Properties

Label 6016.2.a.m.1.7
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $1$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.904794\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.904794 q^{3} -3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} +O(q^{10})\) \(q-0.904794 q^{3} -3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} +6.05981 q^{11} -1.97561 q^{13} +2.80800 q^{15} +0.802172 q^{17} -2.24854 q^{19} -1.54576 q^{21} -2.79083 q^{23} +4.63153 q^{25} +4.68805 q^{27} -9.23320 q^{29} +7.61706 q^{31} -5.48288 q^{33} -5.30202 q^{35} -3.25627 q^{37} +1.78752 q^{39} +3.62554 q^{41} +0.635801 q^{43} +6.76975 q^{45} +1.00000 q^{47} -4.08131 q^{49} -0.725800 q^{51} +4.24499 q^{53} -18.8064 q^{55} +2.03446 q^{57} -2.01915 q^{59} -6.89875 q^{61} -3.72665 q^{63} +6.13125 q^{65} +6.92370 q^{67} +2.52512 q^{69} +11.3962 q^{71} +14.6621 q^{73} -4.19058 q^{75} +10.3527 q^{77} +12.3443 q^{79} +2.30233 q^{81} +0.818293 q^{83} -2.48952 q^{85} +8.35414 q^{87} +10.3807 q^{89} -3.37517 q^{91} -6.89187 q^{93} +6.97827 q^{95} -16.2464 q^{97} -13.2186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9} - 10 q^{11} - 4 q^{13} - 14 q^{15} + 10 q^{17} - 8 q^{19} - 10 q^{21} - 18 q^{23} + 23 q^{25} - 16 q^{27} - 14 q^{29} - 4 q^{31} + 14 q^{33} - 14 q^{35} - 16 q^{37} - 12 q^{39} + 10 q^{41} - 12 q^{43} - 10 q^{45} + 13 q^{47} + 9 q^{49} - 22 q^{51} - 26 q^{53} + 2 q^{55} + 20 q^{57} - 30 q^{59} - 18 q^{61} - 12 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} - 36 q^{71} + 10 q^{73} - 38 q^{75} - 42 q^{77} + 21 q^{81} - 12 q^{83} - 4 q^{85} - 6 q^{87} + 50 q^{89} + 4 q^{91} - 52 q^{93} - 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) \) 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 0
\(3\) −0.904794 −0.522383 −0.261191 0.965287i \(-0.584115\pi\)
−0.261191 + 0.965287i \(0.584115\pi\)
\(4\) 0 0
\(5\) −3.10347 −1.38791 −0.693957 0.720016i \(-0.744134\pi\)
−0.693957 + 0.720016i \(0.744134\pi\)
\(6\) 0 0
\(7\) 1.70842 0.645721 0.322860 0.946447i \(-0.395356\pi\)
0.322860 + 0.946447i \(0.395356\pi\)
\(8\) 0 0
\(9\) −2.18135 −0.727116
\(10\) 0 0
\(11\) 6.05981 1.82710 0.913551 0.406725i \(-0.133329\pi\)
0.913551 + 0.406725i \(0.133329\pi\)
\(12\) 0 0
\(13\) −1.97561 −0.547936 −0.273968 0.961739i \(-0.588336\pi\)
−0.273968 + 0.961739i \(0.588336\pi\)
\(14\) 0 0
\(15\) 2.80800 0.725023
\(16\) 0 0
\(17\) 0.802172 0.194555 0.0972777 0.995257i \(-0.468987\pi\)
0.0972777 + 0.995257i \(0.468987\pi\)
\(18\) 0 0
\(19\) −2.24854 −0.515850 −0.257925 0.966165i \(-0.583039\pi\)
−0.257925 + 0.966165i \(0.583039\pi\)
\(20\) 0 0
\(21\) −1.54576 −0.337314
\(22\) 0 0
\(23\) −2.79083 −0.581928 −0.290964 0.956734i \(-0.593976\pi\)
−0.290964 + 0.956734i \(0.593976\pi\)
\(24\) 0 0
\(25\) 4.63153 0.926306
\(26\) 0 0
\(27\) 4.68805 0.902216
\(28\) 0 0
\(29\) −9.23320 −1.71456 −0.857281 0.514848i \(-0.827848\pi\)
−0.857281 + 0.514848i \(0.827848\pi\)
\(30\) 0 0
\(31\) 7.61706 1.36806 0.684032 0.729452i \(-0.260225\pi\)
0.684032 + 0.729452i \(0.260225\pi\)
\(32\) 0 0
\(33\) −5.48288 −0.954446
\(34\) 0 0
\(35\) −5.30202 −0.896205
\(36\) 0 0
\(37\) −3.25627 −0.535327 −0.267664 0.963512i \(-0.586252\pi\)
−0.267664 + 0.963512i \(0.586252\pi\)
\(38\) 0 0
\(39\) 1.78752 0.286232
\(40\) 0 0
\(41\) 3.62554 0.566214 0.283107 0.959088i \(-0.408635\pi\)
0.283107 + 0.959088i \(0.408635\pi\)
\(42\) 0 0
\(43\) 0.635801 0.0969587 0.0484794 0.998824i \(-0.484562\pi\)
0.0484794 + 0.998824i \(0.484562\pi\)
\(44\) 0 0
\(45\) 6.76975 1.00917
\(46\) 0 0
\(47\) 1.00000 0.145865
\(48\) 0 0
\(49\) −4.08131 −0.583044
\(50\) 0 0
\(51\) −0.725800 −0.101632
\(52\) 0 0
\(53\) 4.24499 0.583094 0.291547 0.956557i \(-0.405830\pi\)
0.291547 + 0.956557i \(0.405830\pi\)
\(54\) 0 0
\(55\) −18.8064 −2.53586
\(56\) 0 0
\(57\) 2.03446 0.269471
\(58\) 0 0
\(59\) −2.01915 −0.262871 −0.131436 0.991325i \(-0.541959\pi\)
−0.131436 + 0.991325i \(0.541959\pi\)
\(60\) 0 0
\(61\) −6.89875 −0.883294 −0.441647 0.897189i \(-0.645606\pi\)
−0.441647 + 0.897189i \(0.645606\pi\)
\(62\) 0 0
\(63\) −3.72665 −0.469514
\(64\) 0 0
\(65\) 6.13125 0.760488
\(66\) 0 0
\(67\) 6.92370 0.845865 0.422933 0.906161i \(-0.361001\pi\)
0.422933 + 0.906161i \(0.361001\pi\)
\(68\) 0 0
\(69\) 2.52512 0.303989
\(70\) 0 0
\(71\) 11.3962 1.35248 0.676240 0.736682i \(-0.263608\pi\)
0.676240 + 0.736682i \(0.263608\pi\)
\(72\) 0 0
\(73\) 14.6621 1.71607 0.858034 0.513593i \(-0.171686\pi\)
0.858034 + 0.513593i \(0.171686\pi\)
\(74\) 0 0
\(75\) −4.19058 −0.483886
\(76\) 0 0
\(77\) 10.3527 1.17980
\(78\) 0 0
\(79\) 12.3443 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(80\) 0 0
\(81\) 2.30233 0.255814
\(82\) 0 0
\(83\) 0.818293 0.0898194 0.0449097 0.998991i \(-0.485700\pi\)
0.0449097 + 0.998991i \(0.485700\pi\)
\(84\) 0 0
\(85\) −2.48952 −0.270026
\(86\) 0 0
\(87\) 8.35414 0.895658
\(88\) 0 0
\(89\) 10.3807 1.10035 0.550175 0.835050i \(-0.314561\pi\)
0.550175 + 0.835050i \(0.314561\pi\)
\(90\) 0 0
\(91\) −3.37517 −0.353813
\(92\) 0 0
\(93\) −6.89187 −0.714653
\(94\) 0 0
\(95\) 6.97827 0.715955
\(96\) 0 0
\(97\) −16.2464 −1.64958 −0.824788 0.565442i \(-0.808705\pi\)
−0.824788 + 0.565442i \(0.808705\pi\)
\(98\) 0 0
\(99\) −13.2186 −1.32851
\(100\) 0 0
\(101\) −6.19583 −0.616508 −0.308254 0.951304i \(-0.599745\pi\)
−0.308254 + 0.951304i \(0.599745\pi\)
\(102\) 0 0
\(103\) −5.69517 −0.561162 −0.280581 0.959830i \(-0.590527\pi\)
−0.280581 + 0.959830i \(0.590527\pi\)
\(104\) 0 0
\(105\) 4.79724 0.468162
\(106\) 0 0
\(107\) −18.8124 −1.81867 −0.909333 0.416069i \(-0.863408\pi\)
−0.909333 + 0.416069i \(0.863408\pi\)
\(108\) 0 0
\(109\) −13.6970 −1.31193 −0.655966 0.754791i \(-0.727738\pi\)
−0.655966 + 0.754791i \(0.727738\pi\)
\(110\) 0 0
\(111\) 2.94625 0.279646
\(112\) 0 0
\(113\) 0.463650 0.0436165 0.0218083 0.999762i \(-0.493058\pi\)
0.0218083 + 0.999762i \(0.493058\pi\)
\(114\) 0 0
\(115\) 8.66126 0.807666
\(116\) 0 0
\(117\) 4.30949 0.398413
\(118\) 0 0
\(119\) 1.37044 0.125628
\(120\) 0 0
\(121\) 25.7213 2.33830
\(122\) 0 0
\(123\) −3.28036 −0.295780
\(124\) 0 0
\(125\) 1.14354 0.102281
\(126\) 0 0
\(127\) −12.4102 −1.10123 −0.550615 0.834759i \(-0.685607\pi\)
−0.550615 + 0.834759i \(0.685607\pi\)
\(128\) 0 0
\(129\) −0.575269 −0.0506496
\(130\) 0 0
\(131\) −19.1000 −1.66877 −0.834386 0.551181i \(-0.814177\pi\)
−0.834386 + 0.551181i \(0.814177\pi\)
\(132\) 0 0
\(133\) −3.84144 −0.333095
\(134\) 0 0
\(135\) −14.5492 −1.25220
\(136\) 0 0
\(137\) 16.1961 1.38372 0.691862 0.722030i \(-0.256791\pi\)
0.691862 + 0.722030i \(0.256791\pi\)
\(138\) 0 0
\(139\) 15.2948 1.29728 0.648642 0.761093i \(-0.275337\pi\)
0.648642 + 0.761093i \(0.275337\pi\)
\(140\) 0 0
\(141\) −0.904794 −0.0761974
\(142\) 0 0
\(143\) −11.9718 −1.00113
\(144\) 0 0
\(145\) 28.6550 2.37967
\(146\) 0 0
\(147\) 3.69274 0.304572
\(148\) 0 0
\(149\) −18.9005 −1.54839 −0.774194 0.632949i \(-0.781844\pi\)
−0.774194 + 0.632949i \(0.781844\pi\)
\(150\) 0 0
\(151\) −6.22346 −0.506458 −0.253229 0.967406i \(-0.581493\pi\)
−0.253229 + 0.967406i \(0.581493\pi\)
\(152\) 0 0
\(153\) −1.74982 −0.141464
\(154\) 0 0
\(155\) −23.6393 −1.89876
\(156\) 0 0
\(157\) 18.3932 1.46794 0.733970 0.679182i \(-0.237665\pi\)
0.733970 + 0.679182i \(0.237665\pi\)
\(158\) 0 0
\(159\) −3.84084 −0.304598
\(160\) 0 0
\(161\) −4.76790 −0.375763
\(162\) 0 0
\(163\) 4.29391 0.336325 0.168162 0.985759i \(-0.446217\pi\)
0.168162 + 0.985759i \(0.446217\pi\)
\(164\) 0 0
\(165\) 17.0159 1.32469
\(166\) 0 0
\(167\) −15.3400 −1.18704 −0.593521 0.804818i \(-0.702263\pi\)
−0.593521 + 0.804818i \(0.702263\pi\)
\(168\) 0 0
\(169\) −9.09697 −0.699767
\(170\) 0 0
\(171\) 4.90484 0.375083
\(172\) 0 0
\(173\) 7.97616 0.606417 0.303208 0.952924i \(-0.401942\pi\)
0.303208 + 0.952924i \(0.401942\pi\)
\(174\) 0 0
\(175\) 7.91258 0.598135
\(176\) 0 0
\(177\) 1.82691 0.137319
\(178\) 0 0
\(179\) −10.5311 −0.787129 −0.393564 0.919297i \(-0.628758\pi\)
−0.393564 + 0.919297i \(0.628758\pi\)
\(180\) 0 0
\(181\) 15.3224 1.13891 0.569453 0.822024i \(-0.307155\pi\)
0.569453 + 0.822024i \(0.307155\pi\)
\(182\) 0 0
\(183\) 6.24195 0.461418
\(184\) 0 0
\(185\) 10.1057 0.742988
\(186\) 0 0
\(187\) 4.86101 0.355472
\(188\) 0 0
\(189\) 8.00915 0.582580
\(190\) 0 0
\(191\) −16.6814 −1.20703 −0.603513 0.797353i \(-0.706233\pi\)
−0.603513 + 0.797353i \(0.706233\pi\)
\(192\) 0 0
\(193\) 4.75035 0.341938 0.170969 0.985276i \(-0.445310\pi\)
0.170969 + 0.985276i \(0.445310\pi\)
\(194\) 0 0
\(195\) −5.54751 −0.397266
\(196\) 0 0
\(197\) −3.61648 −0.257663 −0.128832 0.991666i \(-0.541123\pi\)
−0.128832 + 0.991666i \(0.541123\pi\)
\(198\) 0 0
\(199\) −4.05163 −0.287212 −0.143606 0.989635i \(-0.545870\pi\)
−0.143606 + 0.989635i \(0.545870\pi\)
\(200\) 0 0
\(201\) −6.26452 −0.441865
\(202\) 0 0
\(203\) −15.7742 −1.10713
\(204\) 0 0
\(205\) −11.2518 −0.785856
\(206\) 0 0
\(207\) 6.08777 0.423129
\(208\) 0 0
\(209\) −13.6257 −0.942510
\(210\) 0 0
\(211\) −1.38418 −0.0952910 −0.0476455 0.998864i \(-0.515172\pi\)
−0.0476455 + 0.998864i \(0.515172\pi\)
\(212\) 0 0
\(213\) −10.3112 −0.706512
\(214\) 0 0
\(215\) −1.97319 −0.134570
\(216\) 0 0
\(217\) 13.0131 0.883388
\(218\) 0 0
\(219\) −13.2662 −0.896445
\(220\) 0 0
\(221\) −1.58478 −0.106604
\(222\) 0 0
\(223\) −23.9003 −1.60048 −0.800242 0.599677i \(-0.795296\pi\)
−0.800242 + 0.599677i \(0.795296\pi\)
\(224\) 0 0
\(225\) −10.1030 −0.673532
\(226\) 0 0
\(227\) −11.0611 −0.734150 −0.367075 0.930191i \(-0.619641\pi\)
−0.367075 + 0.930191i \(0.619641\pi\)
\(228\) 0 0
\(229\) −26.8042 −1.77127 −0.885637 0.464379i \(-0.846278\pi\)
−0.885637 + 0.464379i \(0.846278\pi\)
\(230\) 0 0
\(231\) −9.36704 −0.616306
\(232\) 0 0
\(233\) −20.2502 −1.32663 −0.663316 0.748340i \(-0.730851\pi\)
−0.663316 + 0.748340i \(0.730851\pi\)
\(234\) 0 0
\(235\) −3.10347 −0.202448
\(236\) 0 0
\(237\) −11.1691 −0.725510
\(238\) 0 0
\(239\) −7.76035 −0.501975 −0.250988 0.967990i \(-0.580755\pi\)
−0.250988 + 0.967990i \(0.580755\pi\)
\(240\) 0 0
\(241\) −11.5432 −0.743564 −0.371782 0.928320i \(-0.621253\pi\)
−0.371782 + 0.928320i \(0.621253\pi\)
\(242\) 0 0
\(243\) −16.1473 −1.03585
\(244\) 0 0
\(245\) 12.6662 0.809216
\(246\) 0 0
\(247\) 4.44223 0.282652
\(248\) 0 0
\(249\) −0.740387 −0.0469201
\(250\) 0 0
\(251\) −3.90956 −0.246769 −0.123385 0.992359i \(-0.539375\pi\)
−0.123385 + 0.992359i \(0.539375\pi\)
\(252\) 0 0
\(253\) −16.9119 −1.06324
\(254\) 0 0
\(255\) 2.25250 0.141057
\(256\) 0 0
\(257\) −5.92556 −0.369626 −0.184813 0.982774i \(-0.559168\pi\)
−0.184813 + 0.982774i \(0.559168\pi\)
\(258\) 0 0
\(259\) −5.56307 −0.345672
\(260\) 0 0
\(261\) 20.1408 1.24669
\(262\) 0 0
\(263\) 0.888642 0.0547960 0.0273980 0.999625i \(-0.491278\pi\)
0.0273980 + 0.999625i \(0.491278\pi\)
\(264\) 0 0
\(265\) −13.1742 −0.809284
\(266\) 0 0
\(267\) −9.39237 −0.574804
\(268\) 0 0
\(269\) −14.2461 −0.868598 −0.434299 0.900769i \(-0.643004\pi\)
−0.434299 + 0.900769i \(0.643004\pi\)
\(270\) 0 0
\(271\) 1.78285 0.108300 0.0541501 0.998533i \(-0.482755\pi\)
0.0541501 + 0.998533i \(0.482755\pi\)
\(272\) 0 0
\(273\) 3.05383 0.184826
\(274\) 0 0
\(275\) 28.0662 1.69245
\(276\) 0 0
\(277\) −10.3947 −0.624557 −0.312278 0.949991i \(-0.601092\pi\)
−0.312278 + 0.949991i \(0.601092\pi\)
\(278\) 0 0
\(279\) −16.6155 −0.994742
\(280\) 0 0
\(281\) 6.11705 0.364913 0.182456 0.983214i \(-0.441595\pi\)
0.182456 + 0.983214i \(0.441595\pi\)
\(282\) 0 0
\(283\) 23.0070 1.36763 0.683813 0.729657i \(-0.260320\pi\)
0.683813 + 0.729657i \(0.260320\pi\)
\(284\) 0 0
\(285\) −6.31389 −0.374003
\(286\) 0 0
\(287\) 6.19393 0.365616
\(288\) 0 0
\(289\) −16.3565 −0.962148
\(290\) 0 0
\(291\) 14.6997 0.861710
\(292\) 0 0
\(293\) −11.0586 −0.646053 −0.323027 0.946390i \(-0.604700\pi\)
−0.323027 + 0.946390i \(0.604700\pi\)
\(294\) 0 0
\(295\) 6.26637 0.364842
\(296\) 0 0
\(297\) 28.4087 1.64844
\(298\) 0 0
\(299\) 5.51359 0.318859
\(300\) 0 0
\(301\) 1.08621 0.0626083
\(302\) 0 0
\(303\) 5.60594 0.322053
\(304\) 0 0
\(305\) 21.4101 1.22594
\(306\) 0 0
\(307\) 6.42157 0.366499 0.183249 0.983066i \(-0.441338\pi\)
0.183249 + 0.983066i \(0.441338\pi\)
\(308\) 0 0
\(309\) 5.15296 0.293141
\(310\) 0 0
\(311\) 17.1247 0.971053 0.485527 0.874222i \(-0.338628\pi\)
0.485527 + 0.874222i \(0.338628\pi\)
\(312\) 0 0
\(313\) −31.8876 −1.80239 −0.901196 0.433412i \(-0.857309\pi\)
−0.901196 + 0.433412i \(0.857309\pi\)
\(314\) 0 0
\(315\) 11.5656 0.651645
\(316\) 0 0
\(317\) −15.2104 −0.854303 −0.427152 0.904180i \(-0.640483\pi\)
−0.427152 + 0.904180i \(0.640483\pi\)
\(318\) 0 0
\(319\) −55.9514 −3.13268
\(320\) 0 0
\(321\) 17.0214 0.950040
\(322\) 0 0
\(323\) −1.80371 −0.100361
\(324\) 0 0
\(325\) −9.15010 −0.507556
\(326\) 0 0
\(327\) 12.3929 0.685331
\(328\) 0 0
\(329\) 1.70842 0.0941881
\(330\) 0 0
\(331\) 13.8340 0.760384 0.380192 0.924908i \(-0.375858\pi\)
0.380192 + 0.924908i \(0.375858\pi\)
\(332\) 0 0
\(333\) 7.10306 0.389245
\(334\) 0 0
\(335\) −21.4875 −1.17399
\(336\) 0 0
\(337\) −10.4625 −0.569930 −0.284965 0.958538i \(-0.591982\pi\)
−0.284965 + 0.958538i \(0.591982\pi\)
\(338\) 0 0
\(339\) −0.419507 −0.0227845
\(340\) 0 0
\(341\) 46.1579 2.49959
\(342\) 0 0
\(343\) −18.9315 −1.02220
\(344\) 0 0
\(345\) −7.83665 −0.421911
\(346\) 0 0
\(347\) 7.13337 0.382939 0.191470 0.981499i \(-0.438675\pi\)
0.191470 + 0.981499i \(0.438675\pi\)
\(348\) 0 0
\(349\) −21.7215 −1.16272 −0.581362 0.813645i \(-0.697480\pi\)
−0.581362 + 0.813645i \(0.697480\pi\)
\(350\) 0 0
\(351\) −9.26176 −0.494356
\(352\) 0 0
\(353\) −7.88265 −0.419551 −0.209776 0.977750i \(-0.567273\pi\)
−0.209776 + 0.977750i \(0.567273\pi\)
\(354\) 0 0
\(355\) −35.3677 −1.87713
\(356\) 0 0
\(357\) −1.23997 −0.0656262
\(358\) 0 0
\(359\) −20.7942 −1.09747 −0.548737 0.835995i \(-0.684891\pi\)
−0.548737 + 0.835995i \(0.684891\pi\)
\(360\) 0 0
\(361\) −13.9441 −0.733899
\(362\) 0 0
\(363\) −23.2725 −1.22149
\(364\) 0 0
\(365\) −45.5034 −2.38176
\(366\) 0 0
\(367\) 3.84224 0.200563 0.100282 0.994959i \(-0.468026\pi\)
0.100282 + 0.994959i \(0.468026\pi\)
\(368\) 0 0
\(369\) −7.90856 −0.411703
\(370\) 0 0
\(371\) 7.25221 0.376516
\(372\) 0 0
\(373\) 4.67398 0.242010 0.121005 0.992652i \(-0.461388\pi\)
0.121005 + 0.992652i \(0.461388\pi\)
\(374\) 0 0
\(375\) −1.03466 −0.0534298
\(376\) 0 0
\(377\) 18.2412 0.939470
\(378\) 0 0
\(379\) 33.5947 1.72564 0.862822 0.505508i \(-0.168695\pi\)
0.862822 + 0.505508i \(0.168695\pi\)
\(380\) 0 0
\(381\) 11.2287 0.575264
\(382\) 0 0
\(383\) −13.7061 −0.700350 −0.350175 0.936684i \(-0.613878\pi\)
−0.350175 + 0.936684i \(0.613878\pi\)
\(384\) 0 0
\(385\) −32.1292 −1.63746
\(386\) 0 0
\(387\) −1.38690 −0.0705003
\(388\) 0 0
\(389\) −5.21325 −0.264322 −0.132161 0.991228i \(-0.542192\pi\)
−0.132161 + 0.991228i \(0.542192\pi\)
\(390\) 0 0
\(391\) −2.23873 −0.113217
\(392\) 0 0
\(393\) 17.2815 0.871738
\(394\) 0 0
\(395\) −38.3103 −1.92760
\(396\) 0 0
\(397\) 36.2484 1.81926 0.909629 0.415422i \(-0.136366\pi\)
0.909629 + 0.415422i \(0.136366\pi\)
\(398\) 0 0
\(399\) 3.47571 0.174003
\(400\) 0 0
\(401\) −16.7780 −0.837852 −0.418926 0.908020i \(-0.637593\pi\)
−0.418926 + 0.908020i \(0.637593\pi\)
\(402\) 0 0
\(403\) −15.0483 −0.749611
\(404\) 0 0
\(405\) −7.14520 −0.355048
\(406\) 0 0
\(407\) −19.7324 −0.978097
\(408\) 0 0
\(409\) 21.5321 1.06469 0.532346 0.846527i \(-0.321311\pi\)
0.532346 + 0.846527i \(0.321311\pi\)
\(410\) 0 0
\(411\) −14.6541 −0.722833
\(412\) 0 0
\(413\) −3.44955 −0.169741
\(414\) 0 0
\(415\) −2.53955 −0.124662
\(416\) 0 0
\(417\) −13.8386 −0.677679
\(418\) 0 0
\(419\) −5.99082 −0.292671 −0.146335 0.989235i \(-0.546748\pi\)
−0.146335 + 0.989235i \(0.546748\pi\)
\(420\) 0 0
\(421\) −27.6128 −1.34577 −0.672884 0.739748i \(-0.734945\pi\)
−0.672884 + 0.739748i \(0.734945\pi\)
\(422\) 0 0
\(423\) −2.18135 −0.106061
\(424\) 0 0
\(425\) 3.71529 0.180218
\(426\) 0 0
\(427\) −11.7859 −0.570362
\(428\) 0 0
\(429\) 10.8320 0.522975
\(430\) 0 0
\(431\) 14.2613 0.686940 0.343470 0.939164i \(-0.388398\pi\)
0.343470 + 0.939164i \(0.388398\pi\)
\(432\) 0 0
\(433\) 33.4057 1.60537 0.802687 0.596400i \(-0.203403\pi\)
0.802687 + 0.596400i \(0.203403\pi\)
\(434\) 0 0
\(435\) −25.9268 −1.24310
\(436\) 0 0
\(437\) 6.27528 0.300188
\(438\) 0 0
\(439\) 11.5898 0.553151 0.276575 0.960992i \(-0.410800\pi\)
0.276575 + 0.960992i \(0.410800\pi\)
\(440\) 0 0
\(441\) 8.90276 0.423941
\(442\) 0 0
\(443\) 30.6610 1.45675 0.728375 0.685179i \(-0.240276\pi\)
0.728375 + 0.685179i \(0.240276\pi\)
\(444\) 0 0
\(445\) −32.2161 −1.52719
\(446\) 0 0
\(447\) 17.1010 0.808851
\(448\) 0 0
\(449\) 2.55670 0.120658 0.0603290 0.998179i \(-0.480785\pi\)
0.0603290 + 0.998179i \(0.480785\pi\)
\(450\) 0 0
\(451\) 21.9701 1.03453
\(452\) 0 0
\(453\) 5.63095 0.264565
\(454\) 0 0
\(455\) 10.4747 0.491063
\(456\) 0 0
\(457\) 0.723510 0.0338444 0.0169222 0.999857i \(-0.494613\pi\)
0.0169222 + 0.999857i \(0.494613\pi\)
\(458\) 0 0
\(459\) 3.76062 0.175531
\(460\) 0 0
\(461\) −3.29442 −0.153436 −0.0767182 0.997053i \(-0.524444\pi\)
−0.0767182 + 0.997053i \(0.524444\pi\)
\(462\) 0 0
\(463\) −0.386079 −0.0179426 −0.00897130 0.999960i \(-0.502856\pi\)
−0.00897130 + 0.999960i \(0.502856\pi\)
\(464\) 0 0
\(465\) 21.3887 0.991878
\(466\) 0 0
\(467\) −18.8400 −0.871813 −0.435906 0.899992i \(-0.643572\pi\)
−0.435906 + 0.899992i \(0.643572\pi\)
\(468\) 0 0
\(469\) 11.8286 0.546193
\(470\) 0 0
\(471\) −16.6421 −0.766826
\(472\) 0 0
\(473\) 3.85283 0.177153
\(474\) 0 0
\(475\) −10.4142 −0.477835
\(476\) 0 0
\(477\) −9.25980 −0.423977
\(478\) 0 0
\(479\) 3.98968 0.182293 0.0911465 0.995837i \(-0.470947\pi\)
0.0911465 + 0.995837i \(0.470947\pi\)
\(480\) 0 0
\(481\) 6.43312 0.293325
\(482\) 0 0
\(483\) 4.31397 0.196292
\(484\) 0 0
\(485\) 50.4203 2.28947
\(486\) 0 0
\(487\) 6.84955 0.310383 0.155191 0.987884i \(-0.450401\pi\)
0.155191 + 0.987884i \(0.450401\pi\)
\(488\) 0 0
\(489\) −3.88510 −0.175690
\(490\) 0 0
\(491\) −9.02535 −0.407308 −0.203654 0.979043i \(-0.565282\pi\)
−0.203654 + 0.979043i \(0.565282\pi\)
\(492\) 0 0
\(493\) −7.40662 −0.333577
\(494\) 0 0
\(495\) 41.0234 1.84386
\(496\) 0 0
\(497\) 19.4694 0.873324
\(498\) 0 0
\(499\) 1.93588 0.0866619 0.0433309 0.999061i \(-0.486203\pi\)
0.0433309 + 0.999061i \(0.486203\pi\)
\(500\) 0 0
\(501\) 13.8795 0.620091
\(502\) 0 0
\(503\) −20.4584 −0.912196 −0.456098 0.889929i \(-0.650753\pi\)
−0.456098 + 0.889929i \(0.650753\pi\)
\(504\) 0 0
\(505\) 19.2286 0.855660
\(506\) 0 0
\(507\) 8.23088 0.365546
\(508\) 0 0
\(509\) −13.5664 −0.601319 −0.300659 0.953732i \(-0.597207\pi\)
−0.300659 + 0.953732i \(0.597207\pi\)
\(510\) 0 0
\(511\) 25.0490 1.10810
\(512\) 0 0
\(513\) −10.5413 −0.465408
\(514\) 0 0
\(515\) 17.6748 0.778845
\(516\) 0 0
\(517\) 6.05981 0.266510
\(518\) 0 0
\(519\) −7.21678 −0.316782
\(520\) 0 0
\(521\) −25.7717 −1.12908 −0.564540 0.825406i \(-0.690946\pi\)
−0.564540 + 0.825406i \(0.690946\pi\)
\(522\) 0 0
\(523\) −1.82465 −0.0797862 −0.0398931 0.999204i \(-0.512702\pi\)
−0.0398931 + 0.999204i \(0.512702\pi\)
\(524\) 0 0
\(525\) −7.15926 −0.312456
\(526\) 0 0
\(527\) 6.11020 0.266164
\(528\) 0 0
\(529\) −15.2113 −0.661360
\(530\) 0 0
\(531\) 4.40447 0.191138
\(532\) 0 0
\(533\) −7.16265 −0.310249
\(534\) 0 0
\(535\) 58.3838 2.52415
\(536\) 0 0
\(537\) 9.52844 0.411182
\(538\) 0 0
\(539\) −24.7320 −1.06528
\(540\) 0 0
\(541\) −36.2752 −1.55959 −0.779796 0.626034i \(-0.784677\pi\)
−0.779796 + 0.626034i \(0.784677\pi\)
\(542\) 0 0
\(543\) −13.8636 −0.594945
\(544\) 0 0
\(545\) 42.5081 1.82085
\(546\) 0 0
\(547\) −38.4379 −1.64349 −0.821744 0.569857i \(-0.806999\pi\)
−0.821744 + 0.569857i \(0.806999\pi\)
\(548\) 0 0
\(549\) 15.0486 0.642258
\(550\) 0 0
\(551\) 20.7612 0.884457
\(552\) 0 0
\(553\) 21.0893 0.896808
\(554\) 0 0
\(555\) −9.14360 −0.388124
\(556\) 0 0
\(557\) −12.9869 −0.550274 −0.275137 0.961405i \(-0.588723\pi\)
−0.275137 + 0.961405i \(0.588723\pi\)
\(558\) 0 0
\(559\) −1.25609 −0.0531271
\(560\) 0 0
\(561\) −4.39821 −0.185693
\(562\) 0 0
\(563\) 18.6370 0.785455 0.392727 0.919655i \(-0.371532\pi\)
0.392727 + 0.919655i \(0.371532\pi\)
\(564\) 0 0
\(565\) −1.43892 −0.0605360
\(566\) 0 0
\(567\) 3.93333 0.165185
\(568\) 0 0
\(569\) −10.6112 −0.444845 −0.222422 0.974950i \(-0.571396\pi\)
−0.222422 + 0.974950i \(0.571396\pi\)
\(570\) 0 0
\(571\) 26.5317 1.11032 0.555158 0.831745i \(-0.312658\pi\)
0.555158 + 0.831745i \(0.312658\pi\)
\(572\) 0 0
\(573\) 15.0932 0.630529
\(574\) 0 0
\(575\) −12.9258 −0.539044
\(576\) 0 0
\(577\) 7.25089 0.301858 0.150929 0.988545i \(-0.451773\pi\)
0.150929 + 0.988545i \(0.451773\pi\)
\(578\) 0 0
\(579\) −4.29809 −0.178623
\(580\) 0 0
\(581\) 1.39799 0.0579982
\(582\) 0 0
\(583\) 25.7238 1.06537
\(584\) 0 0
\(585\) −13.3744 −0.552963
\(586\) 0 0
\(587\) 15.5420 0.641487 0.320743 0.947166i \(-0.396067\pi\)
0.320743 + 0.947166i \(0.396067\pi\)
\(588\) 0 0
\(589\) −17.1272 −0.705716
\(590\) 0 0
\(591\) 3.27217 0.134599
\(592\) 0 0
\(593\) −9.53631 −0.391609 −0.195805 0.980643i \(-0.562732\pi\)
−0.195805 + 0.980643i \(0.562732\pi\)
\(594\) 0 0
\(595\) −4.25314 −0.174362
\(596\) 0 0
\(597\) 3.66589 0.150035
\(598\) 0 0
\(599\) −28.8238 −1.17771 −0.588854 0.808239i \(-0.700421\pi\)
−0.588854 + 0.808239i \(0.700421\pi\)
\(600\) 0 0
\(601\) 25.6563 1.04654 0.523272 0.852166i \(-0.324711\pi\)
0.523272 + 0.852166i \(0.324711\pi\)
\(602\) 0 0
\(603\) −15.1030 −0.615042
\(604\) 0 0
\(605\) −79.8253 −3.24536
\(606\) 0 0
\(607\) 10.1171 0.410639 0.205319 0.978695i \(-0.434177\pi\)
0.205319 + 0.978695i \(0.434177\pi\)
\(608\) 0 0
\(609\) 14.2724 0.578345
\(610\) 0 0
\(611\) −1.97561 −0.0799246
\(612\) 0 0
\(613\) 33.2492 1.34292 0.671461 0.741040i \(-0.265667\pi\)
0.671461 + 0.741040i \(0.265667\pi\)
\(614\) 0 0
\(615\) 10.1805 0.410518
\(616\) 0 0
\(617\) −39.1115 −1.57457 −0.787285 0.616589i \(-0.788514\pi\)
−0.787285 + 0.616589i \(0.788514\pi\)
\(618\) 0 0
\(619\) 19.7327 0.793122 0.396561 0.918008i \(-0.370203\pi\)
0.396561 + 0.918008i \(0.370203\pi\)
\(620\) 0 0
\(621\) −13.0836 −0.525025
\(622\) 0 0
\(623\) 17.7345 0.710519
\(624\) 0 0
\(625\) −26.7066 −1.06826
\(626\) 0 0
\(627\) 12.3285 0.492351
\(628\) 0 0
\(629\) −2.61209 −0.104151
\(630\) 0 0
\(631\) −2.94916 −0.117404 −0.0587022 0.998276i \(-0.518696\pi\)
−0.0587022 + 0.998276i \(0.518696\pi\)
\(632\) 0 0
\(633\) 1.25240 0.0497784
\(634\) 0 0
\(635\) 38.5148 1.52841
\(636\) 0 0
\(637\) 8.06308 0.319471
\(638\) 0 0
\(639\) −24.8591 −0.983409
\(640\) 0 0
\(641\) 40.8533 1.61361 0.806804 0.590819i \(-0.201195\pi\)
0.806804 + 0.590819i \(0.201195\pi\)
\(642\) 0 0
\(643\) −49.5313 −1.95332 −0.976662 0.214784i \(-0.931095\pi\)
−0.976662 + 0.214784i \(0.931095\pi\)
\(644\) 0 0
\(645\) 1.78533 0.0702973
\(646\) 0 0
\(647\) 20.1422 0.791872 0.395936 0.918278i \(-0.370420\pi\)
0.395936 + 0.918278i \(0.370420\pi\)
\(648\) 0 0
\(649\) −12.2357 −0.480292
\(650\) 0 0
\(651\) −11.7742 −0.461467
\(652\) 0 0
\(653\) −31.5708 −1.23546 −0.617731 0.786390i \(-0.711948\pi\)
−0.617731 + 0.786390i \(0.711948\pi\)
\(654\) 0 0
\(655\) 59.2762 2.31611
\(656\) 0 0
\(657\) −31.9831 −1.24778
\(658\) 0 0
\(659\) −24.5255 −0.955379 −0.477689 0.878529i \(-0.658526\pi\)
−0.477689 + 0.878529i \(0.658526\pi\)
\(660\) 0 0
\(661\) −19.1347 −0.744254 −0.372127 0.928182i \(-0.621371\pi\)
−0.372127 + 0.928182i \(0.621371\pi\)
\(662\) 0 0
\(663\) 1.43390 0.0556880
\(664\) 0 0
\(665\) 11.9218 0.462307
\(666\) 0 0
\(667\) 25.7683 0.997753
\(668\) 0 0
\(669\) 21.6249 0.836065
\(670\) 0 0
\(671\) −41.8051 −1.61387
\(672\) 0 0
\(673\) 15.0430 0.579865 0.289932 0.957047i \(-0.406367\pi\)
0.289932 + 0.957047i \(0.406367\pi\)
\(674\) 0 0
\(675\) 21.7128 0.835728
\(676\) 0 0
\(677\) 36.2052 1.39148 0.695740 0.718294i \(-0.255077\pi\)
0.695740 + 0.718294i \(0.255077\pi\)
\(678\) 0 0
\(679\) −27.7557 −1.06517
\(680\) 0 0
\(681\) 10.0080 0.383508
\(682\) 0 0
\(683\) −15.5572 −0.595278 −0.297639 0.954678i \(-0.596199\pi\)
−0.297639 + 0.954678i \(0.596199\pi\)
\(684\) 0 0
\(685\) −50.2640 −1.92049
\(686\) 0 0
\(687\) 24.2523 0.925283
\(688\) 0 0
\(689\) −8.38644 −0.319498
\(690\) 0 0
\(691\) 44.9577 1.71027 0.855135 0.518405i \(-0.173474\pi\)
0.855135 + 0.518405i \(0.173474\pi\)
\(692\) 0 0
\(693\) −22.5828 −0.857850
\(694\) 0 0
\(695\) −47.4668 −1.80052
\(696\) 0 0
\(697\) 2.90831 0.110160
\(698\) 0 0
\(699\) 18.3222 0.693010
\(700\) 0 0
\(701\) −9.29361 −0.351015 −0.175507 0.984478i \(-0.556157\pi\)
−0.175507 + 0.984478i \(0.556157\pi\)
\(702\) 0 0
\(703\) 7.32184 0.276148
\(704\) 0 0
\(705\) 2.80800 0.105755
\(706\) 0 0
\(707\) −10.5851 −0.398092
\(708\) 0 0
\(709\) −38.6450 −1.45134 −0.725672 0.688040i \(-0.758471\pi\)
−0.725672 + 0.688040i \(0.758471\pi\)
\(710\) 0 0
\(711\) −26.9273 −1.00985
\(712\) 0 0
\(713\) −21.2579 −0.796115
\(714\) 0 0
\(715\) 37.1542 1.38949
\(716\) 0 0
\(717\) 7.02152 0.262223
\(718\) 0 0
\(719\) −27.5083 −1.02588 −0.512942 0.858423i \(-0.671445\pi\)
−0.512942 + 0.858423i \(0.671445\pi\)
\(720\) 0 0
\(721\) −9.72973 −0.362354
\(722\) 0 0
\(723\) 10.4442 0.388425
\(724\) 0 0
\(725\) −42.7639 −1.58821
\(726\) 0 0
\(727\) −11.4162 −0.423403 −0.211702 0.977334i \(-0.567900\pi\)
−0.211702 + 0.977334i \(0.567900\pi\)
\(728\) 0 0
\(729\) 7.70298 0.285295
\(730\) 0 0
\(731\) 0.510022 0.0188638
\(732\) 0 0
\(733\) 4.02880 0.148807 0.0744035 0.997228i \(-0.476295\pi\)
0.0744035 + 0.997228i \(0.476295\pi\)
\(734\) 0 0
\(735\) −11.4603 −0.422720
\(736\) 0 0
\(737\) 41.9563 1.54548
\(738\) 0 0
\(739\) 20.6451 0.759441 0.379720 0.925101i \(-0.376020\pi\)
0.379720 + 0.925101i \(0.376020\pi\)
\(740\) 0 0
\(741\) −4.01930 −0.147653
\(742\) 0 0
\(743\) 17.1888 0.630596 0.315298 0.948993i \(-0.397896\pi\)
0.315298 + 0.948993i \(0.397896\pi\)
\(744\) 0 0
\(745\) 58.6571 2.14903
\(746\) 0 0
\(747\) −1.78498 −0.0653091
\(748\) 0 0
\(749\) −32.1395 −1.17435
\(750\) 0 0
\(751\) −31.8442 −1.16201 −0.581006 0.813899i \(-0.697341\pi\)
−0.581006 + 0.813899i \(0.697341\pi\)
\(752\) 0 0
\(753\) 3.53735 0.128908
\(754\) 0 0
\(755\) 19.3143 0.702921
\(756\) 0 0
\(757\) 34.5104 1.25430 0.627151 0.778898i \(-0.284221\pi\)
0.627151 + 0.778898i \(0.284221\pi\)
\(758\) 0 0
\(759\) 15.3018 0.555419
\(760\) 0 0
\(761\) 12.4728 0.452137 0.226068 0.974111i \(-0.427413\pi\)
0.226068 + 0.974111i \(0.427413\pi\)
\(762\) 0 0
\(763\) −23.4001 −0.847142
\(764\) 0 0
\(765\) 5.43051 0.196340
\(766\) 0 0
\(767\) 3.98905 0.144036
\(768\) 0 0
\(769\) 11.7854 0.424993 0.212496 0.977162i \(-0.431841\pi\)
0.212496 + 0.977162i \(0.431841\pi\)
\(770\) 0 0
\(771\) 5.36141 0.193086
\(772\) 0 0
\(773\) −0.175957 −0.00632874 −0.00316437 0.999995i \(-0.501007\pi\)
−0.00316437 + 0.999995i \(0.501007\pi\)
\(774\) 0 0
\(775\) 35.2786 1.26725
\(776\) 0 0
\(777\) 5.03343 0.180573
\(778\) 0 0
\(779\) −8.15216 −0.292081
\(780\) 0 0
\(781\) 69.0587 2.47112
\(782\) 0 0
\(783\) −43.2857 −1.54691
\(784\) 0 0
\(785\) −57.0828 −2.03737
\(786\) 0 0
\(787\) 2.39218 0.0852719 0.0426359 0.999091i \(-0.486424\pi\)
0.0426359 + 0.999091i \(0.486424\pi\)
\(788\) 0 0
\(789\) −0.804038 −0.0286245
\(790\) 0 0
\(791\) 0.792107 0.0281641
\(792\) 0 0
\(793\) 13.6292 0.483988
\(794\) 0 0
\(795\) 11.9199 0.422756
\(796\) 0 0
\(797\) 19.8575 0.703390 0.351695 0.936115i \(-0.385605\pi\)
0.351695 + 0.936115i \(0.385605\pi\)
\(798\) 0 0
\(799\) 0.802172 0.0283788
\(800\) 0 0
\(801\) −22.6439 −0.800082
\(802\) 0 0
\(803\) 88.8495 3.13543
\(804\) 0 0
\(805\) 14.7970 0.521527
\(806\) 0 0
\(807\) 12.8897 0.453741
\(808\) 0 0
\(809\) 2.50199 0.0879652 0.0439826 0.999032i \(-0.485995\pi\)
0.0439826 + 0.999032i \(0.485995\pi\)
\(810\) 0 0
\(811\) −34.4012 −1.20799 −0.603995 0.796988i \(-0.706425\pi\)
−0.603995 + 0.796988i \(0.706425\pi\)
\(812\) 0 0
\(813\) −1.61311 −0.0565741
\(814\) 0 0
\(815\) −13.3260 −0.466790
\(816\) 0 0
\(817\) −1.42962 −0.0500161
\(818\) 0 0
\(819\) 7.36241 0.257264
\(820\) 0 0
\(821\) 37.3196 1.30246 0.651231 0.758879i \(-0.274253\pi\)
0.651231 + 0.758879i \(0.274253\pi\)
\(822\) 0 0
\(823\) −34.3942 −1.19891 −0.599454 0.800410i \(-0.704615\pi\)
−0.599454 + 0.800410i \(0.704615\pi\)
\(824\) 0 0
\(825\) −25.3941 −0.884109
\(826\) 0 0
\(827\) −30.1413 −1.04812 −0.524058 0.851682i \(-0.675583\pi\)
−0.524058 + 0.851682i \(0.675583\pi\)
\(828\) 0 0
\(829\) 52.6152 1.82740 0.913701 0.406387i \(-0.133211\pi\)
0.913701 + 0.406387i \(0.133211\pi\)
\(830\) 0 0
\(831\) 9.40506 0.326258
\(832\) 0 0
\(833\) −3.27391 −0.113434
\(834\) 0 0
\(835\) 47.6071 1.64751
\(836\) 0 0
\(837\) 35.7092 1.23429
\(838\) 0 0
\(839\) 19.8543 0.685448 0.342724 0.939436i \(-0.388651\pi\)
0.342724 + 0.939436i \(0.388651\pi\)
\(840\) 0 0
\(841\) 56.2520 1.93973
\(842\) 0 0
\(843\) −5.53467 −0.190624
\(844\) 0 0
\(845\) 28.2322 0.971216
\(846\) 0 0
\(847\) 43.9427 1.50989
\(848\) 0 0
\(849\) −20.8166 −0.714425
\(850\) 0 0
\(851\) 9.08769 0.311522
\(852\) 0 0
\(853\) −27.6358 −0.946230 −0.473115 0.881001i \(-0.656871\pi\)
−0.473115 + 0.881001i \(0.656871\pi\)
\(854\) 0 0
\(855\) −15.2220 −0.520583
\(856\) 0 0
\(857\) 24.8177 0.847757 0.423879 0.905719i \(-0.360668\pi\)
0.423879 + 0.905719i \(0.360668\pi\)
\(858\) 0 0
\(859\) −22.0555 −0.752525 −0.376263 0.926513i \(-0.622791\pi\)
−0.376263 + 0.926513i \(0.622791\pi\)
\(860\) 0 0
\(861\) −5.60423 −0.190992
\(862\) 0 0
\(863\) 41.7334 1.42062 0.710310 0.703889i \(-0.248555\pi\)
0.710310 + 0.703889i \(0.248555\pi\)
\(864\) 0 0
\(865\) −24.7538 −0.841654
\(866\) 0 0
\(867\) 14.7993 0.502610
\(868\) 0 0
\(869\) 74.8043 2.53756
\(870\) 0 0
\(871\) −13.6785 −0.463480
\(872\) 0 0
\(873\) 35.4391 1.19943
\(874\) 0 0
\(875\) 1.95364 0.0660450
\(876\) 0 0
\(877\) 14.1170 0.476696 0.238348 0.971180i \(-0.423394\pi\)
0.238348 + 0.971180i \(0.423394\pi\)
\(878\) 0 0
\(879\) 10.0058 0.337487
\(880\) 0 0
\(881\) −48.5767 −1.63659 −0.818296 0.574797i \(-0.805081\pi\)
−0.818296 + 0.574797i \(0.805081\pi\)
\(882\) 0 0
\(883\) 45.5828 1.53398 0.766991 0.641658i \(-0.221753\pi\)
0.766991 + 0.641658i \(0.221753\pi\)
\(884\) 0 0
\(885\) −5.66978 −0.190587
\(886\) 0 0
\(887\) 2.54019 0.0852911 0.0426455 0.999090i \(-0.486421\pi\)
0.0426455 + 0.999090i \(0.486421\pi\)
\(888\) 0 0
\(889\) −21.2019 −0.711087
\(890\) 0 0
\(891\) 13.9517 0.467398
\(892\) 0 0
\(893\) −2.24854 −0.0752444
\(894\) 0 0
\(895\) 32.6828 1.09247
\(896\) 0 0
\(897\) −4.98866 −0.166567
\(898\) 0 0
\(899\) −70.3299 −2.34563
\(900\) 0 0
\(901\) 3.40521 0.113444
\(902\) 0 0
\(903\) −0.982799 −0.0327055
\(904\) 0 0
\(905\) −47.5527 −1.58070
\(906\) 0 0
\(907\) −24.5434 −0.814951 −0.407476 0.913216i \(-0.633591\pi\)
−0.407476 + 0.913216i \(0.633591\pi\)
\(908\) 0 0
\(909\) 13.5153 0.448273
\(910\) 0 0
\(911\) −40.2560 −1.33374 −0.666870 0.745174i \(-0.732366\pi\)
−0.666870 + 0.745174i \(0.732366\pi\)
\(912\) 0 0
\(913\) 4.95870 0.164109
\(914\) 0 0
\(915\) −19.3717 −0.640408
\(916\) 0 0
\(917\) −32.6307 −1.07756
\(918\) 0 0
\(919\) 4.39048 0.144829 0.0724143 0.997375i \(-0.476930\pi\)
0.0724143 + 0.997375i \(0.476930\pi\)
\(920\) 0 0
\(921\) −5.81020 −0.191453
\(922\) 0 0
\(923\) −22.5144 −0.741071
\(924\) 0 0
\(925\) −15.0815 −0.495877
\(926\) 0 0
\(927\) 12.4232 0.408030
\(928\) 0 0
\(929\) −45.5577 −1.49470 −0.747350 0.664431i \(-0.768674\pi\)
−0.747350 + 0.664431i \(0.768674\pi\)
\(930\) 0 0
\(931\) 9.17698 0.300763
\(932\) 0 0
\(933\) −15.4943 −0.507262
\(934\) 0 0
\(935\) −15.0860 −0.493365
\(936\) 0 0
\(937\) −39.7060 −1.29714 −0.648570 0.761155i \(-0.724632\pi\)
−0.648570 + 0.761155i \(0.724632\pi\)
\(938\) 0 0
\(939\) 28.8517 0.941538
\(940\) 0 0
\(941\) −36.3212 −1.18404 −0.592018 0.805925i \(-0.701669\pi\)
−0.592018 + 0.805925i \(0.701669\pi\)
\(942\) 0 0
\(943\) −10.1183 −0.329496
\(944\) 0 0
\(945\) −24.8562 −0.808571
\(946\) 0 0
\(947\) −44.5317 −1.44709 −0.723543 0.690279i \(-0.757488\pi\)
−0.723543 + 0.690279i \(0.757488\pi\)
\(948\) 0 0
\(949\) −28.9666 −0.940295
\(950\) 0 0
\(951\) 13.7623 0.446273
\(952\) 0 0
\(953\) 3.29735 0.106812 0.0534058 0.998573i \(-0.482992\pi\)
0.0534058 + 0.998573i \(0.482992\pi\)
\(954\) 0 0
\(955\) 51.7703 1.67525
\(956\) 0 0
\(957\) 50.6245 1.63646
\(958\) 0 0
\(959\) 27.6696 0.893499
\(960\) 0 0
\(961\) 27.0196 0.871601
\(962\) 0 0
\(963\) 41.0365 1.32238
\(964\) 0 0
\(965\) −14.7426 −0.474581
\(966\) 0 0
\(967\) 33.2862 1.07041 0.535207 0.844721i \(-0.320234\pi\)
0.535207 + 0.844721i \(0.320234\pi\)
\(968\) 0 0
\(969\) 1.63199 0.0524270
\(970\) 0 0
\(971\) −29.5093 −0.947000 −0.473500 0.880794i \(-0.657010\pi\)
−0.473500 + 0.880794i \(0.657010\pi\)
\(972\) 0 0
\(973\) 26.1298 0.837684
\(974\) 0 0
\(975\) 8.27895 0.265139
\(976\) 0 0
\(977\) −47.1102 −1.50719 −0.753594 0.657340i \(-0.771682\pi\)
−0.753594 + 0.657340i \(0.771682\pi\)
\(978\) 0 0
\(979\) 62.9049 2.01045
\(980\) 0 0
\(981\) 29.8779 0.953927
\(982\) 0 0
\(983\) 15.3825 0.490627 0.245313 0.969444i \(-0.421109\pi\)
0.245313 + 0.969444i \(0.421109\pi\)
\(984\) 0 0
\(985\) 11.2236 0.357615
\(986\) 0 0
\(987\) −1.54576 −0.0492022
\(988\) 0 0
\(989\) −1.77441 −0.0564230
\(990\) 0 0
\(991\) 37.5123 1.19162 0.595808 0.803127i \(-0.296832\pi\)
0.595808 + 0.803127i \(0.296832\pi\)
\(992\) 0 0
\(993\) −12.5169 −0.397211
\(994\) 0 0
\(995\) 12.5741 0.398626
\(996\) 0 0
\(997\) −2.29299 −0.0726197 −0.0363099 0.999341i \(-0.511560\pi\)
−0.0363099 + 0.999341i \(0.511560\pi\)
\(998\) 0 0
\(999\) −15.2656 −0.482981
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 6016.2.a.m.1.7 13
4.3 odd 2 6016.2.a.o.1.7 yes 13
8.3 odd 2 6016.2.a.n.1.7 yes 13
8.5 even 2 6016.2.a.p.1.7 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.7 13 1.1 even 1 trivial
6016.2.a.n.1.7 yes 13 8.3 odd 2
6016.2.a.o.1.7 yes 13 4.3 odd 2
6016.2.a.p.1.7 yes 13 8.5 even 2