Properties

Label 5625.2.a.j.1.1
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
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] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, 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("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(0\)
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: no (minimal twist has level 1875)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.33826\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.82709 q^{2} +1.33826 q^{4} -1.44512 q^{7} +1.20906 q^{8} +O(q^{10})\) \(q-1.82709 q^{2} +1.33826 q^{4} -1.44512 q^{7} +1.20906 q^{8} +2.12920 q^{11} +5.70353 q^{13} +2.64037 q^{14} -4.88558 q^{16} +4.15622 q^{17} +1.70353 q^{19} -3.89025 q^{22} -0.323478 q^{23} -10.4209 q^{26} -1.93395 q^{28} +8.74724 q^{29} -8.45991 q^{31} +6.50828 q^{32} -7.59378 q^{34} -1.75170 q^{37} -3.11251 q^{38} +6.87802 q^{41} +11.1411 q^{43} +2.84943 q^{44} +0.591023 q^{46} +12.5982 q^{47} -4.91161 q^{49} +7.63282 q^{52} +8.34451 q^{53} -1.74724 q^{56} -15.9820 q^{58} +2.12474 q^{59} -5.38952 q^{61} +15.4570 q^{62} -2.12007 q^{64} +7.13078 q^{67} +5.56210 q^{68} -2.67461 q^{71} -6.28253 q^{73} +3.20052 q^{74} +2.27977 q^{76} -3.07697 q^{77} +8.37092 q^{79} -12.5668 q^{82} -14.5872 q^{83} -20.3558 q^{86} +2.57433 q^{88} -2.68119 q^{89} -8.24232 q^{91} -0.432897 q^{92} -23.0181 q^{94} -8.55105 q^{97} +8.97397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8} + 6 q^{11} + 7 q^{13} + 10 q^{14} - 9 q^{16} + 7 q^{17} - 9 q^{19} + 6 q^{22} - 10 q^{23} + 2 q^{26} + 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} - 10 q^{37} + 6 q^{38} + q^{43} + 9 q^{44} + 5 q^{46} + 23 q^{47} - 3 q^{49} + 13 q^{52} - 2 q^{58} - 4 q^{59} - 43 q^{61} + 10 q^{62} - 7 q^{64} + 8 q^{67} + 3 q^{68} + 27 q^{71} + 15 q^{73} - 5 q^{74} + 9 q^{76} + 15 q^{77} + 10 q^{79} - 20 q^{82} - 3 q^{83} - 24 q^{86} - 3 q^{88} + 9 q^{89} + 5 q^{91} + 15 q^{92} - 22 q^{94} + 13 q^{97} + 42 q^{98}+O(q^{100}) \) 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 \(-0.723549\pi\)
−0.645974 + 0.763359i \(0.723549\pi\)
\(3\) 0 0
\(4\) 1.33826 0.669131
\(5\) 0 0
\(6\) 0 0
\(7\) −1.44512 −0.546206 −0.273103 0.961985i \(-0.588050\pi\)
−0.273103 + 0.961985i \(0.588050\pi\)
\(8\) 1.20906 0.427466
\(9\) 0 0
\(10\) 0 0
\(11\) 2.12920 0.641979 0.320990 0.947083i \(-0.395985\pi\)
0.320990 + 0.947083i \(0.395985\pi\)
\(12\) 0 0
\(13\) 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 \(-0.331856\pi\)
0.504015 + 0.863695i \(0.331856\pi\)
\(18\) 0 0
\(19\) 1.70353 0.390817 0.195409 0.980722i \(-0.437397\pi\)
0.195409 + 0.980722i \(0.437397\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.89025 −0.829404
\(23\) −0.323478 −0.0674497 −0.0337249 0.999431i \(-0.510737\pi\)
−0.0337249 + 0.999431i \(0.510737\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −10.4209 −2.04370
\(27\) 0 0
\(28\) −1.93395 −0.365483
\(29\) 8.74724 1.62432 0.812161 0.583434i \(-0.198291\pi\)
0.812161 + 0.583434i \(0.198291\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −7.59378 −1.30232
\(35\) 0 0
\(36\) 0 0
\(37\) −1.75170 −0.287978 −0.143989 0.989579i \(-0.545993\pi\)
−0.143989 + 0.989579i \(0.545993\pi\)
\(38\) −3.11251 −0.504916
\(39\) 0 0
\(40\) 0 0
\(41\) 6.87802 1.07417 0.537083 0.843529i \(-0.319526\pi\)
0.537083 + 0.843529i \(0.319526\pi\)
\(42\) 0 0
\(43\) 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 \(-0.129142\pi\)
0.918822 + 0.394673i \(0.129142\pi\)
\(48\) 0 0
\(49\) −4.91161 −0.701659
\(50\) 0 0
\(51\) 0 0
\(52\) 7.63282 1.05848
\(53\) 8.34451 1.14621 0.573103 0.819483i \(-0.305739\pi\)
0.573103 + 0.819483i \(0.305739\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.74724 −0.233485
\(57\) 0 0
\(58\) −15.9820 −2.09854
\(59\) 2.12474 0.276617 0.138309 0.990389i \(-0.455833\pi\)
0.138309 + 0.990389i \(0.455833\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −2.12007 −0.265008
\(65\) 0 0
\(66\) 0 0
\(67\) 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 0
\(70\) 0 0
\(71\) −2.67461 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(72\) 0 0
\(73\) −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
\(78\) 0 0
\(79\) 8.37092 0.941802 0.470901 0.882186i \(-0.343929\pi\)
0.470901 + 0.882186i \(0.343929\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.5668 −1.38777
\(83\) −14.5872 −1.60115 −0.800577 0.599230i \(-0.795473\pi\)
−0.800577 + 0.599230i \(0.795473\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −20.3558 −2.19502
\(87\) 0 0
\(88\) 2.57433 0.274424
\(89\) −2.68119 −0.284206 −0.142103 0.989852i \(-0.545386\pi\)
−0.142103 + 0.989852i \(0.545386\pi\)
\(90\) 0 0
\(91\) −8.24232 −0.864030
\(92\) −0.432897 −0.0451327
\(93\) 0 0
\(94\) −23.0181 −2.37414
\(95\) 0 0
\(96\) 0 0
\(97\) −8.55105 −0.868228 −0.434114 0.900858i \(-0.642939\pi\)
−0.434114 + 0.900858i \(0.642939\pi\)
\(98\) 8.97397 0.906507
\(99\) 0 0
\(100\) 0 0
\(101\) −9.85377 −0.980487 −0.490243 0.871586i \(-0.663092\pi\)
−0.490243 + 0.871586i \(0.663092\pi\)
\(102\) 0 0
\(103\) 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 \(-0.527915\pi\)
−0.0875852 + 0.996157i \(0.527915\pi\)
\(108\) 0 0
\(109\) −18.4646 −1.76859 −0.884293 0.466933i \(-0.845359\pi\)
−0.884293 + 0.466933i \(0.845359\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.06027 0.667133
\(113\) 13.3653 1.25730 0.628650 0.777689i \(-0.283608\pi\)
0.628650 + 0.777689i \(0.283608\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 11.7061 1.08688
\(117\) 0 0
\(118\) −3.88209 −0.357375
\(119\) −6.00625 −0.550592
\(120\) 0 0
\(121\) −6.46649 −0.587863
\(122\) 9.84715 0.891519
\(123\) 0 0
\(124\) −11.3216 −1.01671
\(125\) 0 0
\(126\) 0 0
\(127\) −1.01381 −0.0899608 −0.0449804 0.998988i \(-0.514323\pi\)
−0.0449804 + 0.998988i \(0.514323\pi\)
\(128\) −9.14301 −0.808136
\(129\) 0 0
\(130\) 0 0
\(131\) −2.08550 −0.182211 −0.0911055 0.995841i \(-0.529040\pi\)
−0.0911055 + 0.995841i \(0.529040\pi\)
\(132\) 0 0
\(133\) −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 \(-0.757464\pi\)
−0.723492 + 0.690333i \(0.757464\pi\)
\(138\) 0 0
\(139\) 14.8883 1.26281 0.631406 0.775452i \(-0.282478\pi\)
0.631406 + 0.775452i \(0.282478\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.88676 0.410088
\(143\) 12.1440 1.01553
\(144\) 0 0
\(145\) 0 0
\(146\) 11.4788 0.949989
\(147\) 0 0
\(148\) −2.34424 −0.192695
\(149\) 4.70991 0.385851 0.192925 0.981213i \(-0.438202\pi\)
0.192925 + 0.981213i \(0.438202\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 5.62190 0.453025
\(155\) 0 0
\(156\) 0 0
\(157\) 18.5557 1.48091 0.740454 0.672107i \(-0.234611\pi\)
0.740454 + 0.672107i \(0.234611\pi\)
\(158\) −15.2944 −1.21676
\(159\) 0 0
\(160\) 0 0
\(161\) 0.467465 0.0368414
\(162\) 0 0
\(163\) 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 \(-0.558540\pi\)
−0.182873 + 0.983137i \(0.558540\pi\)
\(168\) 0 0
\(169\) 19.5303 1.50233
\(170\) 0 0
\(171\) 0 0
\(172\) 14.9097 1.13685
\(173\) −2.29489 −0.174477 −0.0872385 0.996187i \(-0.527804\pi\)
−0.0872385 + 0.996187i \(0.527804\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −10.4024 −0.784110
\(177\) 0 0
\(178\) 4.89878 0.367179
\(179\) −2.68757 −0.200878 −0.100439 0.994943i \(-0.532025\pi\)
−0.100439 + 0.994943i \(0.532025\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −0.391103 −0.0288325
\(185\) 0 0
\(186\) 0 0
\(187\) 8.84943 0.647135
\(188\) 16.8597 1.22962
\(189\) 0 0
\(190\) 0 0
\(191\) 3.91824 0.283514 0.141757 0.989902i \(-0.454725\pi\)
0.141757 + 0.989902i \(0.454725\pi\)
\(192\) 0 0
\(193\) −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 \(-0.527843\pi\)
−0.0873614 + 0.996177i \(0.527843\pi\)
\(198\) 0 0
\(199\) −10.4345 −0.739680 −0.369840 0.929095i \(-0.620588\pi\)
−0.369840 + 0.929095i \(0.620588\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0037 1.26674
\(203\) −12.6409 −0.887214
\(204\) 0 0
\(205\) 0 0
\(206\) −29.4671 −2.05307
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 3.31065 0.226311
\(215\) 0 0
\(216\) 0 0
\(217\) 12.2256 0.829929
\(218\) 33.7365 2.28492
\(219\) 0 0
\(220\) 0 0
\(221\) 23.7051 1.59458
\(222\) 0 0
\(223\) 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 \(-0.486073\pi\)
0.0437405 + 0.999043i \(0.486073\pi\)
\(228\) 0 0
\(229\) 1.12632 0.0744292 0.0372146 0.999307i \(-0.488151\pi\)
0.0372146 + 0.999307i \(0.488151\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.5759 0.694342
\(233\) −26.6071 −1.74309 −0.871543 0.490319i \(-0.836880\pi\)
−0.871543 + 0.490319i \(0.836880\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.84345 0.185093
\(237\) 0 0
\(238\) 10.9740 0.711337
\(239\) 24.8692 1.60865 0.804326 0.594188i \(-0.202527\pi\)
0.804326 + 0.594188i \(0.202527\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −7.21259 −0.461739
\(245\) 0 0
\(246\) 0 0
\(247\) 9.71616 0.618224
\(248\) −10.2285 −0.649511
\(249\) 0 0
\(250\) 0 0
\(251\) 29.0986 1.83669 0.918343 0.395786i \(-0.129528\pi\)
0.918343 + 0.395786i \(0.129528\pi\)
\(252\) 0 0
\(253\) −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 \(-0.377358\pi\)
0.375828 + 0.926690i \(0.377358\pi\)
\(258\) 0 0
\(259\) 2.53143 0.157296
\(260\) 0 0
\(261\) 0 0
\(262\) 3.81040 0.235407
\(263\) 1.02892 0.0634460 0.0317230 0.999497i \(-0.489901\pi\)
0.0317230 + 0.999497i \(0.489901\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.49797 0.275788
\(267\) 0 0
\(268\) 9.54285 0.582922
\(269\) −23.9072 −1.45765 −0.728824 0.684701i \(-0.759933\pi\)
−0.728824 + 0.684701i \(0.759933\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 30.9445 1.86943
\(275\) 0 0
\(276\) 0 0
\(277\) −17.5802 −1.05629 −0.528147 0.849153i \(-0.677113\pi\)
−0.528147 + 0.849153i \(0.677113\pi\)
\(278\) −27.2024 −1.63149
\(279\) 0 0
\(280\) 0 0
\(281\) −13.2536 −0.790644 −0.395322 0.918543i \(-0.629367\pi\)
−0.395322 + 0.918543i \(0.629367\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 0 0
\(289\) 0.274126 0.0161251
\(290\) 0 0
\(291\) 0 0
\(292\) −8.40767 −0.492022
\(293\) −3.61673 −0.211291 −0.105646 0.994404i \(-0.533691\pi\)
−0.105646 + 0.994404i \(0.533691\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.11791 −0.123101
\(297\) 0 0
\(298\) −8.60543 −0.498499
\(299\) −1.84497 −0.106697
\(300\) 0 0
\(301\) −16.1003 −0.928005
\(302\) 17.5965 1.01257
\(303\) 0 0
\(304\) −8.32275 −0.477342
\(305\) 0 0
\(306\) 0 0
\(307\) 14.8273 0.846238 0.423119 0.906074i \(-0.360935\pi\)
0.423119 + 0.906074i \(0.360935\pi\)
\(308\) −4.11778 −0.234633
\(309\) 0 0
\(310\) 0 0
\(311\) 6.61579 0.375147 0.187574 0.982251i \(-0.439938\pi\)
0.187574 + 0.982251i \(0.439938\pi\)
\(312\) 0 0
\(313\) −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 \(-0.411693\pi\)
0.273879 + 0.961764i \(0.411693\pi\)
\(318\) 0 0
\(319\) 18.6247 1.04278
\(320\) 0 0
\(321\) 0 0
\(322\) −0.854102 −0.0475972
\(323\) 7.08025 0.393956
\(324\) 0 0
\(325\) 0 0
\(326\) −0.825307 −0.0457095
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 8.63570 0.472525
\(335\) 0 0
\(336\) 0 0
\(337\) −20.8191 −1.13409 −0.567045 0.823687i \(-0.691914\pi\)
−0.567045 + 0.823687i \(0.691914\pi\)
\(338\) −35.6836 −1.94093
\(339\) 0 0
\(340\) 0 0
\(341\) −18.0129 −0.975452
\(342\) 0 0
\(343\) 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 \(-0.187291\pi\)
0.831835 + 0.555023i \(0.187291\pi\)
\(348\) 0 0
\(349\) −7.45484 −0.399048 −0.199524 0.979893i \(-0.563940\pi\)
−0.199524 + 0.979893i \(0.563940\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 13.8575 0.738605
\(353\) 3.68948 0.196371 0.0981856 0.995168i \(-0.468696\pi\)
0.0981856 + 0.995168i \(0.468696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.58814 −0.190171
\(357\) 0 0
\(358\) 4.91043 0.259524
\(359\) 9.92989 0.524079 0.262040 0.965057i \(-0.415605\pi\)
0.262040 + 0.965057i \(0.415605\pi\)
\(360\) 0 0
\(361\) −16.0980 −0.847262
\(362\) 43.6835 2.29595
\(363\) 0 0
\(364\) −11.0304 −0.578149
\(365\) 0 0
\(366\) 0 0
\(367\) −12.5507 −0.655142 −0.327571 0.944826i \(-0.606230\pi\)
−0.327571 + 0.944826i \(0.606230\pi\)
\(368\) 1.58038 0.0823828
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0589 −0.626065
\(372\) 0 0
\(373\) 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
\(378\) 0 0
\(379\) 11.2957 0.580223 0.290112 0.956993i \(-0.406308\pi\)
0.290112 + 0.956993i \(0.406308\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −7.15898 −0.366285
\(383\) 3.04968 0.155831 0.0779157 0.996960i \(-0.475173\pi\)
0.0779157 + 0.996960i \(0.475173\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.6171 0.794889
\(387\) 0 0
\(388\) −11.4435 −0.580958
\(389\) 21.3939 1.08472 0.542358 0.840148i \(-0.317532\pi\)
0.542358 + 0.840148i \(0.317532\pi\)
\(390\) 0 0
\(391\) −1.34444 −0.0679914
\(392\) −5.93842 −0.299936
\(393\) 0 0
\(394\) 4.48067 0.225733
\(395\) 0 0
\(396\) 0 0
\(397\) 34.4305 1.72802 0.864009 0.503476i \(-0.167946\pi\)
0.864009 + 0.503476i \(0.167946\pi\)
\(398\) 19.0647 0.955629
\(399\) 0 0
\(400\) 0 0
\(401\) −36.2976 −1.81262 −0.906309 0.422616i \(-0.861112\pi\)
−0.906309 + 0.422616i \(0.861112\pi\)
\(402\) 0 0
\(403\) −48.2514 −2.40357
\(404\) −13.1869 −0.656074
\(405\) 0 0
\(406\) 23.0960 1.14623
\(407\) −3.72974 −0.184876
\(408\) 0 0
\(409\) 24.9896 1.23566 0.617828 0.786313i \(-0.288013\pi\)
0.617828 + 0.786313i \(0.288013\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 21.5833 1.06333
\(413\) −3.07051 −0.151090
\(414\) 0 0
\(415\) 0 0
\(416\) 37.1202 1.81997
\(417\) 0 0
\(418\) −6.62717 −0.324146
\(419\) −2.70313 −0.132057 −0.0660284 0.997818i \(-0.521033\pi\)
−0.0660284 + 0.997818i \(0.521033\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 10.0890 0.489965
\(425\) 0 0
\(426\) 0 0
\(427\) 7.78853 0.376914
\(428\) −2.42490 −0.117212
\(429\) 0 0
\(430\) 0 0
\(431\) −33.6242 −1.61962 −0.809811 0.586691i \(-0.800430\pi\)
−0.809811 + 0.586691i \(0.800430\pi\)
\(432\) 0 0
\(433\) 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
\(438\) 0 0
\(439\) 26.9690 1.28716 0.643580 0.765379i \(-0.277448\pi\)
0.643580 + 0.765379i \(0.277448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −43.3114 −2.06011
\(443\) −16.8670 −0.801374 −0.400687 0.916215i \(-0.631229\pi\)
−0.400687 + 0.916215i \(0.631229\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −13.9171 −0.658992
\(447\) 0 0
\(448\) 3.06376 0.144749
\(449\) −21.5942 −1.01909 −0.509546 0.860443i \(-0.670187\pi\)
−0.509546 + 0.860443i \(0.670187\pi\)
\(450\) 0 0
\(451\) 14.6447 0.689593
\(452\) 17.8862 0.841297
\(453\) 0 0
\(454\) −2.40817 −0.113021
\(455\) 0 0
\(456\) 0 0
\(457\) −5.60699 −0.262284 −0.131142 0.991364i \(-0.541864\pi\)
−0.131142 + 0.991364i \(0.541864\pi\)
\(458\) −2.05788 −0.0961587
\(459\) 0 0
\(460\) 0 0
\(461\) 9.54504 0.444557 0.222278 0.974983i \(-0.428651\pi\)
0.222278 + 0.974983i \(0.428651\pi\)
\(462\) 0 0
\(463\) −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 \(-0.178742\pi\)
0.846439 + 0.532486i \(0.178742\pi\)
\(468\) 0 0
\(469\) −10.3049 −0.475835
\(470\) 0 0
\(471\) 0 0
\(472\) 2.56893 0.118245
\(473\) 23.7217 1.09072
\(474\) 0 0
\(475\) 0 0
\(476\) −8.03793 −0.368418
\(477\) 0 0
\(478\) −45.4382 −2.07830
\(479\) −29.2750 −1.33761 −0.668804 0.743439i \(-0.733193\pi\)
−0.668804 + 0.743439i \(0.733193\pi\)
\(480\) 0 0
\(481\) −9.99091 −0.455546
\(482\) 22.5705 1.02806
\(483\) 0 0
\(484\) −8.65385 −0.393357
\(485\) 0 0
\(486\) 0 0
\(487\) −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 0
\(490\) 0 0
\(491\) 33.1940 1.49803 0.749013 0.662556i \(-0.230528\pi\)
0.749013 + 0.662556i \(0.230528\pi\)
\(492\) 0 0
\(493\) 36.3554 1.63737
\(494\) −17.7523 −0.798714
\(495\) 0 0
\(496\) 41.3316 1.85584
\(497\) 3.86515 0.173376
\(498\) 0 0
\(499\) 4.72872 0.211686 0.105843 0.994383i \(-0.466246\pi\)
0.105843 + 0.994383i \(0.466246\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −53.1657 −2.37290
\(503\) 0.400444 0.0178549 0.00892745 0.999960i \(-0.497158\pi\)
0.00892745 + 0.999960i \(0.497158\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.25841 0.0559431
\(507\) 0 0
\(508\) −1.35674 −0.0601956
\(509\) 27.6104 1.22381 0.611904 0.790932i \(-0.290404\pi\)
0.611904 + 0.790932i \(0.290404\pi\)
\(510\) 0 0
\(511\) 9.07905 0.401633
\(512\) −19.9828 −0.883126
\(513\) 0 0
\(514\) −22.0164 −0.971100
\(515\) 0 0
\(516\) 0 0
\(517\) 26.8242 1.17973
\(518\) −4.62516 −0.203218
\(519\) 0 0
\(520\) 0 0
\(521\) 28.7374 1.25901 0.629505 0.776996i \(-0.283258\pi\)
0.629505 + 0.776996i \(0.283258\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) −22.8954 −0.995451
\(530\) 0 0
\(531\) 0 0
\(532\) −3.29456 −0.142837
\(533\) 39.2290 1.69920
\(534\) 0 0
\(535\) 0 0
\(536\) 8.62152 0.372393
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 27.0498 1.15975
\(545\) 0 0
\(546\) 0 0
\(547\) −36.7888 −1.57297 −0.786487 0.617607i \(-0.788102\pi\)
−0.786487 + 0.617607i \(0.788102\pi\)
\(548\) −22.6655 −0.968221
\(549\) 0 0
\(550\) 0 0
\(551\) 14.9012 0.634813
\(552\) 0 0
\(553\) −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 \(-0.664698\pi\)
−0.494633 + 0.869102i \(0.664698\pi\)
\(558\) 0 0
\(559\) 63.5436 2.68761
\(560\) 0 0
\(561\) 0 0
\(562\) 24.2156 1.02147
\(563\) 29.2369 1.23219 0.616095 0.787672i \(-0.288714\pi\)
0.616095 + 0.787672i \(0.288714\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −22.1071 −0.929232
\(567\) 0 0
\(568\) −3.23376 −0.135685
\(569\) 20.2530 0.849051 0.424526 0.905416i \(-0.360441\pi\)
0.424526 + 0.905416i \(0.360441\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 18.1606 0.758007
\(575\) 0 0
\(576\) 0 0
\(577\) −31.3502 −1.30513 −0.652564 0.757734i \(-0.726306\pi\)
−0.652564 + 0.757734i \(0.726306\pi\)
\(578\) −0.500853 −0.0208327
\(579\) 0 0
\(580\) 0 0
\(581\) 21.0803 0.874559
\(582\) 0 0
\(583\) 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 \(-0.469261\pi\)
0.0964208 + 0.995341i \(0.469261\pi\)
\(588\) 0 0
\(589\) −14.4117 −0.593825
\(590\) 0 0
\(591\) 0 0
\(592\) 8.55809 0.351735
\(593\) 7.37978 0.303051 0.151526 0.988453i \(-0.451581\pi\)
0.151526 + 0.988453i \(0.451581\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.30309 0.258185
\(597\) 0 0
\(598\) 3.37092 0.137847
\(599\) 46.7505 1.91018 0.955088 0.296323i \(-0.0957605\pi\)
0.955088 + 0.296323i \(0.0957605\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −12.8887 −0.524433
\(605\) 0 0
\(606\) 0 0
\(607\) −14.0891 −0.571858 −0.285929 0.958251i \(-0.592302\pi\)
−0.285929 + 0.958251i \(0.592302\pi\)
\(608\) 11.0871 0.449640
\(609\) 0 0
\(610\) 0 0
\(611\) 71.8545 2.90692
\(612\) 0 0
\(613\) −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 \(-0.487376\pi\)
0.0396485 + 0.999214i \(0.487376\pi\)
\(618\) 0 0
\(619\) 14.9714 0.601749 0.300875 0.953664i \(-0.402721\pi\)
0.300875 + 0.953664i \(0.402721\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12.0877 −0.484671
\(623\) 3.87466 0.155235
\(624\) 0 0
\(625\) 0 0
\(626\) 13.1153 0.524192
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) −17.8188 −0.707675
\(635\) 0 0
\(636\) 0 0
\(637\) −28.0136 −1.10994
\(638\) −34.0289 −1.34722
\(639\) 0 0
\(640\) 0 0
\(641\) −12.2381 −0.483374 −0.241687 0.970354i \(-0.577701\pi\)
−0.241687 + 0.970354i \(0.577701\pi\)
\(642\) 0 0
\(643\) −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 \(-0.611109\pi\)
−0.342014 + 0.939695i \(0.611109\pi\)
\(648\) 0 0
\(649\) 4.52400 0.177583
\(650\) 0 0
\(651\) 0 0
\(652\) 0.604500 0.0236740
\(653\) 28.4645 1.11390 0.556950 0.830546i \(-0.311971\pi\)
0.556950 + 0.830546i \(0.311971\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −33.6031 −1.31198
\(657\) 0 0
\(658\) 33.2641 1.29677
\(659\) 1.51333 0.0589509 0.0294754 0.999566i \(-0.490616\pi\)
0.0294754 + 0.999566i \(0.490616\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −17.6368 −0.684439
\(665\) 0 0
\(666\) 0 0
\(667\) −2.82954 −0.109560
\(668\) −6.32526 −0.244732
\(669\) 0 0
\(670\) 0 0
\(671\) −11.4754 −0.443003
\(672\) 0 0
\(673\) 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 \(-0.658019\pi\)
−0.476290 + 0.879288i \(0.658019\pi\)
\(678\) 0 0
\(679\) 12.3573 0.474231
\(680\) 0 0
\(681\) 0 0
\(682\) 32.9112 1.26023
\(683\) 37.3864 1.43055 0.715276 0.698842i \(-0.246301\pi\)
0.715276 + 0.698842i \(0.246301\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −31.4511 −1.20081
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) −56.6229 −2.14938
\(695\) 0 0
\(696\) 0 0
\(697\) 28.5865 1.08279
\(698\) 13.6207 0.515550
\(699\) 0 0
\(700\) 0 0
\(701\) 42.3336 1.59892 0.799459 0.600721i \(-0.205120\pi\)
0.799459 + 0.600721i \(0.205120\pi\)
\(702\) 0 0
\(703\) −2.98409 −0.112547
\(704\) −4.51406 −0.170130
\(705\) 0 0
\(706\) −6.74101 −0.253701
\(707\) 14.2399 0.535548
\(708\) 0 0
\(709\) 18.3593 0.689498 0.344749 0.938695i \(-0.387964\pi\)
0.344749 + 0.938695i \(0.387964\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.24171 −0.121488
\(713\) 2.73659 0.102486
\(714\) 0 0
\(715\) 0 0
\(716\) −3.59667 −0.134414
\(717\) 0 0
\(718\) −18.1428 −0.677084
\(719\) 19.2706 0.718671 0.359336 0.933208i \(-0.383003\pi\)
0.359336 + 0.933208i \(0.383003\pi\)
\(720\) 0 0
\(721\) −23.3068 −0.867992
\(722\) 29.4125 1.09462
\(723\) 0 0
\(724\) −31.9962 −1.18913
\(725\) 0 0
\(726\) 0 0
\(727\) 17.4238 0.646214 0.323107 0.946362i \(-0.395273\pi\)
0.323107 + 0.946362i \(0.395273\pi\)
\(728\) −9.96543 −0.369343
\(729\) 0 0
\(730\) 0 0
\(731\) 46.3048 1.71265
\(732\) 0 0
\(733\) 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
\(738\) 0 0
\(739\) −38.5798 −1.41918 −0.709591 0.704613i \(-0.751120\pi\)
−0.709591 + 0.704613i \(0.751120\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 22.0326 0.808844
\(743\) 33.8580 1.24213 0.621065 0.783759i \(-0.286700\pi\)
0.621065 + 0.783759i \(0.286700\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −27.1462 −0.993895
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) −91.1539 −3.31963
\(755\) 0 0
\(756\) 0 0
\(757\) 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 0
\(760\) 0 0
\(761\) −49.2652 −1.78586 −0.892931 0.450193i \(-0.851355\pi\)
−0.892931 + 0.450193i \(0.851355\pi\)
\(762\) 0 0
\(763\) 26.6836 0.966012
\(764\) 5.24362 0.189708
\(765\) 0 0
\(766\) −5.57205 −0.201326
\(767\) 12.1185 0.437574
\(768\) 0 0
\(769\) −1.26208 −0.0455116 −0.0227558 0.999741i \(-0.507244\pi\)
−0.0227558 + 0.999741i \(0.507244\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.4388 −0.411692
\(773\) 26.9903 0.970772 0.485386 0.874300i \(-0.338679\pi\)
0.485386 + 0.874300i \(0.338679\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −10.3387 −0.371138
\(777\) 0 0
\(778\) −39.0887 −1.40140
\(779\) 11.7169 0.419803
\(780\) 0 0
\(781\) −5.69480 −0.203776
\(782\) 2.45642 0.0878413
\(783\) 0 0
\(784\) 23.9961 0.857003
\(785\) 0 0
\(786\) 0 0
\(787\) 7.62303 0.271732 0.135866 0.990727i \(-0.456618\pi\)
0.135866 + 0.990727i \(0.456618\pi\)
\(788\) −3.28189 −0.116912
\(789\) 0 0
\(790\) 0 0
\(791\) −19.3145 −0.686744
\(792\) 0 0
\(793\) −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 \(-0.528127\pi\)
−0.0882474 + 0.996099i \(0.528127\pi\)
\(798\) 0 0
\(799\) 52.3610 1.85240
\(800\) 0 0
\(801\) 0 0
\(802\) 66.3191 2.34181
\(803\) −13.3768 −0.472057
\(804\) 0 0
\(805\) 0 0
\(806\) 88.1596 3.10529
\(807\) 0 0
\(808\) −11.9138 −0.419125
\(809\) −5.74271 −0.201903 −0.100952 0.994891i \(-0.532189\pi\)
−0.100952 + 0.994891i \(0.532189\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 6.81457 0.238851
\(815\) 0 0
\(816\) 0 0
\(817\) 18.9792 0.664000
\(818\) −45.6583 −1.59640
\(819\) 0 0
\(820\) 0 0
\(821\) 41.8919 1.46204 0.731019 0.682357i \(-0.239045\pi\)
0.731019 + 0.682357i \(0.239045\pi\)
\(822\) 0 0
\(823\) −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 \(-0.541176\pi\)
−0.128998 + 0.991645i \(0.541176\pi\)
\(828\) 0 0
\(829\) 29.7586 1.03356 0.516779 0.856119i \(-0.327131\pi\)
0.516779 + 0.856119i \(0.327131\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −12.0919 −0.419210
\(833\) −20.4137 −0.707294
\(834\) 0 0
\(835\) 0 0
\(836\) 4.85410 0.167883
\(837\) 0 0
\(838\) 4.93887 0.170610
\(839\) −29.5793 −1.02119 −0.510595 0.859821i \(-0.670575\pi\)
−0.510595 + 0.859821i \(0.670575\pi\)
\(840\) 0 0
\(841\) 47.5142 1.63842
\(842\) −48.6912 −1.67801
\(843\) 0 0
\(844\) 25.7773 0.887291
\(845\) 0 0
\(846\) 0 0
\(847\) 9.34488 0.321094
\(848\) −40.7678 −1.39997
\(849\) 0 0
\(850\) 0 0
\(851\) 0.566637 0.0194241
\(852\) 0 0
\(853\) 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 \(-0.819986\pi\)
−0.844305 + 0.535864i \(0.819986\pi\)
\(858\) 0 0
\(859\) 2.82651 0.0964394 0.0482197 0.998837i \(-0.484645\pi\)
0.0482197 + 0.998837i \(0.484645\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 61.4345 2.09247
\(863\) 33.3977 1.13687 0.568436 0.822727i \(-0.307549\pi\)
0.568436 + 0.822727i \(0.307549\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −9.76142 −0.331707
\(867\) 0 0
\(868\) 16.3611 0.555331
\(869\) 17.8234 0.604617
\(870\) 0 0
\(871\) 40.6707 1.37807
\(872\) −22.3247 −0.756011
\(873\) 0 0
\(874\) 1.00683 0.0340564
\(875\) 0 0
\(876\) 0 0
\(877\) 9.28115 0.313402 0.156701 0.987646i \(-0.449914\pi\)
0.156701 + 0.987646i \(0.449914\pi\)
\(878\) −49.2748 −1.66294
\(879\) 0 0
\(880\) 0 0
\(881\) −6.58736 −0.221934 −0.110967 0.993824i \(-0.535395\pi\)
−0.110967 + 0.993824i \(0.535395\pi\)
\(882\) 0 0
\(883\) −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 \(-0.648979\pi\)
−0.451130 + 0.892458i \(0.648979\pi\)
\(888\) 0 0
\(889\) 1.46508 0.0491371
\(890\) 0 0
\(891\) 0 0
\(892\) 10.1936 0.341307
\(893\) 21.4615 0.718183
\(894\) 0 0
\(895\) 0 0
\(896\) 13.2128 0.441408
\(897\) 0 0
\(898\) 39.4545 1.31661
\(899\) −74.0008 −2.46807
\(900\) 0 0
\(901\) 34.6816 1.15541
\(902\) −26.7572 −0.890918
\(903\) 0 0
\(904\) 16.1594 0.537453
\(905\) 0 0
\(906\) 0 0
\(907\) −39.1821 −1.30102 −0.650511 0.759497i \(-0.725445\pi\)
−0.650511 + 0.759497i \(0.725445\pi\)
\(908\) 1.76387 0.0585362
\(909\) 0 0
\(910\) 0 0
\(911\) −54.2926 −1.79880 −0.899398 0.437132i \(-0.855994\pi\)
−0.899398 + 0.437132i \(0.855994\pi\)
\(912\) 0 0
\(913\) −31.0591 −1.02791
\(914\) 10.2445 0.338857
\(915\) 0 0
\(916\) 1.50731 0.0498028
\(917\) 3.01381 0.0995247
\(918\) 0 0
\(919\) −28.0959 −0.926798 −0.463399 0.886150i \(-0.653370\pi\)
−0.463399 + 0.886150i \(0.653370\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −17.4396 −0.574344
\(923\) −15.2547 −0.502116
\(924\) 0 0
\(925\) 0 0
\(926\) 42.9846 1.41256
\(927\) 0 0
\(928\) 56.9295 1.86880
\(929\) 31.3772 1.02945 0.514727 0.857354i \(-0.327893\pi\)
0.514727 + 0.857354i \(0.327893\pi\)
\(930\) 0 0
\(931\) −8.36710 −0.274221
\(932\) −35.6072 −1.16635
\(933\) 0 0
\(934\) −66.8412 −2.18711
\(935\) 0 0
\(936\) 0 0
\(937\) −49.7286 −1.62456 −0.812281 0.583266i \(-0.801774\pi\)
−0.812281 + 0.583266i \(0.801774\pi\)
\(938\) 18.8279 0.614754
\(939\) 0 0
\(940\) 0 0
\(941\) 4.49598 0.146565 0.0732824 0.997311i \(-0.476653\pi\)
0.0732824 + 0.997311i \(0.476653\pi\)
\(942\) 0 0
\(943\) −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 \(-0.344416\pi\)
0.469551 + 0.882906i \(0.344416\pi\)
\(948\) 0 0
\(949\) −35.8326 −1.16318
\(950\) 0 0
\(951\) 0 0
\(952\) −7.26190 −0.235359
\(953\) 46.1922 1.49631 0.748156 0.663523i \(-0.230940\pi\)
0.748156 + 0.663523i \(0.230940\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 33.2814 1.07640
\(957\) 0 0
\(958\) 53.4880 1.72812
\(959\) 24.4754 0.790351
\(960\) 0 0
\(961\) 40.5701 1.30871
\(962\) 18.2543 0.588542
\(963\) 0 0
\(964\) −16.5319 −0.532456
\(965\) 0 0
\(966\) 0 0
\(967\) 8.41820 0.270711 0.135356 0.990797i \(-0.456782\pi\)
0.135356 + 0.990797i \(0.456782\pi\)
\(968\) −7.81835 −0.251291
\(969\) 0 0
\(970\) 0 0
\(971\) −26.4421 −0.848567 −0.424284 0.905529i \(-0.639474\pi\)
−0.424284 + 0.905529i \(0.639474\pi\)
\(972\) 0 0
\(973\) −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 \(-0.719839\pi\)
−0.637034 + 0.770836i \(0.719839\pi\)
\(978\) 0 0
\(979\) −5.70881 −0.182454
\(980\) 0 0
\(981\) 0 0
\(982\) −60.6485 −1.93537
\(983\) 24.1179 0.769242 0.384621 0.923075i \(-0.374332\pi\)
0.384621 + 0.923075i \(0.374332\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −66.4246 −2.11539
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −7.06198 −0.223992
\(995\) 0 0
\(996\) 0 0
\(997\) −13.3642 −0.423249 −0.211624 0.977351i \(-0.567875\pi\)
−0.211624 + 0.977351i \(0.567875\pi\)
\(998\) −8.63980 −0.273488
\(999\) 0 0
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 5625.2.a.j.1.1 4
3.2 odd 2 1875.2.a.g.1.4 yes 4
5.4 even 2 5625.2.a.m.1.4 4
15.2 even 4 1875.2.b.d.1249.7 8
15.8 even 4 1875.2.b.d.1249.2 8
15.14 odd 2 1875.2.a.f.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.f.1.1 4 15.14 odd 2
1875.2.a.g.1.4 yes 4 3.2 odd 2
1875.2.b.d.1249.2 8 15.8 even 4
1875.2.b.d.1249.7 8 15.2 even 4
5625.2.a.j.1.1 4 1.1 even 1 trivial
5625.2.a.m.1.4 4 5.4 even 2