Properties

Label 1875.2.a.g.1.4
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.9719503790\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) 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 1.4
Root \(1.33826\) 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)
 
\(f(q)\) \(=\) \(q+1.82709 q^{2} -1.00000 q^{3} +1.33826 q^{4} -1.82709 q^{6} -1.44512 q^{7} -1.20906 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.82709 q^{2} -1.00000 q^{3} +1.33826 q^{4} -1.82709 q^{6} -1.44512 q^{7} -1.20906 q^{8} +1.00000 q^{9} -2.12920 q^{11} -1.33826 q^{12} +5.70353 q^{13} -2.64037 q^{14} -4.88558 q^{16} -4.15622 q^{17} +1.82709 q^{18} +1.70353 q^{19} +1.44512 q^{21} -3.89025 q^{22} +0.323478 q^{23} +1.20906 q^{24} +10.4209 q^{26} -1.00000 q^{27} -1.93395 q^{28} -8.74724 q^{29} -8.45991 q^{31} -6.50828 q^{32} +2.12920 q^{33} -7.59378 q^{34} +1.33826 q^{36} -1.75170 q^{37} +3.11251 q^{38} -5.70353 q^{39} -6.87802 q^{41} +2.64037 q^{42} +11.1411 q^{43} -2.84943 q^{44} +0.591023 q^{46} -12.5982 q^{47} +4.88558 q^{48} -4.91161 q^{49} +4.15622 q^{51} +7.63282 q^{52} -8.34451 q^{53} -1.82709 q^{54} +1.74724 q^{56} -1.70353 q^{57} -15.9820 q^{58} -2.12474 q^{59} -5.38952 q^{61} -15.4570 q^{62} -1.44512 q^{63} -2.12007 q^{64} +3.89025 q^{66} +7.13078 q^{67} -5.56210 q^{68} -0.323478 q^{69} +2.67461 q^{71} -1.20906 q^{72} -6.28253 q^{73} -3.20052 q^{74} +2.27977 q^{76} +3.07697 q^{77} -10.4209 q^{78} +8.37092 q^{79} +1.00000 q^{81} -12.5668 q^{82} +14.5872 q^{83} +1.93395 q^{84} +20.3558 q^{86} +8.74724 q^{87} +2.57433 q^{88} +2.68119 q^{89} -8.24232 q^{91} +0.432897 q^{92} +8.45991 q^{93} -23.0181 q^{94} +6.50828 q^{96} -8.55105 q^{97} -8.97397 q^{98} -2.12920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} + 5 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} + 5 q^{7} - 3 q^{8} + 4 q^{9} - 6 q^{11} - q^{12} + 7 q^{13} - 10 q^{14} - 9 q^{16} - 7 q^{17} + q^{18} - 9 q^{19} - 5 q^{21} + 6 q^{22} + 10 q^{23} + 3 q^{24} - 2 q^{26} - 4 q^{27} + 5 q^{28} - 28 q^{29} - 10 q^{31} + 6 q^{33} + 7 q^{34} + q^{36} - 10 q^{37} - 6 q^{38} - 7 q^{39} + 10 q^{42} + q^{43} - 9 q^{44} + 5 q^{46} - 23 q^{47} + 9 q^{48} - 3 q^{49} + 7 q^{51} + 13 q^{52} - q^{54} + 9 q^{57} - 2 q^{58} + 4 q^{59} - 43 q^{61} - 10 q^{62} + 5 q^{63} - 7 q^{64} - 6 q^{66} + 8 q^{67} - 3 q^{68} - 10 q^{69} - 27 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} + 9 q^{76} - 15 q^{77} + 2 q^{78} + 10 q^{79} + 4 q^{81} - 20 q^{82} + 3 q^{83} - 5 q^{84} + 24 q^{86} + 28 q^{87} - 3 q^{88} - 9 q^{89} + 5 q^{91} - 15 q^{92} + 10 q^{93} - 22 q^{94} + 13 q^{97} - 42 q^{98} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82709 1.29195 0.645974 0.763359i \(-0.276451\pi\)
0.645974 + 0.763359i \(0.276451\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.33826 0.669131
\(5\) 0 0
\(6\) −1.82709 −0.745907
\(7\) −1.44512 −0.546206 −0.273103 0.961985i \(-0.588050\pi\)
−0.273103 + 0.961985i \(0.588050\pi\)
\(8\) −1.20906 −0.427466
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.12920 −0.641979 −0.320990 0.947083i \(-0.604015\pi\)
−0.320990 + 0.947083i \(0.604015\pi\)
\(12\) −1.33826 −0.386323
\(13\) 5.70353 1.58188 0.790938 0.611897i \(-0.209593\pi\)
0.790938 + 0.611897i \(0.209593\pi\)
\(14\) −2.64037 −0.705670
\(15\) 0 0
\(16\) −4.88558 −1.22139
\(17\) −4.15622 −1.00803 −0.504015 0.863695i \(-0.668144\pi\)
−0.504015 + 0.863695i \(0.668144\pi\)
\(18\) 1.82709 0.430649
\(19\) 1.70353 0.390817 0.195409 0.980722i \(-0.437397\pi\)
0.195409 + 0.980722i \(0.437397\pi\)
\(20\) 0 0
\(21\) 1.44512 0.315352
\(22\) −3.89025 −0.829404
\(23\) 0.323478 0.0674497 0.0337249 0.999431i \(-0.489263\pi\)
0.0337249 + 0.999431i \(0.489263\pi\)
\(24\) 1.20906 0.246798
\(25\) 0 0
\(26\) 10.4209 2.04370
\(27\) −1.00000 −0.192450
\(28\) −1.93395 −0.365483
\(29\) −8.74724 −1.62432 −0.812161 0.583434i \(-0.801709\pi\)
−0.812161 + 0.583434i \(0.801709\pi\)
\(30\) 0 0
\(31\) −8.45991 −1.51944 −0.759722 0.650248i \(-0.774665\pi\)
−0.759722 + 0.650248i \(0.774665\pi\)
\(32\) −6.50828 −1.15051
\(33\) 2.12920 0.370647
\(34\) −7.59378 −1.30232
\(35\) 0 0
\(36\) 1.33826 0.223044
\(37\) −1.75170 −0.287978 −0.143989 0.989579i \(-0.545993\pi\)
−0.143989 + 0.989579i \(0.545993\pi\)
\(38\) 3.11251 0.504916
\(39\) −5.70353 −0.913296
\(40\) 0 0
\(41\) −6.87802 −1.07417 −0.537083 0.843529i \(-0.680474\pi\)
−0.537083 + 0.843529i \(0.680474\pi\)
\(42\) 2.64037 0.407419
\(43\) 11.1411 1.69900 0.849501 0.527587i \(-0.176903\pi\)
0.849501 + 0.527587i \(0.176903\pi\)
\(44\) −2.84943 −0.429568
\(45\) 0 0
\(46\) 0.591023 0.0871416
\(47\) −12.5982 −1.83764 −0.918822 0.394673i \(-0.870858\pi\)
−0.918822 + 0.394673i \(0.870858\pi\)
\(48\) 4.88558 0.705173
\(49\) −4.91161 −0.701659
\(50\) 0 0
\(51\) 4.15622 0.581987
\(52\) 7.63282 1.05848
\(53\) −8.34451 −1.14621 −0.573103 0.819483i \(-0.694261\pi\)
−0.573103 + 0.819483i \(0.694261\pi\)
\(54\) −1.82709 −0.248636
\(55\) 0 0
\(56\) 1.74724 0.233485
\(57\) −1.70353 −0.225639
\(58\) −15.9820 −2.09854
\(59\) −2.12474 −0.276617 −0.138309 0.990389i \(-0.544167\pi\)
−0.138309 + 0.990389i \(0.544167\pi\)
\(60\) 0 0
\(61\) −5.38952 −0.690058 −0.345029 0.938592i \(-0.612131\pi\)
−0.345029 + 0.938592i \(0.612131\pi\)
\(62\) −15.4570 −1.96304
\(63\) −1.44512 −0.182069
\(64\) −2.12007 −0.265008
\(65\) 0 0
\(66\) 3.89025 0.478857
\(67\) 7.13078 0.871164 0.435582 0.900149i \(-0.356543\pi\)
0.435582 + 0.900149i \(0.356543\pi\)
\(68\) −5.56210 −0.674504
\(69\) −0.323478 −0.0389421
\(70\) 0 0
\(71\) 2.67461 0.317418 0.158709 0.987325i \(-0.449267\pi\)
0.158709 + 0.987325i \(0.449267\pi\)
\(72\) −1.20906 −0.142489
\(73\) −6.28253 −0.735315 −0.367657 0.929961i \(-0.619840\pi\)
−0.367657 + 0.929961i \(0.619840\pi\)
\(74\) −3.20052 −0.372053
\(75\) 0 0
\(76\) 2.27977 0.261508
\(77\) 3.07697 0.350653
\(78\) −10.4209 −1.17993
\(79\) 8.37092 0.941802 0.470901 0.882186i \(-0.343929\pi\)
0.470901 + 0.882186i \(0.343929\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.5668 −1.38777
\(83\) 14.5872 1.60115 0.800577 0.599230i \(-0.204527\pi\)
0.800577 + 0.599230i \(0.204527\pi\)
\(84\) 1.93395 0.211012
\(85\) 0 0
\(86\) 20.3558 2.19502
\(87\) 8.74724 0.937802
\(88\) 2.57433 0.274424
\(89\) 2.68119 0.284206 0.142103 0.989852i \(-0.454614\pi\)
0.142103 + 0.989852i \(0.454614\pi\)
\(90\) 0 0
\(91\) −8.24232 −0.864030
\(92\) 0.432897 0.0451327
\(93\) 8.45991 0.877252
\(94\) −23.0181 −2.37414
\(95\) 0 0
\(96\) 6.50828 0.664249
\(97\) −8.55105 −0.868228 −0.434114 0.900858i \(-0.642939\pi\)
−0.434114 + 0.900858i \(0.642939\pi\)
\(98\) −8.97397 −0.906507
\(99\) −2.12920 −0.213993
\(100\) 0 0
\(101\) 9.85377 0.980487 0.490243 0.871586i \(-0.336908\pi\)
0.490243 + 0.871586i \(0.336908\pi\)
\(102\) 7.59378 0.751897
\(103\) 16.1279 1.58913 0.794564 0.607180i \(-0.207699\pi\)
0.794564 + 0.607180i \(0.207699\pi\)
\(104\) −6.89590 −0.676198
\(105\) 0 0
\(106\) −15.2462 −1.48084
\(107\) 1.81198 0.175170 0.0875852 0.996157i \(-0.472085\pi\)
0.0875852 + 0.996157i \(0.472085\pi\)
\(108\) −1.33826 −0.128774
\(109\) −18.4646 −1.76859 −0.884293 0.466933i \(-0.845359\pi\)
−0.884293 + 0.466933i \(0.845359\pi\)
\(110\) 0 0
\(111\) 1.75170 0.166264
\(112\) 7.06027 0.667133
\(113\) −13.3653 −1.25730 −0.628650 0.777689i \(-0.716392\pi\)
−0.628650 + 0.777689i \(0.716392\pi\)
\(114\) −3.11251 −0.291513
\(115\) 0 0
\(116\) −11.7061 −1.08688
\(117\) 5.70353 0.527292
\(118\) −3.88209 −0.357375
\(119\) 6.00625 0.550592
\(120\) 0 0
\(121\) −6.46649 −0.587863
\(122\) −9.84715 −0.891519
\(123\) 6.87802 0.620170
\(124\) −11.3216 −1.01671
\(125\) 0 0
\(126\) −2.64037 −0.235223
\(127\) −1.01381 −0.0899608 −0.0449804 0.998988i \(-0.514323\pi\)
−0.0449804 + 0.998988i \(0.514323\pi\)
\(128\) 9.14301 0.808136
\(129\) −11.1411 −0.980919
\(130\) 0 0
\(131\) 2.08550 0.182211 0.0911055 0.995841i \(-0.470960\pi\)
0.0911055 + 0.995841i \(0.470960\pi\)
\(132\) 2.84943 0.248011
\(133\) −2.46182 −0.213467
\(134\) 13.0286 1.12550
\(135\) 0 0
\(136\) 5.02510 0.430899
\(137\) 16.9365 1.44698 0.723492 0.690333i \(-0.242536\pi\)
0.723492 + 0.690333i \(0.242536\pi\)
\(138\) −0.591023 −0.0503112
\(139\) 14.8883 1.26281 0.631406 0.775452i \(-0.282478\pi\)
0.631406 + 0.775452i \(0.282478\pi\)
\(140\) 0 0
\(141\) 12.5982 1.06096
\(142\) 4.88676 0.410088
\(143\) −12.1440 −1.01553
\(144\) −4.88558 −0.407132
\(145\) 0 0
\(146\) −11.4788 −0.949989
\(147\) 4.91161 0.405103
\(148\) −2.34424 −0.192695
\(149\) −4.70991 −0.385851 −0.192925 0.981213i \(-0.561798\pi\)
−0.192925 + 0.981213i \(0.561798\pi\)
\(150\) 0 0
\(151\) −9.63091 −0.783752 −0.391876 0.920018i \(-0.628174\pi\)
−0.391876 + 0.920018i \(0.628174\pi\)
\(152\) −2.05967 −0.167061
\(153\) −4.15622 −0.336010
\(154\) 5.62190 0.453025
\(155\) 0 0
\(156\) −7.63282 −0.611114
\(157\) 18.5557 1.48091 0.740454 0.672107i \(-0.234611\pi\)
0.740454 + 0.672107i \(0.234611\pi\)
\(158\) 15.2944 1.21676
\(159\) 8.34451 0.661763
\(160\) 0 0
\(161\) −0.467465 −0.0368414
\(162\) 1.82709 0.143550
\(163\) 0.451705 0.0353803 0.0176902 0.999844i \(-0.494369\pi\)
0.0176902 + 0.999844i \(0.494369\pi\)
\(164\) −9.20459 −0.718758
\(165\) 0 0
\(166\) 26.6521 2.06861
\(167\) 4.72648 0.365746 0.182873 0.983137i \(-0.441460\pi\)
0.182873 + 0.983137i \(0.441460\pi\)
\(168\) −1.74724 −0.134802
\(169\) 19.5303 1.50233
\(170\) 0 0
\(171\) 1.70353 0.130272
\(172\) 14.9097 1.13685
\(173\) 2.29489 0.174477 0.0872385 0.996187i \(-0.472196\pi\)
0.0872385 + 0.996187i \(0.472196\pi\)
\(174\) 15.9820 1.21159
\(175\) 0 0
\(176\) 10.4024 0.784110
\(177\) 2.12474 0.159705
\(178\) 4.89878 0.367179
\(179\) 2.68757 0.200878 0.100439 0.994943i \(-0.467975\pi\)
0.100439 + 0.994943i \(0.467975\pi\)
\(180\) 0 0
\(181\) −23.9088 −1.77712 −0.888562 0.458756i \(-0.848295\pi\)
−0.888562 + 0.458756i \(0.848295\pi\)
\(182\) −15.0595 −1.11628
\(183\) 5.38952 0.398405
\(184\) −0.391103 −0.0288325
\(185\) 0 0
\(186\) 15.4570 1.13336
\(187\) 8.84943 0.647135
\(188\) −16.8597 −1.22962
\(189\) 1.44512 0.105117
\(190\) 0 0
\(191\) −3.91824 −0.283514 −0.141757 0.989902i \(-0.545275\pi\)
−0.141757 + 0.989902i \(0.545275\pi\)
\(192\) 2.12007 0.153003
\(193\) −8.54752 −0.615264 −0.307632 0.951505i \(-0.599537\pi\)
−0.307632 + 0.951505i \(0.599537\pi\)
\(194\) −15.6236 −1.12171
\(195\) 0 0
\(196\) −6.57302 −0.469502
\(197\) 2.45235 0.174723 0.0873614 0.996177i \(-0.472157\pi\)
0.0873614 + 0.996177i \(0.472157\pi\)
\(198\) −3.89025 −0.276468
\(199\) −10.4345 −0.739680 −0.369840 0.929095i \(-0.620588\pi\)
−0.369840 + 0.929095i \(0.620588\pi\)
\(200\) 0 0
\(201\) −7.13078 −0.502967
\(202\) 18.0037 1.26674
\(203\) 12.6409 0.887214
\(204\) 5.56210 0.389425
\(205\) 0 0
\(206\) 29.4671 2.05307
\(207\) 0.323478 0.0224832
\(208\) −27.8651 −1.93209
\(209\) −3.62717 −0.250897
\(210\) 0 0
\(211\) 19.2618 1.32604 0.663018 0.748604i \(-0.269275\pi\)
0.663018 + 0.748604i \(0.269275\pi\)
\(212\) −11.1671 −0.766962
\(213\) −2.67461 −0.183261
\(214\) 3.31065 0.226311
\(215\) 0 0
\(216\) 1.20906 0.0822659
\(217\) 12.2256 0.829929
\(218\) −33.7365 −2.28492
\(219\) 6.28253 0.424534
\(220\) 0 0
\(221\) −23.7051 −1.59458
\(222\) 3.20052 0.214805
\(223\) 7.61706 0.510076 0.255038 0.966931i \(-0.417912\pi\)
0.255038 + 0.966931i \(0.417912\pi\)
\(224\) 9.40528 0.628417
\(225\) 0 0
\(226\) −24.4196 −1.62437
\(227\) −1.31803 −0.0874810 −0.0437405 0.999043i \(-0.513927\pi\)
−0.0437405 + 0.999043i \(0.513927\pi\)
\(228\) −2.27977 −0.150982
\(229\) 1.12632 0.0744292 0.0372146 0.999307i \(-0.488151\pi\)
0.0372146 + 0.999307i \(0.488151\pi\)
\(230\) 0 0
\(231\) −3.07697 −0.202450
\(232\) 10.5759 0.694342
\(233\) 26.6071 1.74309 0.871543 0.490319i \(-0.163120\pi\)
0.871543 + 0.490319i \(0.163120\pi\)
\(234\) 10.4209 0.681234
\(235\) 0 0
\(236\) −2.84345 −0.185093
\(237\) −8.37092 −0.543750
\(238\) 10.9740 0.711337
\(239\) −24.8692 −1.60865 −0.804326 0.594188i \(-0.797473\pi\)
−0.804326 + 0.594188i \(0.797473\pi\)
\(240\) 0 0
\(241\) −12.3532 −0.795743 −0.397871 0.917441i \(-0.630251\pi\)
−0.397871 + 0.917441i \(0.630251\pi\)
\(242\) −11.8149 −0.759488
\(243\) −1.00000 −0.0641500
\(244\) −7.21259 −0.461739
\(245\) 0 0
\(246\) 12.5668 0.801228
\(247\) 9.71616 0.618224
\(248\) 10.2285 0.649511
\(249\) −14.5872 −0.924426
\(250\) 0 0
\(251\) −29.0986 −1.83669 −0.918343 0.395786i \(-0.870472\pi\)
−0.918343 + 0.395786i \(0.870472\pi\)
\(252\) −1.93395 −0.121828
\(253\) −0.688750 −0.0433013
\(254\) −1.85232 −0.116225
\(255\) 0 0
\(256\) 20.9452 1.30908
\(257\) −12.0500 −0.751656 −0.375828 0.926690i \(-0.622642\pi\)
−0.375828 + 0.926690i \(0.622642\pi\)
\(258\) −20.3558 −1.26730
\(259\) 2.53143 0.157296
\(260\) 0 0
\(261\) −8.74724 −0.541440
\(262\) 3.81040 0.235407
\(263\) −1.02892 −0.0634460 −0.0317230 0.999497i \(-0.510099\pi\)
−0.0317230 + 0.999497i \(0.510099\pi\)
\(264\) −2.57433 −0.158439
\(265\) 0 0
\(266\) −4.49797 −0.275788
\(267\) −2.68119 −0.164086
\(268\) 9.54285 0.582922
\(269\) 23.9072 1.45765 0.728824 0.684701i \(-0.240067\pi\)
0.728824 + 0.684701i \(0.240067\pi\)
\(270\) 0 0
\(271\) 3.72214 0.226104 0.113052 0.993589i \(-0.463937\pi\)
0.113052 + 0.993589i \(0.463937\pi\)
\(272\) 20.3055 1.23120
\(273\) 8.24232 0.498848
\(274\) 30.9445 1.86943
\(275\) 0 0
\(276\) −0.432897 −0.0260574
\(277\) −17.5802 −1.05629 −0.528147 0.849153i \(-0.677113\pi\)
−0.528147 + 0.849153i \(0.677113\pi\)
\(278\) 27.2024 1.63149
\(279\) −8.45991 −0.506481
\(280\) 0 0
\(281\) 13.2536 0.790644 0.395322 0.918543i \(-0.370633\pi\)
0.395322 + 0.918543i \(0.370633\pi\)
\(282\) 23.0181 1.37071
\(283\) 12.0996 0.719249 0.359624 0.933097i \(-0.382905\pi\)
0.359624 + 0.933097i \(0.382905\pi\)
\(284\) 3.57933 0.212394
\(285\) 0 0
\(286\) −22.1882 −1.31201
\(287\) 9.93960 0.586716
\(288\) −6.50828 −0.383504
\(289\) 0.274126 0.0161251
\(290\) 0 0
\(291\) 8.55105 0.501272
\(292\) −8.40767 −0.492022
\(293\) 3.61673 0.211291 0.105646 0.994404i \(-0.466309\pi\)
0.105646 + 0.994404i \(0.466309\pi\)
\(294\) 8.97397 0.523372
\(295\) 0 0
\(296\) 2.11791 0.123101
\(297\) 2.12920 0.123549
\(298\) −8.60543 −0.498499
\(299\) 1.84497 0.106697
\(300\) 0 0
\(301\) −16.1003 −0.928005
\(302\) −17.5965 −1.01257
\(303\) −9.85377 −0.566084
\(304\) −8.32275 −0.477342
\(305\) 0 0
\(306\) −7.59378 −0.434108
\(307\) 14.8273 0.846238 0.423119 0.906074i \(-0.360935\pi\)
0.423119 + 0.906074i \(0.360935\pi\)
\(308\) 4.11778 0.234633
\(309\) −16.1279 −0.917484
\(310\) 0 0
\(311\) −6.61579 −0.375147 −0.187574 0.982251i \(-0.560062\pi\)
−0.187574 + 0.982251i \(0.560062\pi\)
\(312\) 6.89590 0.390403
\(313\) −7.17823 −0.405737 −0.202869 0.979206i \(-0.565026\pi\)
−0.202869 + 0.979206i \(0.565026\pi\)
\(314\) 33.9030 1.91326
\(315\) 0 0
\(316\) 11.2025 0.630189
\(317\) −9.75255 −0.547758 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(318\) 15.2462 0.854963
\(319\) 18.6247 1.04278
\(320\) 0 0
\(321\) −1.81198 −0.101135
\(322\) −0.854102 −0.0475972
\(323\) −7.08025 −0.393956
\(324\) 1.33826 0.0743478
\(325\) 0 0
\(326\) 0.825307 0.0457095
\(327\) 18.4646 1.02109
\(328\) 8.31592 0.459170
\(329\) 18.2060 1.00373
\(330\) 0 0
\(331\) 6.68822 0.367618 0.183809 0.982962i \(-0.441157\pi\)
0.183809 + 0.982962i \(0.441157\pi\)
\(332\) 19.5215 1.07138
\(333\) −1.75170 −0.0959928
\(334\) 8.63570 0.472525
\(335\) 0 0
\(336\) −7.06027 −0.385169
\(337\) −20.8191 −1.13409 −0.567045 0.823687i \(-0.691914\pi\)
−0.567045 + 0.823687i \(0.691914\pi\)
\(338\) 35.6836 1.94093
\(339\) 13.3653 0.725902
\(340\) 0 0
\(341\) 18.0129 0.975452
\(342\) 3.11251 0.168305
\(343\) 17.2138 0.929456
\(344\) −13.4702 −0.726266
\(345\) 0 0
\(346\) 4.19297 0.225415
\(347\) −30.9907 −1.66367 −0.831835 0.555023i \(-0.812709\pi\)
−0.831835 + 0.555023i \(0.812709\pi\)
\(348\) 11.7061 0.627512
\(349\) −7.45484 −0.399048 −0.199524 0.979893i \(-0.563940\pi\)
−0.199524 + 0.979893i \(0.563940\pi\)
\(350\) 0 0
\(351\) −5.70353 −0.304432
\(352\) 13.8575 0.738605
\(353\) −3.68948 −0.196371 −0.0981856 0.995168i \(-0.531304\pi\)
−0.0981856 + 0.995168i \(0.531304\pi\)
\(354\) 3.88209 0.206331
\(355\) 0 0
\(356\) 3.58814 0.190171
\(357\) −6.00625 −0.317884
\(358\) 4.91043 0.259524
\(359\) −9.92989 −0.524079 −0.262040 0.965057i \(-0.584395\pi\)
−0.262040 + 0.965057i \(0.584395\pi\)
\(360\) 0 0
\(361\) −16.0980 −0.847262
\(362\) −43.6835 −2.29595
\(363\) 6.46649 0.339403
\(364\) −11.0304 −0.578149
\(365\) 0 0
\(366\) 9.84715 0.514719
\(367\) −12.5507 −0.655142 −0.327571 0.944826i \(-0.606230\pi\)
−0.327571 + 0.944826i \(0.606230\pi\)
\(368\) −1.58038 −0.0823828
\(369\) −6.87802 −0.358056
\(370\) 0 0
\(371\) 12.0589 0.626065
\(372\) 11.3216 0.586996
\(373\) 14.8576 0.769299 0.384650 0.923063i \(-0.374322\pi\)
0.384650 + 0.923063i \(0.374322\pi\)
\(374\) 16.1687 0.836064
\(375\) 0 0
\(376\) 15.2320 0.785530
\(377\) −49.8902 −2.56947
\(378\) 2.64037 0.135806
\(379\) 11.2957 0.580223 0.290112 0.956993i \(-0.406308\pi\)
0.290112 + 0.956993i \(0.406308\pi\)
\(380\) 0 0
\(381\) 1.01381 0.0519389
\(382\) −7.15898 −0.366285
\(383\) −3.04968 −0.155831 −0.0779157 0.996960i \(-0.524827\pi\)
−0.0779157 + 0.996960i \(0.524827\pi\)
\(384\) −9.14301 −0.466577
\(385\) 0 0
\(386\) −15.6171 −0.794889
\(387\) 11.1411 0.566334
\(388\) −11.4435 −0.580958
\(389\) −21.3939 −1.08472 −0.542358 0.840148i \(-0.682468\pi\)
−0.542358 + 0.840148i \(0.682468\pi\)
\(390\) 0 0
\(391\) −1.34444 −0.0679914
\(392\) 5.93842 0.299936
\(393\) −2.08550 −0.105200
\(394\) 4.48067 0.225733
\(395\) 0 0
\(396\) −2.84943 −0.143189
\(397\) 34.4305 1.72802 0.864009 0.503476i \(-0.167946\pi\)
0.864009 + 0.503476i \(0.167946\pi\)
\(398\) −19.0647 −0.955629
\(399\) 2.46182 0.123245
\(400\) 0 0
\(401\) 36.2976 1.81262 0.906309 0.422616i \(-0.138888\pi\)
0.906309 + 0.422616i \(0.138888\pi\)
\(402\) −13.0286 −0.649807
\(403\) −48.2514 −2.40357
\(404\) 13.1869 0.656074
\(405\) 0 0
\(406\) 23.0960 1.14623
\(407\) 3.72974 0.184876
\(408\) −5.02510 −0.248780
\(409\) 24.9896 1.23566 0.617828 0.786313i \(-0.288013\pi\)
0.617828 + 0.786313i \(0.288013\pi\)
\(410\) 0 0
\(411\) −16.9365 −0.835416
\(412\) 21.5833 1.06333
\(413\) 3.07051 0.151090
\(414\) 0.591023 0.0290472
\(415\) 0 0
\(416\) −37.1202 −1.81997
\(417\) −14.8883 −0.729085
\(418\) −6.62717 −0.324146
\(419\) 2.70313 0.132057 0.0660284 0.997818i \(-0.478967\pi\)
0.0660284 + 0.997818i \(0.478967\pi\)
\(420\) 0 0
\(421\) 26.6496 1.29882 0.649411 0.760438i \(-0.275016\pi\)
0.649411 + 0.760438i \(0.275016\pi\)
\(422\) 35.1930 1.71317
\(423\) −12.5982 −0.612548
\(424\) 10.0890 0.489965
\(425\) 0 0
\(426\) −4.88676 −0.236764
\(427\) 7.78853 0.376914
\(428\) 2.42490 0.117212
\(429\) 12.1440 0.586317
\(430\) 0 0
\(431\) 33.6242 1.61962 0.809811 0.586691i \(-0.199570\pi\)
0.809811 + 0.586691i \(0.199570\pi\)
\(432\) 4.88558 0.235058
\(433\) 5.34260 0.256749 0.128375 0.991726i \(-0.459024\pi\)
0.128375 + 0.991726i \(0.459024\pi\)
\(434\) 22.3373 1.07223
\(435\) 0 0
\(436\) −24.7104 −1.18341
\(437\) 0.551055 0.0263605
\(438\) 11.4788 0.548476
\(439\) 26.9690 1.28716 0.643580 0.765379i \(-0.277448\pi\)
0.643580 + 0.765379i \(0.277448\pi\)
\(440\) 0 0
\(441\) −4.91161 −0.233886
\(442\) −43.3114 −2.06011
\(443\) 16.8670 0.801374 0.400687 0.916215i \(-0.368771\pi\)
0.400687 + 0.916215i \(0.368771\pi\)
\(444\) 2.34424 0.111253
\(445\) 0 0
\(446\) 13.9171 0.658992
\(447\) 4.70991 0.222771
\(448\) 3.06376 0.144749
\(449\) 21.5942 1.01909 0.509546 0.860443i \(-0.329813\pi\)
0.509546 + 0.860443i \(0.329813\pi\)
\(450\) 0 0
\(451\) 14.6447 0.689593
\(452\) −17.8862 −0.841297
\(453\) 9.63091 0.452500
\(454\) −2.40817 −0.113021
\(455\) 0 0
\(456\) 2.05967 0.0964528
\(457\) −5.60699 −0.262284 −0.131142 0.991364i \(-0.541864\pi\)
−0.131142 + 0.991364i \(0.541864\pi\)
\(458\) 2.05788 0.0961587
\(459\) 4.15622 0.193996
\(460\) 0 0
\(461\) −9.54504 −0.444557 −0.222278 0.974983i \(-0.571349\pi\)
−0.222278 + 0.974983i \(0.571349\pi\)
\(462\) −5.62190 −0.261554
\(463\) −23.5262 −1.09336 −0.546678 0.837343i \(-0.684108\pi\)
−0.546678 + 0.837343i \(0.684108\pi\)
\(464\) 42.7353 1.98394
\(465\) 0 0
\(466\) 48.6135 2.25198
\(467\) −36.5834 −1.69288 −0.846439 0.532486i \(-0.821258\pi\)
−0.846439 + 0.532486i \(0.821258\pi\)
\(468\) 7.63282 0.352827
\(469\) −10.3049 −0.475835
\(470\) 0 0
\(471\) −18.5557 −0.855003
\(472\) 2.56893 0.118245
\(473\) −23.7217 −1.09072
\(474\) −15.2944 −0.702496
\(475\) 0 0
\(476\) 8.03793 0.368418
\(477\) −8.34451 −0.382069
\(478\) −45.4382 −2.07830
\(479\) 29.2750 1.33761 0.668804 0.743439i \(-0.266807\pi\)
0.668804 + 0.743439i \(0.266807\pi\)
\(480\) 0 0
\(481\) −9.99091 −0.455546
\(482\) −22.5705 −1.02806
\(483\) 0.467465 0.0212704
\(484\) −8.65385 −0.393357
\(485\) 0 0
\(486\) −1.82709 −0.0828785
\(487\) −21.2869 −0.964600 −0.482300 0.876006i \(-0.660199\pi\)
−0.482300 + 0.876006i \(0.660199\pi\)
\(488\) 6.51624 0.294976
\(489\) −0.451705 −0.0204268
\(490\) 0 0
\(491\) −33.1940 −1.49803 −0.749013 0.662556i \(-0.769472\pi\)
−0.749013 + 0.662556i \(0.769472\pi\)
\(492\) 9.20459 0.414975
\(493\) 36.3554 1.63737
\(494\) 17.7523 0.798714
\(495\) 0 0
\(496\) 41.3316 1.85584
\(497\) −3.86515 −0.173376
\(498\) −26.6521 −1.19431
\(499\) 4.72872 0.211686 0.105843 0.994383i \(-0.466246\pi\)
0.105843 + 0.994383i \(0.466246\pi\)
\(500\) 0 0
\(501\) −4.72648 −0.211163
\(502\) −53.1657 −2.37290
\(503\) −0.400444 −0.0178549 −0.00892745 0.999960i \(-0.502842\pi\)
−0.00892745 + 0.999960i \(0.502842\pi\)
\(504\) 1.74724 0.0778282
\(505\) 0 0
\(506\) −1.25841 −0.0559431
\(507\) −19.5303 −0.867371
\(508\) −1.35674 −0.0601956
\(509\) −27.6104 −1.22381 −0.611904 0.790932i \(-0.709596\pi\)
−0.611904 + 0.790932i \(0.709596\pi\)
\(510\) 0 0
\(511\) 9.07905 0.401633
\(512\) 19.9828 0.883126
\(513\) −1.70353 −0.0752128
\(514\) −22.0164 −0.971100
\(515\) 0 0
\(516\) −14.9097 −0.656363
\(517\) 26.8242 1.17973
\(518\) 4.62516 0.203218
\(519\) −2.29489 −0.100734
\(520\) 0 0
\(521\) −28.7374 −1.25901 −0.629505 0.776996i \(-0.716742\pi\)
−0.629505 + 0.776996i \(0.716742\pi\)
\(522\) −15.9820 −0.699513
\(523\) 18.0660 0.789969 0.394985 0.918688i \(-0.370750\pi\)
0.394985 + 0.918688i \(0.370750\pi\)
\(524\) 2.79094 0.121923
\(525\) 0 0
\(526\) −1.87993 −0.0819690
\(527\) 35.1612 1.53165
\(528\) −10.4024 −0.452706
\(529\) −22.8954 −0.995451
\(530\) 0 0
\(531\) −2.12474 −0.0922058
\(532\) −3.29456 −0.142837
\(533\) −39.2290 −1.69920
\(534\) −4.89878 −0.211991
\(535\) 0 0
\(536\) −8.62152 −0.372393
\(537\) −2.68757 −0.115977
\(538\) 43.6807 1.88321
\(539\) 10.4578 0.450451
\(540\) 0 0
\(541\) −5.26032 −0.226159 −0.113079 0.993586i \(-0.536071\pi\)
−0.113079 + 0.993586i \(0.536071\pi\)
\(542\) 6.80068 0.292114
\(543\) 23.9088 1.02602
\(544\) 27.0498 1.15975
\(545\) 0 0
\(546\) 15.0595 0.644486
\(547\) −36.7888 −1.57297 −0.786487 0.617607i \(-0.788102\pi\)
−0.786487 + 0.617607i \(0.788102\pi\)
\(548\) 22.6655 0.968221
\(549\) −5.38952 −0.230019
\(550\) 0 0
\(551\) −14.9012 −0.634813
\(552\) 0.391103 0.0166464
\(553\) −12.0970 −0.514418
\(554\) −32.1207 −1.36468
\(555\) 0 0
\(556\) 19.9245 0.844986
\(557\) 23.3475 0.989266 0.494633 0.869102i \(-0.335302\pi\)
0.494633 + 0.869102i \(0.335302\pi\)
\(558\) −15.4570 −0.654348
\(559\) 63.5436 2.68761
\(560\) 0 0
\(561\) −8.84943 −0.373623
\(562\) 24.2156 1.02147
\(563\) −29.2369 −1.23219 −0.616095 0.787672i \(-0.711286\pi\)
−0.616095 + 0.787672i \(0.711286\pi\)
\(564\) 16.8597 0.709923
\(565\) 0 0
\(566\) 22.1071 0.929232
\(567\) −1.44512 −0.0606895
\(568\) −3.23376 −0.135685
\(569\) −20.2530 −0.849051 −0.424526 0.905416i \(-0.639559\pi\)
−0.424526 + 0.905416i \(0.639559\pi\)
\(570\) 0 0
\(571\) −4.57564 −0.191484 −0.0957422 0.995406i \(-0.530522\pi\)
−0.0957422 + 0.995406i \(0.530522\pi\)
\(572\) −16.2518 −0.679523
\(573\) 3.91824 0.163687
\(574\) 18.1606 0.758007
\(575\) 0 0
\(576\) −2.12007 −0.0883361
\(577\) −31.3502 −1.30513 −0.652564 0.757734i \(-0.726306\pi\)
−0.652564 + 0.757734i \(0.726306\pi\)
\(578\) 0.500853 0.0208327
\(579\) 8.54752 0.355223
\(580\) 0 0
\(581\) −21.0803 −0.874559
\(582\) 15.6236 0.647617
\(583\) 17.7672 0.735841
\(584\) 7.59594 0.314322
\(585\) 0 0
\(586\) 6.60809 0.272978
\(587\) −4.67218 −0.192842 −0.0964208 0.995341i \(-0.530739\pi\)
−0.0964208 + 0.995341i \(0.530739\pi\)
\(588\) 6.57302 0.271067
\(589\) −14.4117 −0.593825
\(590\) 0 0
\(591\) −2.45235 −0.100876
\(592\) 8.55809 0.351735
\(593\) −7.37978 −0.303051 −0.151526 0.988453i \(-0.548419\pi\)
−0.151526 + 0.988453i \(0.548419\pi\)
\(594\) 3.89025 0.159619
\(595\) 0 0
\(596\) −6.30309 −0.258185
\(597\) 10.4345 0.427055
\(598\) 3.37092 0.137847
\(599\) −46.7505 −1.91018 −0.955088 0.296323i \(-0.904239\pi\)
−0.955088 + 0.296323i \(0.904239\pi\)
\(600\) 0 0
\(601\) 32.3225 1.31846 0.659230 0.751941i \(-0.270882\pi\)
0.659230 + 0.751941i \(0.270882\pi\)
\(602\) −29.4167 −1.19893
\(603\) 7.13078 0.290388
\(604\) −12.8887 −0.524433
\(605\) 0 0
\(606\) −18.0037 −0.731352
\(607\) −14.0891 −0.571858 −0.285929 0.958251i \(-0.592302\pi\)
−0.285929 + 0.958251i \(0.592302\pi\)
\(608\) −11.0871 −0.449640
\(609\) −12.6409 −0.512233
\(610\) 0 0
\(611\) −71.8545 −2.90692
\(612\) −5.56210 −0.224835
\(613\) −12.2959 −0.496625 −0.248313 0.968680i \(-0.579876\pi\)
−0.248313 + 0.968680i \(0.579876\pi\)
\(614\) 27.0908 1.09330
\(615\) 0 0
\(616\) −3.72023 −0.149892
\(617\) −1.96969 −0.0792969 −0.0396485 0.999214i \(-0.512624\pi\)
−0.0396485 + 0.999214i \(0.512624\pi\)
\(618\) −29.4671 −1.18534
\(619\) 14.9714 0.601749 0.300875 0.953664i \(-0.402721\pi\)
0.300875 + 0.953664i \(0.402721\pi\)
\(620\) 0 0
\(621\) −0.323478 −0.0129807
\(622\) −12.0877 −0.484671
\(623\) −3.87466 −0.155235
\(624\) 27.8651 1.11550
\(625\) 0 0
\(626\) −13.1153 −0.524192
\(627\) 3.62717 0.144855
\(628\) 24.8324 0.990921
\(629\) 7.28046 0.290291
\(630\) 0 0
\(631\) 18.0729 0.719472 0.359736 0.933054i \(-0.382867\pi\)
0.359736 + 0.933054i \(0.382867\pi\)
\(632\) −10.1209 −0.402588
\(633\) −19.2618 −0.765587
\(634\) −17.8188 −0.707675
\(635\) 0 0
\(636\) 11.1671 0.442806
\(637\) −28.0136 −1.10994
\(638\) 34.0289 1.34722
\(639\) 2.67461 0.105806
\(640\) 0 0
\(641\) 12.2381 0.483374 0.241687 0.970354i \(-0.422299\pi\)
0.241687 + 0.970354i \(0.422299\pi\)
\(642\) −3.31065 −0.130661
\(643\) −16.4453 −0.648540 −0.324270 0.945964i \(-0.605119\pi\)
−0.324270 + 0.945964i \(0.605119\pi\)
\(644\) −0.625591 −0.0246517
\(645\) 0 0
\(646\) −12.9363 −0.508971
\(647\) 17.3991 0.684028 0.342014 0.939695i \(-0.388891\pi\)
0.342014 + 0.939695i \(0.388891\pi\)
\(648\) −1.20906 −0.0474962
\(649\) 4.52400 0.177583
\(650\) 0 0
\(651\) −12.2256 −0.479160
\(652\) 0.604500 0.0236740
\(653\) −28.4645 −1.11390 −0.556950 0.830546i \(-0.688029\pi\)
−0.556950 + 0.830546i \(0.688029\pi\)
\(654\) 33.7365 1.31920
\(655\) 0 0
\(656\) 33.6031 1.31198
\(657\) −6.28253 −0.245105
\(658\) 33.2641 1.29677
\(659\) −1.51333 −0.0589509 −0.0294754 0.999566i \(-0.509384\pi\)
−0.0294754 + 0.999566i \(0.509384\pi\)
\(660\) 0 0
\(661\) 26.7842 1.04179 0.520893 0.853622i \(-0.325599\pi\)
0.520893 + 0.853622i \(0.325599\pi\)
\(662\) 12.2200 0.474943
\(663\) 23.7051 0.920630
\(664\) −17.6368 −0.684439
\(665\) 0 0
\(666\) −3.20052 −0.124018
\(667\) −2.82954 −0.109560
\(668\) 6.32526 0.244732
\(669\) −7.61706 −0.294492
\(670\) 0 0
\(671\) 11.4754 0.443003
\(672\) −9.40528 −0.362817
\(673\) 9.31141 0.358929 0.179464 0.983764i \(-0.442564\pi\)
0.179464 + 0.983764i \(0.442564\pi\)
\(674\) −38.0385 −1.46519
\(675\) 0 0
\(676\) 26.1366 1.00526
\(677\) 24.7854 0.952579 0.476290 0.879288i \(-0.341981\pi\)
0.476290 + 0.879288i \(0.341981\pi\)
\(678\) 24.4196 0.937828
\(679\) 12.3573 0.474231
\(680\) 0 0
\(681\) 1.31803 0.0505072
\(682\) 32.9112 1.26023
\(683\) −37.3864 −1.43055 −0.715276 0.698842i \(-0.753699\pi\)
−0.715276 + 0.698842i \(0.753699\pi\)
\(684\) 2.27977 0.0871693
\(685\) 0 0
\(686\) 31.4511 1.20081
\(687\) −1.12632 −0.0429717
\(688\) −54.4307 −2.07515
\(689\) −47.5932 −1.81316
\(690\) 0 0
\(691\) −32.8909 −1.25123 −0.625615 0.780132i \(-0.715152\pi\)
−0.625615 + 0.780132i \(0.715152\pi\)
\(692\) 3.07116 0.116748
\(693\) 3.07697 0.116884
\(694\) −56.6229 −2.14938
\(695\) 0 0
\(696\) −10.5759 −0.400879
\(697\) 28.5865 1.08279
\(698\) −13.6207 −0.515550
\(699\) −26.6071 −1.00637
\(700\) 0 0
\(701\) −42.3336 −1.59892 −0.799459 0.600721i \(-0.794880\pi\)
−0.799459 + 0.600721i \(0.794880\pi\)
\(702\) −10.4209 −0.393311
\(703\) −2.98409 −0.112547
\(704\) 4.51406 0.170130
\(705\) 0 0
\(706\) −6.74101 −0.253701
\(707\) −14.2399 −0.535548
\(708\) 2.84345 0.106864
\(709\) 18.3593 0.689498 0.344749 0.938695i \(-0.387964\pi\)
0.344749 + 0.938695i \(0.387964\pi\)
\(710\) 0 0
\(711\) 8.37092 0.313934
\(712\) −3.24171 −0.121488
\(713\) −2.73659 −0.102486
\(714\) −10.9740 −0.410690
\(715\) 0 0
\(716\) 3.59667 0.134414
\(717\) 24.8692 0.928756
\(718\) −18.1428 −0.677084
\(719\) −19.2706 −0.718671 −0.359336 0.933208i \(-0.616997\pi\)
−0.359336 + 0.933208i \(0.616997\pi\)
\(720\) 0 0
\(721\) −23.3068 −0.867992
\(722\) −29.4125 −1.09462
\(723\) 12.3532 0.459422
\(724\) −31.9962 −1.18913
\(725\) 0 0
\(726\) 11.8149 0.438491
\(727\) 17.4238 0.646214 0.323107 0.946362i \(-0.395273\pi\)
0.323107 + 0.946362i \(0.395273\pi\)
\(728\) 9.96543 0.369343
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −46.3048 −1.71265
\(732\) 7.21259 0.266585
\(733\) 24.2010 0.893883 0.446942 0.894563i \(-0.352513\pi\)
0.446942 + 0.894563i \(0.352513\pi\)
\(734\) −22.9313 −0.846410
\(735\) 0 0
\(736\) −2.10528 −0.0776018
\(737\) −15.1829 −0.559269
\(738\) −12.5668 −0.462589
\(739\) −38.5798 −1.41918 −0.709591 0.704613i \(-0.751120\pi\)
−0.709591 + 0.704613i \(0.751120\pi\)
\(740\) 0 0
\(741\) −9.71616 −0.356932
\(742\) 22.0326 0.808844
\(743\) −33.8580 −1.24213 −0.621065 0.783759i \(-0.713300\pi\)
−0.621065 + 0.783759i \(0.713300\pi\)
\(744\) −10.2285 −0.374995
\(745\) 0 0
\(746\) 27.1462 0.993895
\(747\) 14.5872 0.533718
\(748\) 11.8429 0.433018
\(749\) −2.61853 −0.0956791
\(750\) 0 0
\(751\) −45.9138 −1.67542 −0.837710 0.546115i \(-0.816106\pi\)
−0.837710 + 0.546115i \(0.816106\pi\)
\(752\) 61.5497 2.24449
\(753\) 29.0986 1.06041
\(754\) −91.1539 −3.31963
\(755\) 0 0
\(756\) 1.93395 0.0703372
\(757\) 10.3561 0.376400 0.188200 0.982131i \(-0.439735\pi\)
0.188200 + 0.982131i \(0.439735\pi\)
\(758\) 20.6383 0.749618
\(759\) 0.688750 0.0250000
\(760\) 0 0
\(761\) 49.2652 1.78586 0.892931 0.450193i \(-0.148645\pi\)
0.892931 + 0.450193i \(0.148645\pi\)
\(762\) 1.85232 0.0671024
\(763\) 26.6836 0.966012
\(764\) −5.24362 −0.189708
\(765\) 0 0
\(766\) −5.57205 −0.201326
\(767\) −12.1185 −0.437574
\(768\) −20.9452 −0.755797
\(769\) −1.26208 −0.0455116 −0.0227558 0.999741i \(-0.507244\pi\)
−0.0227558 + 0.999741i \(0.507244\pi\)
\(770\) 0 0
\(771\) 12.0500 0.433969
\(772\) −11.4388 −0.411692
\(773\) −26.9903 −0.970772 −0.485386 0.874300i \(-0.661321\pi\)
−0.485386 + 0.874300i \(0.661321\pi\)
\(774\) 20.3558 0.731674
\(775\) 0 0
\(776\) 10.3387 0.371138
\(777\) −2.53143 −0.0908146
\(778\) −39.0887 −1.40140
\(779\) −11.7169 −0.419803
\(780\) 0 0
\(781\) −5.69480 −0.203776
\(782\) −2.45642 −0.0878413
\(783\) 8.74724 0.312601
\(784\) 23.9961 0.857003
\(785\) 0 0
\(786\) −3.81040 −0.135912
\(787\) 7.62303 0.271732 0.135866 0.990727i \(-0.456618\pi\)
0.135866 + 0.990727i \(0.456618\pi\)
\(788\) 3.28189 0.116912
\(789\) 1.02892 0.0366306
\(790\) 0 0
\(791\) 19.3145 0.686744
\(792\) 2.57433 0.0914748
\(793\) −30.7393 −1.09159
\(794\) 62.9077 2.23251
\(795\) 0 0
\(796\) −13.9641 −0.494943
\(797\) 4.98266 0.176495 0.0882474 0.996099i \(-0.471873\pi\)
0.0882474 + 0.996099i \(0.471873\pi\)
\(798\) 4.49797 0.159226
\(799\) 52.3610 1.85240
\(800\) 0 0
\(801\) 2.68119 0.0947353
\(802\) 66.3191 2.34181
\(803\) 13.3768 0.472057
\(804\) −9.54285 −0.336550
\(805\) 0 0
\(806\) −88.1596 −3.10529
\(807\) −23.9072 −0.841574
\(808\) −11.9138 −0.419125
\(809\) 5.74271 0.201903 0.100952 0.994891i \(-0.467811\pi\)
0.100952 + 0.994891i \(0.467811\pi\)
\(810\) 0 0
\(811\) 38.8021 1.36253 0.681264 0.732038i \(-0.261431\pi\)
0.681264 + 0.732038i \(0.261431\pi\)
\(812\) 16.9168 0.593662
\(813\) −3.72214 −0.130541
\(814\) 6.81457 0.238851
\(815\) 0 0
\(816\) −20.3055 −0.710835
\(817\) 18.9792 0.664000
\(818\) 45.6583 1.59640
\(819\) −8.24232 −0.288010
\(820\) 0 0
\(821\) −41.8919 −1.46204 −0.731019 0.682357i \(-0.760955\pi\)
−0.731019 + 0.682357i \(0.760955\pi\)
\(822\) −30.9445 −1.07931
\(823\) −29.2212 −1.01859 −0.509294 0.860593i \(-0.670093\pi\)
−0.509294 + 0.860593i \(0.670093\pi\)
\(824\) −19.4995 −0.679299
\(825\) 0 0
\(826\) 5.61010 0.195200
\(827\) 7.41934 0.257996 0.128998 0.991645i \(-0.458824\pi\)
0.128998 + 0.991645i \(0.458824\pi\)
\(828\) 0.432897 0.0150442
\(829\) 29.7586 1.03356 0.516779 0.856119i \(-0.327131\pi\)
0.516779 + 0.856119i \(0.327131\pi\)
\(830\) 0 0
\(831\) 17.5802 0.609852
\(832\) −12.0919 −0.419210
\(833\) 20.4137 0.707294
\(834\) −27.2024 −0.941940
\(835\) 0 0
\(836\) −4.85410 −0.167883
\(837\) 8.45991 0.292417
\(838\) 4.93887 0.170610
\(839\) 29.5793 1.02119 0.510595 0.859821i \(-0.329425\pi\)
0.510595 + 0.859821i \(0.329425\pi\)
\(840\) 0 0
\(841\) 47.5142 1.63842
\(842\) 48.6912 1.67801
\(843\) −13.2536 −0.456479
\(844\) 25.7773 0.887291
\(845\) 0 0
\(846\) −23.0181 −0.791380
\(847\) 9.34488 0.321094
\(848\) 40.7678 1.39997
\(849\) −12.0996 −0.415258
\(850\) 0 0
\(851\) −0.566637 −0.0194241
\(852\) −3.57933 −0.122626
\(853\) 37.5298 1.28499 0.642497 0.766288i \(-0.277898\pi\)
0.642497 + 0.766288i \(0.277898\pi\)
\(854\) 14.2304 0.486953
\(855\) 0 0
\(856\) −2.19078 −0.0748794
\(857\) 49.4333 1.68861 0.844305 0.535864i \(-0.180014\pi\)
0.844305 + 0.535864i \(0.180014\pi\)
\(858\) 22.1882 0.757492
\(859\) 2.82651 0.0964394 0.0482197 0.998837i \(-0.484645\pi\)
0.0482197 + 0.998837i \(0.484645\pi\)
\(860\) 0 0
\(861\) −9.93960 −0.338741
\(862\) 61.4345 2.09247
\(863\) −33.3977 −1.13687 −0.568436 0.822727i \(-0.692451\pi\)
−0.568436 + 0.822727i \(0.692451\pi\)
\(864\) 6.50828 0.221416
\(865\) 0 0
\(866\) 9.76142 0.331707
\(867\) −0.274126 −0.00930980
\(868\) 16.3611 0.555331
\(869\) −17.8234 −0.604617
\(870\) 0 0
\(871\) 40.6707 1.37807
\(872\) 22.3247 0.756011
\(873\) −8.55105 −0.289409
\(874\) 1.00683 0.0340564
\(875\) 0 0
\(876\) 8.40767 0.284069
\(877\) 9.28115 0.313402 0.156701 0.987646i \(-0.449914\pi\)
0.156701 + 0.987646i \(0.449914\pi\)
\(878\) 49.2748 1.66294
\(879\) −3.61673 −0.121989
\(880\) 0 0
\(881\) 6.58736 0.221934 0.110967 0.993824i \(-0.464605\pi\)
0.110967 + 0.993824i \(0.464605\pi\)
\(882\) −8.97397 −0.302169
\(883\) −4.18883 −0.140965 −0.0704827 0.997513i \(-0.522454\pi\)
−0.0704827 + 0.997513i \(0.522454\pi\)
\(884\) −31.7236 −1.06698
\(885\) 0 0
\(886\) 30.8175 1.03533
\(887\) 26.8716 0.902260 0.451130 0.892458i \(-0.351021\pi\)
0.451130 + 0.892458i \(0.351021\pi\)
\(888\) −2.11791 −0.0710724
\(889\) 1.46508 0.0491371
\(890\) 0 0
\(891\) −2.12920 −0.0713310
\(892\) 10.1936 0.341307
\(893\) −21.4615 −0.718183
\(894\) 8.60543 0.287809
\(895\) 0 0
\(896\) −13.2128 −0.441408
\(897\) −1.84497 −0.0616016
\(898\) 39.4545 1.31661
\(899\) 74.0008 2.46807
\(900\) 0 0
\(901\) 34.6816 1.15541
\(902\) 26.7572 0.890918
\(903\) 16.1003 0.535784
\(904\) 16.1594 0.537453
\(905\) 0 0
\(906\) 17.5965 0.584606
\(907\) −39.1821 −1.30102 −0.650511 0.759497i \(-0.725445\pi\)
−0.650511 + 0.759497i \(0.725445\pi\)
\(908\) −1.76387 −0.0585362
\(909\) 9.85377 0.326829
\(910\) 0 0
\(911\) 54.2926 1.79880 0.899398 0.437132i \(-0.144006\pi\)
0.899398 + 0.437132i \(0.144006\pi\)
\(912\) 8.32275 0.275594
\(913\) −31.0591 −1.02791
\(914\) −10.2445 −0.338857
\(915\) 0 0
\(916\) 1.50731 0.0498028
\(917\) −3.01381 −0.0995247
\(918\) 7.59378 0.250632
\(919\) −28.0959 −0.926798 −0.463399 0.886150i \(-0.653370\pi\)
−0.463399 + 0.886150i \(0.653370\pi\)
\(920\) 0 0
\(921\) −14.8273 −0.488576
\(922\) −17.4396 −0.574344
\(923\) 15.2547 0.502116
\(924\) −4.11778 −0.135465
\(925\) 0 0
\(926\) −42.9846 −1.41256
\(927\) 16.1279 0.529710
\(928\) 56.9295 1.86880
\(929\) −31.3772 −1.02945 −0.514727 0.857354i \(-0.672107\pi\)
−0.514727 + 0.857354i \(0.672107\pi\)
\(930\) 0 0
\(931\) −8.36710 −0.274221
\(932\) 35.6072 1.16635
\(933\) 6.61579 0.216591
\(934\) −66.8412 −2.18711
\(935\) 0 0
\(936\) −6.89590 −0.225399
\(937\) −49.7286 −1.62456 −0.812281 0.583266i \(-0.801774\pi\)
−0.812281 + 0.583266i \(0.801774\pi\)
\(938\) −18.8279 −0.614754
\(939\) 7.17823 0.234253
\(940\) 0 0
\(941\) −4.49598 −0.146565 −0.0732824 0.997311i \(-0.523347\pi\)
−0.0732824 + 0.997311i \(0.523347\pi\)
\(942\) −33.9030 −1.10462
\(943\) −2.22489 −0.0724523
\(944\) 10.3806 0.337859
\(945\) 0 0
\(946\) −43.3417 −1.40916
\(947\) −28.8993 −0.939101 −0.469551 0.882906i \(-0.655584\pi\)
−0.469551 + 0.882906i \(0.655584\pi\)
\(948\) −11.2025 −0.363840
\(949\) −35.8326 −1.16318
\(950\) 0 0
\(951\) 9.75255 0.316248
\(952\) −7.26190 −0.235359
\(953\) −46.1922 −1.49631 −0.748156 0.663523i \(-0.769060\pi\)
−0.748156 + 0.663523i \(0.769060\pi\)
\(954\) −15.2462 −0.493613
\(955\) 0 0
\(956\) −33.2814 −1.07640
\(957\) −18.6247 −0.602050
\(958\) 53.4880 1.72812
\(959\) −24.4754 −0.790351
\(960\) 0 0
\(961\) 40.5701 1.30871
\(962\) −18.2543 −0.588542
\(963\) 1.81198 0.0583901
\(964\) −16.5319 −0.532456
\(965\) 0 0
\(966\) 0.854102 0.0274803
\(967\) 8.41820 0.270711 0.135356 0.990797i \(-0.456782\pi\)
0.135356 + 0.990797i \(0.456782\pi\)
\(968\) 7.81835 0.251291
\(969\) 7.08025 0.227450
\(970\) 0 0
\(971\) 26.4421 0.848567 0.424284 0.905529i \(-0.360526\pi\)
0.424284 + 0.905529i \(0.360526\pi\)
\(972\) −1.33826 −0.0429247
\(973\) −21.5155 −0.689756
\(974\) −38.8931 −1.24621
\(975\) 0 0
\(976\) 26.3309 0.842833
\(977\) 39.8235 1.27407 0.637034 0.770836i \(-0.280161\pi\)
0.637034 + 0.770836i \(0.280161\pi\)
\(978\) −0.825307 −0.0263904
\(979\) −5.70881 −0.182454
\(980\) 0 0
\(981\) −18.4646 −0.589529
\(982\) −60.6485 −1.93537
\(983\) −24.1179 −0.769242 −0.384621 0.923075i \(-0.625668\pi\)
−0.384621 + 0.923075i \(0.625668\pi\)
\(984\) −8.31592 −0.265102
\(985\) 0 0
\(986\) 66.4246 2.11539
\(987\) −18.2060 −0.579505
\(988\) 13.0028 0.413673
\(989\) 3.60390 0.114597
\(990\) 0 0
\(991\) −10.4044 −0.330506 −0.165253 0.986251i \(-0.552844\pi\)
−0.165253 + 0.986251i \(0.552844\pi\)
\(992\) 55.0595 1.74814
\(993\) −6.68822 −0.212244
\(994\) −7.06198 −0.223992
\(995\) 0 0
\(996\) −19.5215 −0.618562
\(997\) −13.3642 −0.423249 −0.211624 0.977351i \(-0.567875\pi\)
−0.211624 + 0.977351i \(0.567875\pi\)
\(998\) 8.63980 0.273488
\(999\) 1.75170 0.0554215
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.g.1.4 yes 4
3.2 odd 2 5625.2.a.j.1.1 4
5.2 odd 4 1875.2.b.d.1249.7 8
5.3 odd 4 1875.2.b.d.1249.2 8
5.4 even 2 1875.2.a.f.1.1 4
15.14 odd 2 5625.2.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.f.1.1 4 5.4 even 2
1875.2.a.g.1.4 yes 4 1.1 even 1 trivial
1875.2.b.d.1249.2 8 5.3 odd 4
1875.2.b.d.1249.7 8 5.2 odd 4
5625.2.a.j.1.1 4 3.2 odd 2
5625.2.a.m.1.4 4 15.14 odd 2