)
\(824\) 37.8885 1.31991
\(825\) 0 0
\(826\) −20.3262 −0.707240
\(827\) −36.9443 −1.28468 −0.642339 0.766421i \(-0.722036\pi\)
−0.642339 + 0.766421i \(0.722036\pi\)
\(828\) −5.23607 −0.181966
\(829\) 55.6656 1.93335 0.966674 0.256012i \(-0.0824086\pi\)
0.966674 + 0.256012i \(0.0824086\pi\)
\(830\) 0 0
\(831\) −5.58359 −0.193693
\(832\) −1.52786 −0.0529692
\(833\) −2.47214 −0.0856544
\(834\) −9.74265 −0.337360
\(835\) 0 0
\(836\) −6.47214 −0.223844
\(837\) 10.4721 0.361970
\(838\) −24.2918 −0.839146
\(839\) −38.2361 −1.32006 −0.660028 0.751241i \(-0.729456\pi\)
−0.660028 + 0.751241i \(0.729456\pi\)
\(840\) 0 0
\(841\) 26.8328 0.925270
\(842\) −7.12461 −0.245530
\(843\) −7.88854 −0.271696
\(844\) −9.41641 −0.324126
\(845\) 0 0
\(846\) −0.291796 −0.0100322
\(847\) −11.1803 −0.384161
\(848\) −9.27051 −0.318351
\(849\) −9.88854 −0.339374
\(850\) 0 0
\(851\) 8.94427 0.306606
\(852\) 14.8541 0.508893
\(853\) −36.8885 −1.26304 −0.631520 0.775360i \(-0.717569\pi\)
−0.631520 + 0.775360i \(0.717569\pi\)
\(854\) 10.9787 0.375684
\(855\) 0 0
\(856\) −36.3050 −1.24088
\(857\) 32.7771 1.11964 0.559822 0.828613i \(-0.310870\pi\)
0.559822 + 0.828613i \(0.310870\pi\)
\(858\) −16.0000 −0.546231
\(859\) −21.2918 −0.726467 −0.363233 0.931698i \(-0.618327\pi\)
−0.363233 + 0.931698i \(0.618327\pi\)
\(860\) 0 0
\(861\) −11.1803 −0.381025
\(862\) 3.20163 0.109048
\(863\) 34.8197 1.18528 0.592638 0.805469i \(-0.298087\pi\)
0.592638 + 0.805469i \(0.298087\pi\)
\(864\) 5.61803 0.191129
\(865\) 0 0
\(866\) 12.0689 0.410117
\(867\) 15.4721 0.525461
\(868\) −37.8885 −1.28602
\(869\) 9.16718 0.310975
\(870\) 0 0
\(871\) −70.8328 −2.40008
\(872\) −17.2361 −0.583687
\(873\) −1.70820 −0.0578139
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 5.61803 0.189816
\(877\) −44.9443 −1.51766 −0.758830 0.651289i \(-0.774229\pi\)
−0.758830 + 0.651289i \(0.774229\pi\)
\(878\) 10.3607 0.349656
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) −0.944272 −0.0318133 −0.0159067 0.999873i \(-0.505063\pi\)
−0.0159067 + 0.999873i \(0.505063\pi\)
\(882\) 1.23607 0.0416206
\(883\) 45.6525 1.53633 0.768164 0.640253i \(-0.221171\pi\)
0.768164 + 0.640253i \(0.221171\pi\)
\(884\) −12.9443 −0.435363
\(885\) 0 0
\(886\) −8.29180 −0.278568
\(887\) −18.8328 −0.632344 −0.316172 0.948702i \(-0.602398\pi\)
−0.316172 + 0.948702i \(0.602398\pi\)
\(888\) −6.18034 −0.207399
\(889\) −24.0689 −0.807244
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −14.9443 −0.500371
\(893\) −0.472136 −0.0157994
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −25.4508 −0.850253
\(897\) −20.9443 −0.699309
\(898\) 15.7426 0.525339
\(899\) −78.2492 −2.60976
\(900\) 0 0
\(901\) −6.18034 −0.205897
\(902\) −12.3607 −0.411566
\(903\) 8.94427 0.297647
\(904\) −27.8885 −0.927559
\(905\) 0 0
\(906\) 7.23607 0.240402
\(907\) −29.8197 −0.990146 −0.495073 0.868852i \(-0.664859\pi\)
−0.495073 + 0.868852i \(0.664859\pi\)
\(908\) −45.5066 −1.51019
\(909\) −14.1803 −0.470332
\(910\) 0 0
\(911\) 11.7639 0.389756 0.194878 0.980827i \(-0.437569\pi\)
0.194878 + 0.980827i \(0.437569\pi\)
\(912\) 1.85410 0.0613955
\(913\) 27.7771 0.919287
\(914\) −5.49342 −0.181706
\(915\) 0 0
\(916\) 7.23607 0.239086
\(917\) 15.5279 0.512775
\(918\) 0.763932 0.0252135
\(919\) 41.2918 1.36209 0.681045 0.732241i \(-0.261526\pi\)
0.681045 + 0.732241i \(0.261526\pi\)
\(920\) 0 0
\(921\) 3.05573 0.100690
\(922\) 1.12461 0.0370371
\(923\) 59.4164 1.95571
\(924\) 14.4721 0.476098
\(925\) 0 0
\(926\) −5.30495 −0.174332
\(927\) 16.9443 0.556523
\(928\) −41.9787 −1.37802
\(929\) 47.5967 1.56160 0.780799 0.624782i \(-0.214812\pi\)
0.780799 + 0.624782i \(0.214812\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −4.76393 −0.156048
\(933\) −4.00000 −0.130954
\(934\) 16.1803 0.529437
\(935\) 0 0
\(936\) 14.4721 0.473037
\(937\) 39.9443 1.30492 0.652461 0.757822i \(-0.273737\pi\)
0.652461 + 0.757822i \(0.273737\pi\)
\(938\) −15.1246 −0.493836
\(939\) 30.9443 1.00983
\(940\) 0 0
\(941\) 36.4721 1.18896 0.594479 0.804111i \(-0.297358\pi\)
0.594479 + 0.804111i \(0.297358\pi\)
\(942\) 4.32624 0.140956
\(943\) −16.1803 −0.526904
\(944\) −27.2705 −0.887579
\(945\) 0 0
\(946\) 9.88854 0.321504
\(947\) −36.9443 −1.20053 −0.600264 0.799802i \(-0.704938\pi\)
−0.600264 + 0.799802i \(0.704938\pi\)
\(948\) −3.70820 −0.120437
\(949\) 22.4721 0.729476
\(950\) 0 0
\(951\) 7.94427 0.257611
\(952\) −6.18034 −0.200306
\(953\) 53.2492 1.72491 0.862456 0.506132i \(-0.168925\pi\)
0.862456 + 0.506132i \(0.168925\pi\)
\(954\) 3.09017 0.100048
\(955\) 0 0
\(956\) −0.472136 −0.0152700
\(957\) 29.8885 0.966159
\(958\) 23.9574 0.774029
\(959\) −9.59675 −0.309895
\(960\) 0 0
\(961\) 78.6656 2.53760
\(962\) −11.0557 −0.356451
\(963\) −16.2361 −0.523200
\(964\) 26.1803 0.843212
\(965\) 0 0
\(966\) −4.47214 −0.143889
\(967\) 22.5967 0.726662 0.363331 0.931660i \(-0.381639\pi\)
0.363331 + 0.931660i \(0.381639\pi\)
\(968\) 11.1803 0.359350
\(969\) 1.23607 0.0397082
\(970\) 0 0
\(971\) 21.0689 0.676133 0.338066 0.941122i \(-0.390227\pi\)
0.338066 + 0.941122i \(0.390227\pi\)
\(972\) 1.61803 0.0518985
\(973\) 35.2492 1.13004
\(974\) −14.8328 −0.475274
\(975\) 0 0
\(976\) 14.7295 0.471479
\(977\) 5.05573 0.161747 0.0808735 0.996724i \(-0.474229\pi\)
0.0808735 + 0.996724i \(0.474229\pi\)
\(978\) −6.90983 −0.220952
\(979\) 39.7771 1.27128
\(980\) 0 0
\(981\) −7.70820 −0.246104
\(982\) 5.12461 0.163533
\(983\) 57.3050 1.82774 0.913872 0.406002i \(-0.133078\pi\)
0.913872 + 0.406002i \(0.133078\pi\)
\(984\) 11.1803 0.356416
\(985\) 0 0
\(986\) −5.70820 −0.181786
\(987\) 1.05573 0.0336042
\(988\) 10.4721 0.333163
\(989\) 12.9443 0.411604
\(990\) 0 0
\(991\) −37.0557 −1.17711 −0.588557 0.808456i \(-0.700304\pi\)
−0.588557 + 0.808456i \(0.700304\pi\)
\(992\) 58.8328 1.86794
\(993\) 26.3607 0.836531
\(994\) 12.6869 0.402405
\(995\) 0 0
\(996\) −11.2361 −0.356028
\(997\) 12.4721 0.394997 0.197498 0.980303i \(-0.436718\pi\)
0.197498 + 0.980303i \(0.436718\pi\)
\(998\) 19.2705 0.609997
\(999\) −2.76393 −0.0874469
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 1425.2.a.q.1.1 yes 2
3.2 odd 2 4275.2.a.s.1.2 2
5.2 odd 4 1425.2.c.m.799.2 4
5.3 odd 4 1425.2.c.m.799.3 4
5.4 even 2 1425.2.a.n.1.2 2
15.14 odd 2 4275.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.n.1.2 2 5.4 even 2
1425.2.a.q.1.1 yes 2 1.1 even 1 trivial
1425.2.c.m.799.2 4 5.2 odd 4
1425.2.c.m.799.3 4 5.3 odd 4
4275.2.a.s.1.2 2 3.2 odd 2
4275.2.a.v.1.1 2 15.14 odd 2