Properties

Label 7616.2.a.bp.1.1
Level $7616$
Weight $2$
Character 7616.1
Self dual yes
Analytic conductor $60.814$
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] = [7616,2,Mod(1,7616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7616, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7616.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.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: \(60.8140661794\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.10522\) of defining polynomial
Character \(\chi\) \(=\) 7616.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.10522 q^{3} +3.40632 q^{5} +1.00000 q^{7} +1.43195 q^{9} +O(q^{10})\) \(q-2.10522 q^{3} +3.40632 q^{5} +1.00000 q^{7} +1.43195 q^{9} -1.23607 q^{11} +6.47214 q^{13} -7.17104 q^{15} +1.00000 q^{17} -2.21044 q^{19} -2.10522 q^{21} -1.39781 q^{23} +6.60299 q^{25} +3.30110 q^{27} +0.633874 q^{29} -0.965861 q^{31} +2.60219 q^{33} +3.40632 q^{35} +8.31040 q^{37} -13.6253 q^{39} -5.60299 q^{41} +11.0796 q^{43} +4.87766 q^{45} -7.67652 q^{47} +1.00000 q^{49} -2.10522 q^{51} -11.7167 q^{53} -4.21044 q^{55} +4.65345 q^{57} -0.602193 q^{59} +9.84352 q^{61} +1.43195 q^{63} +22.0461 q^{65} +5.30636 q^{67} +2.94269 q^{69} +3.33603 q^{71} +14.0239 q^{73} -13.9007 q^{75} -1.23607 q^{77} +0.323477 q^{79} -11.2454 q^{81} +13.8870 q^{83} +3.40632 q^{85} -1.33444 q^{87} -14.7509 q^{89} +6.47214 q^{91} +2.03335 q^{93} -7.52945 q^{95} -5.90855 q^{97} -1.76998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 8 q^{13} - 12 q^{15} + 4 q^{17} + 14 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 12 q^{27} - 4 q^{29} - 5 q^{31} + 8 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 7 q^{41} + 19 q^{43} + 2 q^{45} - 8 q^{47} + 4 q^{49} + 3 q^{51} - 5 q^{53} + 6 q^{55} + 44 q^{57} + 23 q^{61} + 7 q^{63} - 8 q^{65} + 15 q^{67} + 2 q^{69} - 2 q^{71} - 5 q^{73} + 10 q^{75} + 4 q^{77} + 24 q^{79} - 8 q^{81} + 10 q^{83} + q^{85} - 16 q^{87} - 16 q^{89} + 8 q^{91} + 20 q^{93} - 22 q^{95} - 15 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.10522 −1.21545 −0.607724 0.794148i \(-0.707917\pi\)
−0.607724 + 0.794148i \(0.707917\pi\)
\(4\) 0 0
\(5\) 3.40632 1.52335 0.761675 0.647959i \(-0.224377\pi\)
0.761675 + 0.647959i \(0.224377\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.43195 0.477315
\(10\) 0 0
\(11\) −1.23607 −0.372689 −0.186344 0.982485i \(-0.559664\pi\)
−0.186344 + 0.982485i \(0.559664\pi\)
\(12\) 0 0
\(13\) 6.47214 1.79505 0.897524 0.440966i \(-0.145364\pi\)
0.897524 + 0.440966i \(0.145364\pi\)
\(14\) 0 0
\(15\) −7.17104 −1.85155
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −2.21044 −0.507109 −0.253555 0.967321i \(-0.581600\pi\)
−0.253555 + 0.967321i \(0.581600\pi\)
\(20\) 0 0
\(21\) −2.10522 −0.459396
\(22\) 0 0
\(23\) −1.39781 −0.291463 −0.145731 0.989324i \(-0.546554\pi\)
−0.145731 + 0.989324i \(0.546554\pi\)
\(24\) 0 0
\(25\) 6.60299 1.32060
\(26\) 0 0
\(27\) 3.30110 0.635296
\(28\) 0 0
\(29\) 0.633874 0.117708 0.0588538 0.998267i \(-0.481255\pi\)
0.0588538 + 0.998267i \(0.481255\pi\)
\(30\) 0 0
\(31\) −0.965861 −0.173474 −0.0867368 0.996231i \(-0.527644\pi\)
−0.0867368 + 0.996231i \(0.527644\pi\)
\(32\) 0 0
\(33\) 2.60219 0.452984
\(34\) 0 0
\(35\) 3.40632 0.575772
\(36\) 0 0
\(37\) 8.31040 1.36622 0.683110 0.730315i \(-0.260627\pi\)
0.683110 + 0.730315i \(0.260627\pi\)
\(38\) 0 0
\(39\) −13.6253 −2.18179
\(40\) 0 0
\(41\) −5.60299 −0.875039 −0.437520 0.899209i \(-0.644143\pi\)
−0.437520 + 0.899209i \(0.644143\pi\)
\(42\) 0 0
\(43\) 11.0796 1.68962 0.844811 0.535065i \(-0.179713\pi\)
0.844811 + 0.535065i \(0.179713\pi\)
\(44\) 0 0
\(45\) 4.87766 0.727119
\(46\) 0 0
\(47\) −7.67652 −1.11974 −0.559868 0.828582i \(-0.689148\pi\)
−0.559868 + 0.828582i \(0.689148\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.10522 −0.294790
\(52\) 0 0
\(53\) −11.7167 −1.60941 −0.804707 0.593672i \(-0.797678\pi\)
−0.804707 + 0.593672i \(0.797678\pi\)
\(54\) 0 0
\(55\) −4.21044 −0.567735
\(56\) 0 0
\(57\) 4.65345 0.616365
\(58\) 0 0
\(59\) −0.602193 −0.0783989 −0.0391995 0.999231i \(-0.512481\pi\)
−0.0391995 + 0.999231i \(0.512481\pi\)
\(60\) 0 0
\(61\) 9.84352 1.26033 0.630167 0.776460i \(-0.282986\pi\)
0.630167 + 0.776460i \(0.282986\pi\)
\(62\) 0 0
\(63\) 1.43195 0.180408
\(64\) 0 0
\(65\) 22.0461 2.73449
\(66\) 0 0
\(67\) 5.30636 0.648275 0.324137 0.946010i \(-0.394926\pi\)
0.324137 + 0.946010i \(0.394926\pi\)
\(68\) 0 0
\(69\) 2.94269 0.354258
\(70\) 0 0
\(71\) 3.33603 0.395914 0.197957 0.980211i \(-0.436569\pi\)
0.197957 + 0.980211i \(0.436569\pi\)
\(72\) 0 0
\(73\) 14.0239 1.64137 0.820684 0.571382i \(-0.193592\pi\)
0.820684 + 0.571382i \(0.193592\pi\)
\(74\) 0 0
\(75\) −13.9007 −1.60512
\(76\) 0 0
\(77\) −1.23607 −0.140863
\(78\) 0 0
\(79\) 0.323477 0.0363940 0.0181970 0.999834i \(-0.494207\pi\)
0.0181970 + 0.999834i \(0.494207\pi\)
\(80\) 0 0
\(81\) −11.2454 −1.24949
\(82\) 0 0
\(83\) 13.8870 1.52429 0.762146 0.647405i \(-0.224146\pi\)
0.762146 + 0.647405i \(0.224146\pi\)
\(84\) 0 0
\(85\) 3.40632 0.369467
\(86\) 0 0
\(87\) −1.33444 −0.143067
\(88\) 0 0
\(89\) −14.7509 −1.56359 −0.781794 0.623537i \(-0.785695\pi\)
−0.781794 + 0.623537i \(0.785695\pi\)
\(90\) 0 0
\(91\) 6.47214 0.678464
\(92\) 0 0
\(93\) 2.03335 0.210848
\(94\) 0 0
\(95\) −7.52945 −0.772505
\(96\) 0 0
\(97\) −5.90855 −0.599922 −0.299961 0.953951i \(-0.596974\pi\)
−0.299961 + 0.953951i \(0.596974\pi\)
\(98\) 0 0
\(99\) −1.76998 −0.177890
\(100\) 0 0
\(101\) −5.08485 −0.505961 −0.252981 0.967471i \(-0.581411\pi\)
−0.252981 + 0.967471i \(0.581411\pi\)
\(102\) 0 0
\(103\) 5.28477 0.520724 0.260362 0.965511i \(-0.416158\pi\)
0.260362 + 0.965511i \(0.416158\pi\)
\(104\) 0 0
\(105\) −7.17104 −0.699822
\(106\) 0 0
\(107\) −9.25309 −0.894530 −0.447265 0.894402i \(-0.647602\pi\)
−0.447265 + 0.894402i \(0.647602\pi\)
\(108\) 0 0
\(109\) 16.1000 1.54210 0.771048 0.636777i \(-0.219733\pi\)
0.771048 + 0.636777i \(0.219733\pi\)
\(110\) 0 0
\(111\) −17.4952 −1.66057
\(112\) 0 0
\(113\) 10.5509 0.992548 0.496274 0.868166i \(-0.334701\pi\)
0.496274 + 0.868166i \(0.334701\pi\)
\(114\) 0 0
\(115\) −4.76137 −0.444000
\(116\) 0 0
\(117\) 9.26775 0.856804
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) 11.7955 1.06357
\(124\) 0 0
\(125\) 5.46027 0.488382
\(126\) 0 0
\(127\) 17.9272 1.59078 0.795389 0.606100i \(-0.207267\pi\)
0.795389 + 0.606100i \(0.207267\pi\)
\(128\) 0 0
\(129\) −23.3250 −2.05365
\(130\) 0 0
\(131\) 1.49777 0.130860 0.0654302 0.997857i \(-0.479158\pi\)
0.0654302 + 0.997857i \(0.479158\pi\)
\(132\) 0 0
\(133\) −2.21044 −0.191669
\(134\) 0 0
\(135\) 11.2446 0.967779
\(136\) 0 0
\(137\) 13.6030 1.16218 0.581091 0.813839i \(-0.302626\pi\)
0.581091 + 0.813839i \(0.302626\pi\)
\(138\) 0 0
\(139\) −6.28073 −0.532724 −0.266362 0.963873i \(-0.585822\pi\)
−0.266362 + 0.963873i \(0.585822\pi\)
\(140\) 0 0
\(141\) 16.1608 1.36098
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 2.15918 0.179310
\(146\) 0 0
\(147\) −2.10522 −0.173636
\(148\) 0 0
\(149\) −21.7004 −1.77776 −0.888882 0.458136i \(-0.848517\pi\)
−0.888882 + 0.458136i \(0.848517\pi\)
\(150\) 0 0
\(151\) −4.60745 −0.374949 −0.187475 0.982269i \(-0.560030\pi\)
−0.187475 + 0.982269i \(0.560030\pi\)
\(152\) 0 0
\(153\) 1.43195 0.115766
\(154\) 0 0
\(155\) −3.29003 −0.264261
\(156\) 0 0
\(157\) 4.53391 0.361846 0.180923 0.983497i \(-0.442092\pi\)
0.180923 + 0.983497i \(0.442092\pi\)
\(158\) 0 0
\(159\) 24.6662 1.95616
\(160\) 0 0
\(161\) −1.39781 −0.110163
\(162\) 0 0
\(163\) −12.2421 −0.958877 −0.479438 0.877576i \(-0.659160\pi\)
−0.479438 + 0.877576i \(0.659160\pi\)
\(164\) 0 0
\(165\) 8.86389 0.690053
\(166\) 0 0
\(167\) −1.34129 −0.103792 −0.0518959 0.998652i \(-0.516526\pi\)
−0.0518959 + 0.998652i \(0.516526\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) −3.16523 −0.242051
\(172\) 0 0
\(173\) −1.83300 −0.139361 −0.0696803 0.997569i \(-0.522198\pi\)
−0.0696803 + 0.997569i \(0.522198\pi\)
\(174\) 0 0
\(175\) 6.60299 0.499139
\(176\) 0 0
\(177\) 1.26775 0.0952898
\(178\) 0 0
\(179\) 20.2343 1.51238 0.756191 0.654351i \(-0.227058\pi\)
0.756191 + 0.654351i \(0.227058\pi\)
\(180\) 0 0
\(181\) 6.03010 0.448214 0.224107 0.974565i \(-0.428054\pi\)
0.224107 + 0.974565i \(0.428054\pi\)
\(182\) 0 0
\(183\) −20.7228 −1.53187
\(184\) 0 0
\(185\) 28.3078 2.08123
\(186\) 0 0
\(187\) −1.23607 −0.0903902
\(188\) 0 0
\(189\) 3.30110 0.240119
\(190\) 0 0
\(191\) 10.9146 0.789753 0.394876 0.918734i \(-0.370787\pi\)
0.394876 + 0.918734i \(0.370787\pi\)
\(192\) 0 0
\(193\) −19.1717 −1.38001 −0.690006 0.723804i \(-0.742392\pi\)
−0.690006 + 0.723804i \(0.742392\pi\)
\(194\) 0 0
\(195\) −46.4119 −3.32363
\(196\) 0 0
\(197\) −2.89557 −0.206301 −0.103151 0.994666i \(-0.532892\pi\)
−0.103151 + 0.994666i \(0.532892\pi\)
\(198\) 0 0
\(199\) −26.1830 −1.85607 −0.928033 0.372498i \(-0.878501\pi\)
−0.928033 + 0.372498i \(0.878501\pi\)
\(200\) 0 0
\(201\) −11.1710 −0.787944
\(202\) 0 0
\(203\) 0.633874 0.0444893
\(204\) 0 0
\(205\) −19.0855 −1.33299
\(206\) 0 0
\(207\) −2.00158 −0.139120
\(208\) 0 0
\(209\) 2.73225 0.188994
\(210\) 0 0
\(211\) −2.11048 −0.145291 −0.0726456 0.997358i \(-0.523144\pi\)
−0.0726456 + 0.997358i \(0.523144\pi\)
\(212\) 0 0
\(213\) −7.02307 −0.481213
\(214\) 0 0
\(215\) 37.7406 2.57389
\(216\) 0 0
\(217\) −0.965861 −0.0655669
\(218\) 0 0
\(219\) −29.5233 −1.99500
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 0 0
\(223\) 18.8126 1.25979 0.629893 0.776682i \(-0.283099\pi\)
0.629893 + 0.776682i \(0.283099\pi\)
\(224\) 0 0
\(225\) 9.45512 0.630341
\(226\) 0 0
\(227\) −17.3737 −1.15313 −0.576565 0.817051i \(-0.695607\pi\)
−0.576565 + 0.817051i \(0.695607\pi\)
\(228\) 0 0
\(229\) 2.71523 0.179428 0.0897138 0.995968i \(-0.471405\pi\)
0.0897138 + 0.995968i \(0.471405\pi\)
\(230\) 0 0
\(231\) 2.60219 0.171212
\(232\) 0 0
\(233\) 0.715233 0.0468565 0.0234282 0.999726i \(-0.492542\pi\)
0.0234282 + 0.999726i \(0.492542\pi\)
\(234\) 0 0
\(235\) −26.1487 −1.70575
\(236\) 0 0
\(237\) −0.680990 −0.0442351
\(238\) 0 0
\(239\) −8.60745 −0.556770 −0.278385 0.960470i \(-0.589799\pi\)
−0.278385 + 0.960470i \(0.589799\pi\)
\(240\) 0 0
\(241\) 6.31241 0.406618 0.203309 0.979115i \(-0.434830\pi\)
0.203309 + 0.979115i \(0.434830\pi\)
\(242\) 0 0
\(243\) 13.7707 0.883389
\(244\) 0 0
\(245\) 3.40632 0.217622
\(246\) 0 0
\(247\) −14.3063 −0.910285
\(248\) 0 0
\(249\) −29.2351 −1.85270
\(250\) 0 0
\(251\) −26.5711 −1.67715 −0.838577 0.544783i \(-0.816612\pi\)
−0.838577 + 0.544783i \(0.816612\pi\)
\(252\) 0 0
\(253\) 1.72778 0.108625
\(254\) 0 0
\(255\) −7.17104 −0.449068
\(256\) 0 0
\(257\) 6.75085 0.421107 0.210553 0.977582i \(-0.432473\pi\)
0.210553 + 0.977582i \(0.432473\pi\)
\(258\) 0 0
\(259\) 8.31040 0.516383
\(260\) 0 0
\(261\) 0.907674 0.0561836
\(262\) 0 0
\(263\) −1.20439 −0.0742657 −0.0371328 0.999310i \(-0.511822\pi\)
−0.0371328 + 0.999310i \(0.511822\pi\)
\(264\) 0 0
\(265\) −39.9108 −2.45170
\(266\) 0 0
\(267\) 31.0538 1.90046
\(268\) 0 0
\(269\) −18.0673 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(270\) 0 0
\(271\) −19.7464 −1.19951 −0.599754 0.800185i \(-0.704735\pi\)
−0.599754 + 0.800185i \(0.704735\pi\)
\(272\) 0 0
\(273\) −13.6253 −0.824638
\(274\) 0 0
\(275\) −8.16174 −0.492171
\(276\) 0 0
\(277\) 7.99744 0.480519 0.240260 0.970709i \(-0.422767\pi\)
0.240260 + 0.970709i \(0.422767\pi\)
\(278\) 0 0
\(279\) −1.38306 −0.0828016
\(280\) 0 0
\(281\) 3.07890 0.183672 0.0918359 0.995774i \(-0.470726\pi\)
0.0918359 + 0.995774i \(0.470726\pi\)
\(282\) 0 0
\(283\) 28.6927 1.70560 0.852801 0.522236i \(-0.174902\pi\)
0.852801 + 0.522236i \(0.174902\pi\)
\(284\) 0 0
\(285\) 15.8511 0.938940
\(286\) 0 0
\(287\) −5.60299 −0.330734
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 12.4388 0.729175
\(292\) 0 0
\(293\) 2.14866 0.125526 0.0627630 0.998028i \(-0.480009\pi\)
0.0627630 + 0.998028i \(0.480009\pi\)
\(294\) 0 0
\(295\) −2.05126 −0.119429
\(296\) 0 0
\(297\) −4.08038 −0.236768
\(298\) 0 0
\(299\) −9.04679 −0.523190
\(300\) 0 0
\(301\) 11.0796 0.638617
\(302\) 0 0
\(303\) 10.7047 0.614970
\(304\) 0 0
\(305\) 33.5301 1.91993
\(306\) 0 0
\(307\) 16.0291 0.914830 0.457415 0.889253i \(-0.348775\pi\)
0.457415 + 0.889253i \(0.348775\pi\)
\(308\) 0 0
\(309\) −11.1256 −0.632913
\(310\) 0 0
\(311\) −34.4505 −1.95351 −0.976756 0.214356i \(-0.931235\pi\)
−0.976756 + 0.214356i \(0.931235\pi\)
\(312\) 0 0
\(313\) 32.5585 1.84031 0.920157 0.391551i \(-0.128061\pi\)
0.920157 + 0.391551i \(0.128061\pi\)
\(314\) 0 0
\(315\) 4.87766 0.274825
\(316\) 0 0
\(317\) −2.82571 −0.158708 −0.0793539 0.996847i \(-0.525286\pi\)
−0.0793539 + 0.996847i \(0.525286\pi\)
\(318\) 0 0
\(319\) −0.783512 −0.0438682
\(320\) 0 0
\(321\) 19.4798 1.08725
\(322\) 0 0
\(323\) −2.21044 −0.122992
\(324\) 0 0
\(325\) 42.7354 2.37053
\(326\) 0 0
\(327\) −33.8939 −1.87434
\(328\) 0 0
\(329\) −7.67652 −0.423220
\(330\) 0 0
\(331\) 16.5822 0.911439 0.455720 0.890123i \(-0.349382\pi\)
0.455720 + 0.890123i \(0.349382\pi\)
\(332\) 0 0
\(333\) 11.9000 0.652118
\(334\) 0 0
\(335\) 18.0751 0.987549
\(336\) 0 0
\(337\) 15.7553 0.858247 0.429123 0.903246i \(-0.358823\pi\)
0.429123 + 0.903246i \(0.358823\pi\)
\(338\) 0 0
\(339\) −22.2120 −1.20639
\(340\) 0 0
\(341\) 1.19387 0.0646516
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 10.0237 0.539659
\(346\) 0 0
\(347\) −12.6630 −0.679785 −0.339893 0.940464i \(-0.610391\pi\)
−0.339893 + 0.940464i \(0.610391\pi\)
\(348\) 0 0
\(349\) 25.1547 1.34650 0.673250 0.739415i \(-0.264898\pi\)
0.673250 + 0.739415i \(0.264898\pi\)
\(350\) 0 0
\(351\) 21.3651 1.14039
\(352\) 0 0
\(353\) 28.5801 1.52116 0.760581 0.649243i \(-0.224914\pi\)
0.760581 + 0.649243i \(0.224914\pi\)
\(354\) 0 0
\(355\) 11.3636 0.603115
\(356\) 0 0
\(357\) −2.10522 −0.111420
\(358\) 0 0
\(359\) 25.4038 1.34076 0.670379 0.742018i \(-0.266131\pi\)
0.670379 + 0.742018i \(0.266131\pi\)
\(360\) 0 0
\(361\) −14.1140 −0.742840
\(362\) 0 0
\(363\) 19.9409 1.04663
\(364\) 0 0
\(365\) 47.7697 2.50038
\(366\) 0 0
\(367\) 4.87073 0.254250 0.127125 0.991887i \(-0.459425\pi\)
0.127125 + 0.991887i \(0.459425\pi\)
\(368\) 0 0
\(369\) −8.02317 −0.417670
\(370\) 0 0
\(371\) −11.7167 −0.608301
\(372\) 0 0
\(373\) −23.2388 −1.20326 −0.601629 0.798776i \(-0.705481\pi\)
−0.601629 + 0.798776i \(0.705481\pi\)
\(374\) 0 0
\(375\) −11.4951 −0.593603
\(376\) 0 0
\(377\) 4.10252 0.211291
\(378\) 0 0
\(379\) 15.8658 0.814971 0.407486 0.913212i \(-0.366406\pi\)
0.407486 + 0.913212i \(0.366406\pi\)
\(380\) 0 0
\(381\) −37.7406 −1.93351
\(382\) 0 0
\(383\) 2.06828 0.105684 0.0528421 0.998603i \(-0.483172\pi\)
0.0528421 + 0.998603i \(0.483172\pi\)
\(384\) 0 0
\(385\) −4.21044 −0.214584
\(386\) 0 0
\(387\) 15.8654 0.806482
\(388\) 0 0
\(389\) −3.51679 −0.178308 −0.0891542 0.996018i \(-0.528416\pi\)
−0.0891542 + 0.996018i \(0.528416\pi\)
\(390\) 0 0
\(391\) −1.39781 −0.0706901
\(392\) 0 0
\(393\) −3.15313 −0.159054
\(394\) 0 0
\(395\) 1.10187 0.0554408
\(396\) 0 0
\(397\) −5.47459 −0.274762 −0.137381 0.990518i \(-0.543868\pi\)
−0.137381 + 0.990518i \(0.543868\pi\)
\(398\) 0 0
\(399\) 4.65345 0.232964
\(400\) 0 0
\(401\) −6.76137 −0.337647 −0.168823 0.985646i \(-0.553997\pi\)
−0.168823 + 0.985646i \(0.553997\pi\)
\(402\) 0 0
\(403\) −6.25118 −0.311393
\(404\) 0 0
\(405\) −38.3053 −1.90340
\(406\) 0 0
\(407\) −10.2722 −0.509175
\(408\) 0 0
\(409\) −14.6378 −0.723793 −0.361897 0.932218i \(-0.617871\pi\)
−0.361897 + 0.932218i \(0.617871\pi\)
\(410\) 0 0
\(411\) −28.6373 −1.41257
\(412\) 0 0
\(413\) −0.602193 −0.0296320
\(414\) 0 0
\(415\) 47.3034 2.32203
\(416\) 0 0
\(417\) 13.2223 0.647499
\(418\) 0 0
\(419\) 29.5736 1.44476 0.722382 0.691494i \(-0.243047\pi\)
0.722382 + 0.691494i \(0.243047\pi\)
\(420\) 0 0
\(421\) −12.0223 −0.585930 −0.292965 0.956123i \(-0.594642\pi\)
−0.292965 + 0.956123i \(0.594642\pi\)
\(422\) 0 0
\(423\) −10.9924 −0.534467
\(424\) 0 0
\(425\) 6.60299 0.320292
\(426\) 0 0
\(427\) 9.84352 0.476361
\(428\) 0 0
\(429\) 16.8417 0.813127
\(430\) 0 0
\(431\) 11.1034 0.534834 0.267417 0.963581i \(-0.413830\pi\)
0.267417 + 0.963581i \(0.413830\pi\)
\(432\) 0 0
\(433\) 17.2165 0.827372 0.413686 0.910420i \(-0.364241\pi\)
0.413686 + 0.910420i \(0.364241\pi\)
\(434\) 0 0
\(435\) −4.54554 −0.217942
\(436\) 0 0
\(437\) 3.08976 0.147803
\(438\) 0 0
\(439\) −25.1681 −1.20121 −0.600603 0.799548i \(-0.705073\pi\)
−0.600603 + 0.799548i \(0.705073\pi\)
\(440\) 0 0
\(441\) 1.43195 0.0681879
\(442\) 0 0
\(443\) 31.1531 1.48013 0.740065 0.672536i \(-0.234795\pi\)
0.740065 + 0.672536i \(0.234795\pi\)
\(444\) 0 0
\(445\) −50.2461 −2.38189
\(446\) 0 0
\(447\) 45.6841 2.16078
\(448\) 0 0
\(449\) 10.7444 0.507057 0.253529 0.967328i \(-0.418409\pi\)
0.253529 + 0.967328i \(0.418409\pi\)
\(450\) 0 0
\(451\) 6.92567 0.326117
\(452\) 0 0
\(453\) 9.69969 0.455731
\(454\) 0 0
\(455\) 22.0461 1.03354
\(456\) 0 0
\(457\) −30.0863 −1.40738 −0.703690 0.710508i \(-0.748465\pi\)
−0.703690 + 0.710508i \(0.748465\pi\)
\(458\) 0 0
\(459\) 3.30110 0.154082
\(460\) 0 0
\(461\) −8.29082 −0.386142 −0.193071 0.981185i \(-0.561845\pi\)
−0.193071 + 0.981185i \(0.561845\pi\)
\(462\) 0 0
\(463\) −11.2795 −0.524203 −0.262102 0.965040i \(-0.584416\pi\)
−0.262102 + 0.965040i \(0.584416\pi\)
\(464\) 0 0
\(465\) 6.92622 0.321196
\(466\) 0 0
\(467\) 38.2582 1.77038 0.885188 0.465233i \(-0.154029\pi\)
0.885188 + 0.465233i \(0.154029\pi\)
\(468\) 0 0
\(469\) 5.30636 0.245025
\(470\) 0 0
\(471\) −9.54488 −0.439805
\(472\) 0 0
\(473\) −13.6951 −0.629702
\(474\) 0 0
\(475\) −14.5955 −0.669687
\(476\) 0 0
\(477\) −16.7777 −0.768198
\(478\) 0 0
\(479\) 33.0812 1.51152 0.755759 0.654850i \(-0.227268\pi\)
0.755759 + 0.654850i \(0.227268\pi\)
\(480\) 0 0
\(481\) 53.7860 2.45243
\(482\) 0 0
\(483\) 2.94269 0.133897
\(484\) 0 0
\(485\) −20.1264 −0.913892
\(486\) 0 0
\(487\) 25.3757 1.14988 0.574941 0.818195i \(-0.305025\pi\)
0.574941 + 0.818195i \(0.305025\pi\)
\(488\) 0 0
\(489\) 25.7723 1.16547
\(490\) 0 0
\(491\) 30.4505 1.37421 0.687107 0.726556i \(-0.258880\pi\)
0.687107 + 0.726556i \(0.258880\pi\)
\(492\) 0 0
\(493\) 0.633874 0.0285483
\(494\) 0 0
\(495\) −6.02912 −0.270989
\(496\) 0 0
\(497\) 3.33603 0.149641
\(498\) 0 0
\(499\) 11.9016 0.532790 0.266395 0.963864i \(-0.414167\pi\)
0.266395 + 0.963864i \(0.414167\pi\)
\(500\) 0 0
\(501\) 2.82370 0.126154
\(502\) 0 0
\(503\) −8.90567 −0.397084 −0.198542 0.980092i \(-0.563621\pi\)
−0.198542 + 0.980092i \(0.563621\pi\)
\(504\) 0 0
\(505\) −17.3206 −0.770756
\(506\) 0 0
\(507\) −60.8167 −2.70096
\(508\) 0 0
\(509\) 27.1838 1.20490 0.602451 0.798156i \(-0.294191\pi\)
0.602451 + 0.798156i \(0.294191\pi\)
\(510\) 0 0
\(511\) 14.0239 0.620379
\(512\) 0 0
\(513\) −7.29687 −0.322165
\(514\) 0 0
\(515\) 18.0016 0.793245
\(516\) 0 0
\(517\) 9.48870 0.417313
\(518\) 0 0
\(519\) 3.85887 0.169386
\(520\) 0 0
\(521\) 7.10981 0.311487 0.155743 0.987798i \(-0.450223\pi\)
0.155743 + 0.987798i \(0.450223\pi\)
\(522\) 0 0
\(523\) −30.4581 −1.33184 −0.665919 0.746024i \(-0.731961\pi\)
−0.665919 + 0.746024i \(0.731961\pi\)
\(524\) 0 0
\(525\) −13.9007 −0.606678
\(526\) 0 0
\(527\) −0.965861 −0.0420735
\(528\) 0 0
\(529\) −21.0461 −0.915049
\(530\) 0 0
\(531\) −0.862309 −0.0374210
\(532\) 0 0
\(533\) −36.2633 −1.57074
\(534\) 0 0
\(535\) −31.5189 −1.36268
\(536\) 0 0
\(537\) −42.5976 −1.83822
\(538\) 0 0
\(539\) −1.23607 −0.0532412
\(540\) 0 0
\(541\) 1.88794 0.0811688 0.0405844 0.999176i \(-0.487078\pi\)
0.0405844 + 0.999176i \(0.487078\pi\)
\(542\) 0 0
\(543\) −12.6947 −0.544781
\(544\) 0 0
\(545\) 54.8415 2.34915
\(546\) 0 0
\(547\) 13.8844 0.593654 0.296827 0.954931i \(-0.404072\pi\)
0.296827 + 0.954931i \(0.404072\pi\)
\(548\) 0 0
\(549\) 14.0954 0.601577
\(550\) 0 0
\(551\) −1.40114 −0.0596906
\(552\) 0 0
\(553\) 0.323477 0.0137556
\(554\) 0 0
\(555\) −59.5942 −2.52963
\(556\) 0 0
\(557\) 18.5183 0.784644 0.392322 0.919828i \(-0.371672\pi\)
0.392322 + 0.919828i \(0.371672\pi\)
\(558\) 0 0
\(559\) 71.7086 3.03295
\(560\) 0 0
\(561\) 2.60219 0.109865
\(562\) 0 0
\(563\) −36.1932 −1.52536 −0.762681 0.646775i \(-0.776117\pi\)
−0.762681 + 0.646775i \(0.776117\pi\)
\(564\) 0 0
\(565\) 35.9398 1.51200
\(566\) 0 0
\(567\) −11.2454 −0.472261
\(568\) 0 0
\(569\) 28.9137 1.21213 0.606063 0.795417i \(-0.292748\pi\)
0.606063 + 0.795417i \(0.292748\pi\)
\(570\) 0 0
\(571\) 15.1214 0.632813 0.316406 0.948624i \(-0.397524\pi\)
0.316406 + 0.948624i \(0.397524\pi\)
\(572\) 0 0
\(573\) −22.9776 −0.959904
\(574\) 0 0
\(575\) −9.22970 −0.384905
\(576\) 0 0
\(577\) −18.1608 −0.756042 −0.378021 0.925797i \(-0.623395\pi\)
−0.378021 + 0.925797i \(0.623395\pi\)
\(578\) 0 0
\(579\) 40.3607 1.67733
\(580\) 0 0
\(581\) 13.8870 0.576128
\(582\) 0 0
\(583\) 14.4827 0.599810
\(584\) 0 0
\(585\) 31.5689 1.30521
\(586\) 0 0
\(587\) 45.0538 1.85957 0.929784 0.368105i \(-0.119993\pi\)
0.929784 + 0.368105i \(0.119993\pi\)
\(588\) 0 0
\(589\) 2.13497 0.0879701
\(590\) 0 0
\(591\) 6.09581 0.250748
\(592\) 0 0
\(593\) 6.43139 0.264106 0.132053 0.991243i \(-0.457843\pi\)
0.132053 + 0.991243i \(0.457843\pi\)
\(594\) 0 0
\(595\) 3.40632 0.139645
\(596\) 0 0
\(597\) 55.1210 2.25595
\(598\) 0 0
\(599\) 41.0655 1.67789 0.838946 0.544215i \(-0.183172\pi\)
0.838946 + 0.544215i \(0.183172\pi\)
\(600\) 0 0
\(601\) 39.2626 1.60156 0.800778 0.598961i \(-0.204420\pi\)
0.800778 + 0.598961i \(0.204420\pi\)
\(602\) 0 0
\(603\) 7.59841 0.309431
\(604\) 0 0
\(605\) −32.2651 −1.31176
\(606\) 0 0
\(607\) 11.9954 0.486879 0.243440 0.969916i \(-0.421724\pi\)
0.243440 + 0.969916i \(0.421724\pi\)
\(608\) 0 0
\(609\) −1.33444 −0.0540744
\(610\) 0 0
\(611\) −49.6835 −2.00998
\(612\) 0 0
\(613\) −28.0774 −1.13404 −0.567018 0.823706i \(-0.691903\pi\)
−0.567018 + 0.823706i \(0.691903\pi\)
\(614\) 0 0
\(615\) 40.1792 1.62018
\(616\) 0 0
\(617\) −6.08688 −0.245049 −0.122524 0.992466i \(-0.539099\pi\)
−0.122524 + 0.992466i \(0.539099\pi\)
\(618\) 0 0
\(619\) 3.24659 0.130491 0.0652456 0.997869i \(-0.479217\pi\)
0.0652456 + 0.997869i \(0.479217\pi\)
\(620\) 0 0
\(621\) −4.61429 −0.185165
\(622\) 0 0
\(623\) −14.7509 −0.590980
\(624\) 0 0
\(625\) −14.4155 −0.576621
\(626\) 0 0
\(627\) −5.75199 −0.229712
\(628\) 0 0
\(629\) 8.31040 0.331357
\(630\) 0 0
\(631\) 6.53093 0.259992 0.129996 0.991515i \(-0.458504\pi\)
0.129996 + 0.991515i \(0.458504\pi\)
\(632\) 0 0
\(633\) 4.44302 0.176594
\(634\) 0 0
\(635\) 61.0655 2.42331
\(636\) 0 0
\(637\) 6.47214 0.256435
\(638\) 0 0
\(639\) 4.77701 0.188976
\(640\) 0 0
\(641\) −6.98745 −0.275988 −0.137994 0.990433i \(-0.544065\pi\)
−0.137994 + 0.990433i \(0.544065\pi\)
\(642\) 0 0
\(643\) 21.1348 0.833475 0.416737 0.909027i \(-0.363173\pi\)
0.416737 + 0.909027i \(0.363173\pi\)
\(644\) 0 0
\(645\) −79.4522 −3.12843
\(646\) 0 0
\(647\) −34.2892 −1.34805 −0.674024 0.738709i \(-0.735436\pi\)
−0.674024 + 0.738709i \(0.735436\pi\)
\(648\) 0 0
\(649\) 0.744352 0.0292184
\(650\) 0 0
\(651\) 2.03335 0.0796932
\(652\) 0 0
\(653\) −20.4234 −0.799231 −0.399615 0.916683i \(-0.630856\pi\)
−0.399615 + 0.916683i \(0.630856\pi\)
\(654\) 0 0
\(655\) 5.10187 0.199346
\(656\) 0 0
\(657\) 20.0814 0.783450
\(658\) 0 0
\(659\) −9.16578 −0.357048 −0.178524 0.983936i \(-0.557132\pi\)
−0.178524 + 0.983936i \(0.557132\pi\)
\(660\) 0 0
\(661\) −33.1919 −1.29102 −0.645508 0.763754i \(-0.723354\pi\)
−0.645508 + 0.763754i \(0.723354\pi\)
\(662\) 0 0
\(663\) −13.6253 −0.529161
\(664\) 0 0
\(665\) −7.52945 −0.291979
\(666\) 0 0
\(667\) −0.886034 −0.0343074
\(668\) 0 0
\(669\) −39.6047 −1.53121
\(670\) 0 0
\(671\) −12.1673 −0.469712
\(672\) 0 0
\(673\) 44.8277 1.72798 0.863990 0.503509i \(-0.167958\pi\)
0.863990 + 0.503509i \(0.167958\pi\)
\(674\) 0 0
\(675\) 21.7971 0.838970
\(676\) 0 0
\(677\) 17.9276 0.689013 0.344506 0.938784i \(-0.388046\pi\)
0.344506 + 0.938784i \(0.388046\pi\)
\(678\) 0 0
\(679\) −5.90855 −0.226749
\(680\) 0 0
\(681\) 36.5753 1.40157
\(682\) 0 0
\(683\) −35.3742 −1.35356 −0.676778 0.736187i \(-0.736624\pi\)
−0.676778 + 0.736187i \(0.736624\pi\)
\(684\) 0 0
\(685\) 46.3361 1.77041
\(686\) 0 0
\(687\) −5.71616 −0.218085
\(688\) 0 0
\(689\) −75.8322 −2.88898
\(690\) 0 0
\(691\) −32.6537 −1.24221 −0.621103 0.783729i \(-0.713315\pi\)
−0.621103 + 0.783729i \(0.713315\pi\)
\(692\) 0 0
\(693\) −1.76998 −0.0672361
\(694\) 0 0
\(695\) −21.3941 −0.811526
\(696\) 0 0
\(697\) −5.60299 −0.212228
\(698\) 0 0
\(699\) −1.50572 −0.0569516
\(700\) 0 0
\(701\) 12.3696 0.467194 0.233597 0.972334i \(-0.424950\pi\)
0.233597 + 0.972334i \(0.424950\pi\)
\(702\) 0 0
\(703\) −18.3696 −0.692823
\(704\) 0 0
\(705\) 55.0486 2.07325
\(706\) 0 0
\(707\) −5.08485 −0.191235
\(708\) 0 0
\(709\) −20.9482 −0.786727 −0.393363 0.919383i \(-0.628689\pi\)
−0.393363 + 0.919383i \(0.628689\pi\)
\(710\) 0 0
\(711\) 0.463202 0.0173714
\(712\) 0 0
\(713\) 1.35009 0.0505611
\(714\) 0 0
\(715\) −27.2505 −1.01911
\(716\) 0 0
\(717\) 18.1206 0.676725
\(718\) 0 0
\(719\) 28.0239 1.04511 0.522557 0.852604i \(-0.324978\pi\)
0.522557 + 0.852604i \(0.324978\pi\)
\(720\) 0 0
\(721\) 5.28477 0.196815
\(722\) 0 0
\(723\) −13.2890 −0.494223
\(724\) 0 0
\(725\) 4.18546 0.155444
\(726\) 0 0
\(727\) 23.9844 0.889531 0.444765 0.895647i \(-0.353287\pi\)
0.444765 + 0.895647i \(0.353287\pi\)
\(728\) 0 0
\(729\) 4.74583 0.175772
\(730\) 0 0
\(731\) 11.0796 0.409793
\(732\) 0 0
\(733\) 31.0864 1.14820 0.574102 0.818784i \(-0.305351\pi\)
0.574102 + 0.818784i \(0.305351\pi\)
\(734\) 0 0
\(735\) −7.17104 −0.264508
\(736\) 0 0
\(737\) −6.55902 −0.241604
\(738\) 0 0
\(739\) −12.0841 −0.444519 −0.222260 0.974988i \(-0.571343\pi\)
−0.222260 + 0.974988i \(0.571343\pi\)
\(740\) 0 0
\(741\) 30.1178 1.10640
\(742\) 0 0
\(743\) −52.8588 −1.93920 −0.969600 0.244695i \(-0.921312\pi\)
−0.969600 + 0.244695i \(0.921312\pi\)
\(744\) 0 0
\(745\) −73.9184 −2.70816
\(746\) 0 0
\(747\) 19.8854 0.727568
\(748\) 0 0
\(749\) −9.25309 −0.338100
\(750\) 0 0
\(751\) −30.6499 −1.11843 −0.559216 0.829022i \(-0.688898\pi\)
−0.559216 + 0.829022i \(0.688898\pi\)
\(752\) 0 0
\(753\) 55.9380 2.03849
\(754\) 0 0
\(755\) −15.6944 −0.571179
\(756\) 0 0
\(757\) 3.57276 0.129854 0.0649271 0.997890i \(-0.479319\pi\)
0.0649271 + 0.997890i \(0.479319\pi\)
\(758\) 0 0
\(759\) −3.63736 −0.132028
\(760\) 0 0
\(761\) −4.64537 −0.168395 −0.0841973 0.996449i \(-0.526833\pi\)
−0.0841973 + 0.996449i \(0.526833\pi\)
\(762\) 0 0
\(763\) 16.1000 0.582858
\(764\) 0 0
\(765\) 4.87766 0.176352
\(766\) 0 0
\(767\) −3.89748 −0.140730
\(768\) 0 0
\(769\) −13.1361 −0.473700 −0.236850 0.971546i \(-0.576115\pi\)
−0.236850 + 0.971546i \(0.576115\pi\)
\(770\) 0 0
\(771\) −14.2120 −0.511833
\(772\) 0 0
\(773\) 44.0372 1.58391 0.791954 0.610581i \(-0.209064\pi\)
0.791954 + 0.610581i \(0.209064\pi\)
\(774\) 0 0
\(775\) −6.37756 −0.229089
\(776\) 0 0
\(777\) −17.4952 −0.627637
\(778\) 0 0
\(779\) 12.3850 0.443740
\(780\) 0 0
\(781\) −4.12356 −0.147552
\(782\) 0 0
\(783\) 2.09248 0.0747792
\(784\) 0 0
\(785\) 15.4439 0.551218
\(786\) 0 0
\(787\) 39.3178 1.40153 0.700765 0.713393i \(-0.252842\pi\)
0.700765 + 0.713393i \(0.252842\pi\)
\(788\) 0 0
\(789\) 2.53550 0.0902661
\(790\) 0 0
\(791\) 10.5509 0.375148
\(792\) 0 0
\(793\) 63.7086 2.26236
\(794\) 0 0
\(795\) 84.0210 2.97992
\(796\) 0 0
\(797\) 32.5987 1.15470 0.577352 0.816496i \(-0.304086\pi\)
0.577352 + 0.816496i \(0.304086\pi\)
\(798\) 0 0
\(799\) −7.67652 −0.271576
\(800\) 0 0
\(801\) −21.1224 −0.746324
\(802\) 0 0
\(803\) −17.3344 −0.611719
\(804\) 0 0
\(805\) −4.76137 −0.167816
\(806\) 0 0
\(807\) 38.0356 1.33892
\(808\) 0 0
\(809\) −56.0461 −1.97048 −0.985239 0.171187i \(-0.945240\pi\)
−0.985239 + 0.171187i \(0.945240\pi\)
\(810\) 0 0
\(811\) 9.39490 0.329900 0.164950 0.986302i \(-0.447254\pi\)
0.164950 + 0.986302i \(0.447254\pi\)
\(812\) 0 0
\(813\) 41.5705 1.45794
\(814\) 0 0
\(815\) −41.7005 −1.46071
\(816\) 0 0
\(817\) −24.4907 −0.856822
\(818\) 0 0
\(819\) 9.26775 0.323841
\(820\) 0 0
\(821\) 7.40016 0.258267 0.129134 0.991627i \(-0.458780\pi\)
0.129134 + 0.991627i \(0.458780\pi\)
\(822\) 0 0
\(823\) 39.2675 1.36878 0.684390 0.729116i \(-0.260068\pi\)
0.684390 + 0.729116i \(0.260068\pi\)
\(824\) 0 0
\(825\) 17.1822 0.598209
\(826\) 0 0
\(827\) 30.5052 1.06077 0.530385 0.847757i \(-0.322047\pi\)
0.530385 + 0.847757i \(0.322047\pi\)
\(828\) 0 0
\(829\) 46.0461 1.59925 0.799624 0.600501i \(-0.205032\pi\)
0.799624 + 0.600501i \(0.205032\pi\)
\(830\) 0 0
\(831\) −16.8364 −0.584047
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −4.56885 −0.158111
\(836\) 0 0
\(837\) −3.18840 −0.110207
\(838\) 0 0
\(839\) −43.1903 −1.49110 −0.745548 0.666452i \(-0.767812\pi\)
−0.745548 + 0.666452i \(0.767812\pi\)
\(840\) 0 0
\(841\) −28.5982 −0.986145
\(842\) 0 0
\(843\) −6.48176 −0.223244
\(844\) 0 0
\(845\) 98.4035 3.38518
\(846\) 0 0
\(847\) −9.47214 −0.325466
\(848\) 0 0
\(849\) −60.4043 −2.07307
\(850\) 0 0
\(851\) −11.6163 −0.398203
\(852\) 0 0
\(853\) −31.9186 −1.09287 −0.546437 0.837500i \(-0.684016\pi\)
−0.546437 + 0.837500i \(0.684016\pi\)
\(854\) 0 0
\(855\) −10.7818 −0.368728
\(856\) 0 0
\(857\) 11.6363 0.397490 0.198745 0.980051i \(-0.436314\pi\)
0.198745 + 0.980051i \(0.436314\pi\)
\(858\) 0 0
\(859\) 44.9205 1.53267 0.766335 0.642442i \(-0.222078\pi\)
0.766335 + 0.642442i \(0.222078\pi\)
\(860\) 0 0
\(861\) 11.7955 0.401990
\(862\) 0 0
\(863\) −22.2676 −0.757999 −0.379000 0.925397i \(-0.623732\pi\)
−0.379000 + 0.925397i \(0.623732\pi\)
\(864\) 0 0
\(865\) −6.24379 −0.212295
\(866\) 0 0
\(867\) −2.10522 −0.0714970
\(868\) 0 0
\(869\) −0.399840 −0.0135636
\(870\) 0 0
\(871\) 34.3435 1.16368
\(872\) 0 0
\(873\) −8.46072 −0.286352
\(874\) 0 0
\(875\) 5.46027 0.184591
\(876\) 0 0
\(877\) −18.2095 −0.614890 −0.307445 0.951566i \(-0.599474\pi\)
−0.307445 + 0.951566i \(0.599474\pi\)
\(878\) 0 0
\(879\) −4.52340 −0.152570
\(880\) 0 0
\(881\) 31.7548 1.06985 0.534923 0.844901i \(-0.320341\pi\)
0.534923 + 0.844901i \(0.320341\pi\)
\(882\) 0 0
\(883\) −57.7644 −1.94393 −0.971964 0.235130i \(-0.924448\pi\)
−0.971964 + 0.235130i \(0.924448\pi\)
\(884\) 0 0
\(885\) 4.31835 0.145160
\(886\) 0 0
\(887\) −21.9442 −0.736813 −0.368407 0.929665i \(-0.620097\pi\)
−0.368407 + 0.929665i \(0.620097\pi\)
\(888\) 0 0
\(889\) 17.9272 0.601257
\(890\) 0 0
\(891\) 13.9000 0.465669
\(892\) 0 0
\(893\) 16.9685 0.567828
\(894\) 0 0
\(895\) 68.9244 2.30389
\(896\) 0 0
\(897\) 19.0455 0.635910
\(898\) 0 0
\(899\) −0.612234 −0.0204192
\(900\) 0 0
\(901\) −11.7167 −0.390340
\(902\) 0 0
\(903\) −23.3250 −0.776206
\(904\) 0 0
\(905\) 20.5404 0.682786
\(906\) 0 0
\(907\) −10.0883 −0.334978 −0.167489 0.985874i \(-0.553566\pi\)
−0.167489 + 0.985874i \(0.553566\pi\)
\(908\) 0 0
\(909\) −7.28123 −0.241503
\(910\) 0 0
\(911\) 53.3155 1.76642 0.883210 0.468977i \(-0.155377\pi\)
0.883210 + 0.468977i \(0.155377\pi\)
\(912\) 0 0
\(913\) −17.1652 −0.568086
\(914\) 0 0
\(915\) −70.5883 −2.33358
\(916\) 0 0
\(917\) 1.49777 0.0494606
\(918\) 0 0
\(919\) −14.3247 −0.472529 −0.236264 0.971689i \(-0.575923\pi\)
−0.236264 + 0.971689i \(0.575923\pi\)
\(920\) 0 0
\(921\) −33.7448 −1.11193
\(922\) 0 0
\(923\) 21.5912 0.710684
\(924\) 0 0
\(925\) 54.8734 1.80423
\(926\) 0 0
\(927\) 7.56750 0.248549
\(928\) 0 0
\(929\) 30.0091 0.984567 0.492284 0.870435i \(-0.336162\pi\)
0.492284 + 0.870435i \(0.336162\pi\)
\(930\) 0 0
\(931\) −2.21044 −0.0724442
\(932\) 0 0
\(933\) 72.5259 2.37439
\(934\) 0 0
\(935\) −4.21044 −0.137696
\(936\) 0 0
\(937\) −45.1807 −1.47599 −0.737994 0.674807i \(-0.764227\pi\)
−0.737994 + 0.674807i \(0.764227\pi\)
\(938\) 0 0
\(939\) −68.5427 −2.23681
\(940\) 0 0
\(941\) 37.4598 1.22116 0.610578 0.791956i \(-0.290937\pi\)
0.610578 + 0.791956i \(0.290937\pi\)
\(942\) 0 0
\(943\) 7.83189 0.255041
\(944\) 0 0
\(945\) 11.2446 0.365786
\(946\) 0 0
\(947\) −1.33347 −0.0433318 −0.0216659 0.999765i \(-0.506897\pi\)
−0.0216659 + 0.999765i \(0.506897\pi\)
\(948\) 0 0
\(949\) 90.7643 2.94633
\(950\) 0 0
\(951\) 5.94874 0.192901
\(952\) 0 0
\(953\) 4.14271 0.134196 0.0670978 0.997746i \(-0.478626\pi\)
0.0670978 + 0.997746i \(0.478626\pi\)
\(954\) 0 0
\(955\) 37.1786 1.20307
\(956\) 0 0
\(957\) 1.64946 0.0533196
\(958\) 0 0
\(959\) 13.6030 0.439263
\(960\) 0 0
\(961\) −30.0671 −0.969907
\(962\) 0 0
\(963\) −13.2499 −0.426973
\(964\) 0 0
\(965\) −65.3050 −2.10224
\(966\) 0 0
\(967\) −23.9265 −0.769423 −0.384712 0.923037i \(-0.625699\pi\)
−0.384712 + 0.923037i \(0.625699\pi\)
\(968\) 0 0
\(969\) 4.65345 0.149490
\(970\) 0 0
\(971\) 41.7634 1.34025 0.670126 0.742248i \(-0.266240\pi\)
0.670126 + 0.742248i \(0.266240\pi\)
\(972\) 0 0
\(973\) −6.28073 −0.201351
\(974\) 0 0
\(975\) −89.9674 −2.88126
\(976\) 0 0
\(977\) −35.4143 −1.13300 −0.566501 0.824061i \(-0.691703\pi\)
−0.566501 + 0.824061i \(0.691703\pi\)
\(978\) 0 0
\(979\) 18.2331 0.582731
\(980\) 0 0
\(981\) 23.0543 0.736066
\(982\) 0 0
\(983\) 45.7062 1.45780 0.728901 0.684620i \(-0.240032\pi\)
0.728901 + 0.684620i \(0.240032\pi\)
\(984\) 0 0
\(985\) −9.86324 −0.314269
\(986\) 0 0
\(987\) 16.1608 0.514403
\(988\) 0 0
\(989\) −15.4871 −0.492462
\(990\) 0 0
\(991\) −13.3586 −0.424351 −0.212176 0.977232i \(-0.568055\pi\)
−0.212176 + 0.977232i \(0.568055\pi\)
\(992\) 0 0
\(993\) −34.9091 −1.10781
\(994\) 0 0
\(995\) −89.1877 −2.82744
\(996\) 0 0
\(997\) 31.8346 1.00821 0.504106 0.863642i \(-0.331822\pi\)
0.504106 + 0.863642i \(0.331822\pi\)
\(998\) 0 0
\(999\) 27.4334 0.867955
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 7616.2.a.bp.1.1 4
4.3 odd 2 7616.2.a.bj.1.4 4
8.3 odd 2 952.2.a.g.1.1 4
8.5 even 2 1904.2.a.q.1.4 4
24.11 even 2 8568.2.a.bj.1.4 4
56.27 even 2 6664.2.a.o.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.1 4 8.3 odd 2
1904.2.a.q.1.4 4 8.5 even 2
6664.2.a.o.1.4 4 56.27 even 2
7616.2.a.bj.1.4 4 4.3 odd 2
7616.2.a.bp.1.1 4 1.1 even 1 trivial
8568.2.a.bj.1.4 4 24.11 even 2