Properties

Label 9680.2.a.cq.1.3
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.219687\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.60020 q^{3} -1.00000 q^{5} -4.33225 q^{7} -0.439374 q^{9} +O(q^{10})\) \(q+1.60020 q^{3} -1.00000 q^{5} -4.33225 q^{7} -0.439374 q^{9} -0.439374 q^{13} -1.60020 q^{15} +7.10387 q^{19} -6.93244 q^{21} +3.90348 q^{23} +1.00000 q^{25} -5.50367 q^{27} +4.00000 q^{29} -8.22512 q^{31} +4.33225 q^{35} +0.878747 q^{37} -0.703084 q^{39} +10.8359 q^{41} +10.4114 q^{43} +0.439374 q^{45} +0.478943 q^{47} +11.7684 q^{49} -1.63977 q^{53} +11.3676 q^{57} +2.70308 q^{59} -12.0363 q^{61} +1.90348 q^{63} +0.439374 q^{65} -5.67933 q^{67} +6.24632 q^{69} -16.0321 q^{71} +8.00000 q^{73} +1.60020 q^{75} +3.78575 q^{79} -7.48883 q^{81} -3.50269 q^{83} +6.40078 q^{87} -11.2077 q^{89} +1.90348 q^{91} -13.1618 q^{93} -7.10387 q^{95} -14.9688 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} + 4 q^{13} - 2 q^{15} - 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} + 16 q^{29} - 16 q^{31} + 6 q^{35} - 8 q^{37} + 8 q^{39} + 8 q^{41} + 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{57} - 4 q^{61} - 12 q^{63} - 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} + 32 q^{73} + 2 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} + 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100}) \) 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
\(3\) 1.60020 0.923873 0.461937 0.886913i \(-0.347155\pi\)
0.461937 + 0.886913i \(0.347155\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.33225 −1.63744 −0.818718 0.574196i \(-0.805315\pi\)
−0.818718 + 0.574196i \(0.805315\pi\)
\(8\) 0 0
\(9\) −0.439374 −0.146458
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −0.439374 −0.121860 −0.0609302 0.998142i \(-0.519407\pi\)
−0.0609302 + 0.998142i \(0.519407\pi\)
\(14\) 0 0
\(15\) −1.60020 −0.413169
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.10387 1.62974 0.814869 0.579645i \(-0.196809\pi\)
0.814869 + 0.579645i \(0.196809\pi\)
\(20\) 0 0
\(21\) −6.93244 −1.51278
\(22\) 0 0
\(23\) 3.90348 0.813931 0.406965 0.913444i \(-0.366587\pi\)
0.406965 + 0.913444i \(0.366587\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.50367 −1.05918
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −8.22512 −1.47728 −0.738638 0.674103i \(-0.764530\pi\)
−0.738638 + 0.674103i \(0.764530\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.33225 0.732283
\(36\) 0 0
\(37\) 0.878747 0.144465 0.0722326 0.997388i \(-0.476988\pi\)
0.0722326 + 0.997388i \(0.476988\pi\)
\(38\) 0 0
\(39\) −0.703084 −0.112584
\(40\) 0 0
\(41\) 10.8359 1.69229 0.846143 0.532956i \(-0.178919\pi\)
0.846143 + 0.532956i \(0.178919\pi\)
\(42\) 0 0
\(43\) 10.4114 1.58772 0.793861 0.608100i \(-0.208068\pi\)
0.793861 + 0.608100i \(0.208068\pi\)
\(44\) 0 0
\(45\) 0.439374 0.0654980
\(46\) 0 0
\(47\) 0.478943 0.0698610 0.0349305 0.999390i \(-0.488879\pi\)
0.0349305 + 0.999390i \(0.488879\pi\)
\(48\) 0 0
\(49\) 11.7684 1.68119
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.63977 −0.225239 −0.112620 0.993638i \(-0.535924\pi\)
−0.112620 + 0.993638i \(0.535924\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 11.3676 1.50567
\(58\) 0 0
\(59\) 2.70308 0.351912 0.175956 0.984398i \(-0.443698\pi\)
0.175956 + 0.984398i \(0.443698\pi\)
\(60\) 0 0
\(61\) −12.0363 −1.54109 −0.770546 0.637385i \(-0.780016\pi\)
−0.770546 + 0.637385i \(0.780016\pi\)
\(62\) 0 0
\(63\) 1.90348 0.239815
\(64\) 0 0
\(65\) 0.439374 0.0544976
\(66\) 0 0
\(67\) −5.67933 −0.693841 −0.346921 0.937895i \(-0.612773\pi\)
−0.346921 + 0.937895i \(0.612773\pi\)
\(68\) 0 0
\(69\) 6.24632 0.751969
\(70\) 0 0
\(71\) −16.0321 −1.90266 −0.951328 0.308179i \(-0.900280\pi\)
−0.951328 + 0.308179i \(0.900280\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 1.60020 0.184775
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.78575 0.425930 0.212965 0.977060i \(-0.431688\pi\)
0.212965 + 0.977060i \(0.431688\pi\)
\(80\) 0 0
\(81\) −7.48883 −0.832092
\(82\) 0 0
\(83\) −3.50269 −0.384470 −0.192235 0.981349i \(-0.561574\pi\)
−0.192235 + 0.981349i \(0.561574\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.40078 0.686236
\(88\) 0 0
\(89\) −11.2077 −1.18802 −0.594009 0.804459i \(-0.702456\pi\)
−0.594009 + 0.804459i \(0.702456\pi\)
\(90\) 0 0
\(91\) 1.90348 0.199538
\(92\) 0 0
\(93\) −13.1618 −1.36482
\(94\) 0 0
\(95\) −7.10387 −0.728841
\(96\) 0 0
\(97\) −14.9688 −1.51985 −0.759923 0.650013i \(-0.774763\pi\)
−0.759923 + 0.650013i \(0.774763\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.46834 −0.345113 −0.172556 0.985000i \(-0.555203\pi\)
−0.172556 + 0.985000i \(0.555203\pi\)
\(102\) 0 0
\(103\) −1.85402 −0.182682 −0.0913410 0.995820i \(-0.529115\pi\)
−0.0913410 + 0.995820i \(0.529115\pi\)
\(104\) 0 0
\(105\) 6.93244 0.676537
\(106\) 0 0
\(107\) −16.5079 −1.59588 −0.797940 0.602737i \(-0.794077\pi\)
−0.797940 + 0.602737i \(0.794077\pi\)
\(108\) 0 0
\(109\) −0.949828 −0.0909770 −0.0454885 0.998965i \(-0.514484\pi\)
−0.0454885 + 0.998965i \(0.514484\pi\)
\(110\) 0 0
\(111\) 1.40617 0.133468
\(112\) 0 0
\(113\) −0.477965 −0.0449631 −0.0224816 0.999747i \(-0.507157\pi\)
−0.0224816 + 0.999747i \(0.507157\pi\)
\(114\) 0 0
\(115\) −3.90348 −0.364001
\(116\) 0 0
\(117\) 0.193049 0.0178474
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 17.3396 1.56346
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.82955 −0.428554 −0.214277 0.976773i \(-0.568739\pi\)
−0.214277 + 0.976773i \(0.568739\pi\)
\(128\) 0 0
\(129\) 16.6603 1.46685
\(130\) 0 0
\(131\) −17.1533 −1.49869 −0.749346 0.662178i \(-0.769632\pi\)
−0.749346 + 0.662178i \(0.769632\pi\)
\(132\) 0 0
\(133\) −30.7757 −2.66859
\(134\) 0 0
\(135\) 5.50367 0.473681
\(136\) 0 0
\(137\) 8.56797 0.732011 0.366005 0.930613i \(-0.380725\pi\)
0.366005 + 0.930613i \(0.380725\pi\)
\(138\) 0 0
\(139\) −8.19687 −0.695249 −0.347625 0.937634i \(-0.613012\pi\)
−0.347625 + 0.937634i \(0.613012\pi\)
\(140\) 0 0
\(141\) 0.766403 0.0645428
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 18.8317 1.55321
\(148\) 0 0
\(149\) −18.2440 −1.49461 −0.747305 0.664481i \(-0.768653\pi\)
−0.747305 + 0.664481i \(0.768653\pi\)
\(150\) 0 0
\(151\) 6.88961 0.560669 0.280334 0.959902i \(-0.409555\pi\)
0.280334 + 0.959902i \(0.409555\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.22512 0.660657
\(156\) 0 0
\(157\) −8.15828 −0.651101 −0.325551 0.945525i \(-0.605550\pi\)
−0.325551 + 0.945525i \(0.605550\pi\)
\(158\) 0 0
\(159\) −2.62395 −0.208092
\(160\) 0 0
\(161\) −16.9108 −1.33276
\(162\) 0 0
\(163\) 9.42453 0.738186 0.369093 0.929392i \(-0.379668\pi\)
0.369093 + 0.929392i \(0.379668\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.26045 0.561830 0.280915 0.959733i \(-0.409362\pi\)
0.280915 + 0.959733i \(0.409362\pi\)
\(168\) 0 0
\(169\) −12.8070 −0.985150
\(170\) 0 0
\(171\) −3.12125 −0.238688
\(172\) 0 0
\(173\) 13.8455 1.05266 0.526329 0.850281i \(-0.323568\pi\)
0.526329 + 0.850281i \(0.323568\pi\)
\(174\) 0 0
\(175\) −4.33225 −0.327487
\(176\) 0 0
\(177\) 4.32546 0.325122
\(178\) 0 0
\(179\) −19.0073 −1.42068 −0.710338 0.703861i \(-0.751458\pi\)
−0.710338 + 0.703861i \(0.751458\pi\)
\(180\) 0 0
\(181\) 20.9850 1.55980 0.779901 0.625902i \(-0.215269\pi\)
0.779901 + 0.625902i \(0.215269\pi\)
\(182\) 0 0
\(183\) −19.2604 −1.42377
\(184\) 0 0
\(185\) −0.878747 −0.0646068
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 23.8433 1.73434
\(190\) 0 0
\(191\) −2.97527 −0.215283 −0.107642 0.994190i \(-0.534330\pi\)
−0.107642 + 0.994190i \(0.534330\pi\)
\(192\) 0 0
\(193\) 22.5506 1.62323 0.811613 0.584195i \(-0.198590\pi\)
0.811613 + 0.584195i \(0.198590\pi\)
\(194\) 0 0
\(195\) 0.703084 0.0503489
\(196\) 0 0
\(197\) 10.2162 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(198\) 0 0
\(199\) 5.40812 0.383372 0.191686 0.981456i \(-0.438605\pi\)
0.191686 + 0.981456i \(0.438605\pi\)
\(200\) 0 0
\(201\) −9.08805 −0.641021
\(202\) 0 0
\(203\) −17.3290 −1.21626
\(204\) 0 0
\(205\) −10.8359 −0.756813
\(206\) 0 0
\(207\) −1.71508 −0.119207
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −27.4981 −1.89305 −0.946525 0.322632i \(-0.895433\pi\)
−0.946525 + 0.322632i \(0.895433\pi\)
\(212\) 0 0
\(213\) −25.6544 −1.75781
\(214\) 0 0
\(215\) −10.4114 −0.710051
\(216\) 0 0
\(217\) 35.6332 2.41894
\(218\) 0 0
\(219\) 12.8016 0.865050
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −27.3898 −1.83415 −0.917077 0.398710i \(-0.869458\pi\)
−0.917077 + 0.398710i \(0.869458\pi\)
\(224\) 0 0
\(225\) −0.439374 −0.0292916
\(226\) 0 0
\(227\) −15.1319 −1.00434 −0.502168 0.864770i \(-0.667464\pi\)
−0.502168 + 0.864770i \(0.667464\pi\)
\(228\) 0 0
\(229\) 15.2869 1.01018 0.505092 0.863065i \(-0.331458\pi\)
0.505092 + 0.863065i \(0.331458\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.1251 −0.859852 −0.429926 0.902864i \(-0.641460\pi\)
−0.429926 + 0.902864i \(0.641460\pi\)
\(234\) 0 0
\(235\) −0.478943 −0.0312428
\(236\) 0 0
\(237\) 6.05793 0.393505
\(238\) 0 0
\(239\) −7.37410 −0.476991 −0.238495 0.971144i \(-0.576654\pi\)
−0.238495 + 0.971144i \(0.576654\pi\)
\(240\) 0 0
\(241\) −15.3633 −0.989640 −0.494820 0.868996i \(-0.664766\pi\)
−0.494820 + 0.868996i \(0.664766\pi\)
\(242\) 0 0
\(243\) 4.52742 0.290434
\(244\) 0 0
\(245\) −11.7684 −0.751853
\(246\) 0 0
\(247\) −3.12125 −0.198601
\(248\) 0 0
\(249\) −5.60499 −0.355202
\(250\) 0 0
\(251\) −17.3611 −1.09582 −0.547910 0.836537i \(-0.684576\pi\)
−0.547910 + 0.836537i \(0.684576\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.8475 1.86184 0.930918 0.365228i \(-0.119009\pi\)
0.930918 + 0.365228i \(0.119009\pi\)
\(258\) 0 0
\(259\) −3.80695 −0.236552
\(260\) 0 0
\(261\) −1.75749 −0.108786
\(262\) 0 0
\(263\) −11.4255 −0.704527 −0.352264 0.935901i \(-0.614588\pi\)
−0.352264 + 0.935901i \(0.614588\pi\)
\(264\) 0 0
\(265\) 1.63977 0.100730
\(266\) 0 0
\(267\) −17.9346 −1.09758
\(268\) 0 0
\(269\) −23.3255 −1.42218 −0.711089 0.703102i \(-0.751798\pi\)
−0.711089 + 0.703102i \(0.751798\pi\)
\(270\) 0 0
\(271\) −2.04946 −0.124496 −0.0622478 0.998061i \(-0.519827\pi\)
−0.0622478 + 0.998061i \(0.519827\pi\)
\(272\) 0 0
\(273\) 3.04593 0.184348
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.377621 0.0226890 0.0113445 0.999936i \(-0.496389\pi\)
0.0113445 + 0.999936i \(0.496389\pi\)
\(278\) 0 0
\(279\) 3.61390 0.216359
\(280\) 0 0
\(281\) 19.2795 1.15012 0.575060 0.818111i \(-0.304979\pi\)
0.575060 + 0.818111i \(0.304979\pi\)
\(282\) 0 0
\(283\) 10.5786 0.628831 0.314415 0.949286i \(-0.398192\pi\)
0.314415 + 0.949286i \(0.398192\pi\)
\(284\) 0 0
\(285\) −11.3676 −0.673357
\(286\) 0 0
\(287\) −46.9439 −2.77101
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −23.9529 −1.40415
\(292\) 0 0
\(293\) 22.9900 1.34309 0.671544 0.740965i \(-0.265632\pi\)
0.671544 + 0.740965i \(0.265632\pi\)
\(294\) 0 0
\(295\) −2.70308 −0.157380
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.71508 −0.0991859
\(300\) 0 0
\(301\) −45.1047 −2.59979
\(302\) 0 0
\(303\) −5.55002 −0.318841
\(304\) 0 0
\(305\) 12.0363 0.689197
\(306\) 0 0
\(307\) −30.0835 −1.71696 −0.858478 0.512851i \(-0.828589\pi\)
−0.858478 + 0.512851i \(0.828589\pi\)
\(308\) 0 0
\(309\) −2.96679 −0.168775
\(310\) 0 0
\(311\) 17.2819 0.979968 0.489984 0.871732i \(-0.337003\pi\)
0.489984 + 0.871732i \(0.337003\pi\)
\(312\) 0 0
\(313\) −6.71156 −0.379360 −0.189680 0.981846i \(-0.560745\pi\)
−0.189680 + 0.981846i \(0.560745\pi\)
\(314\) 0 0
\(315\) −1.90348 −0.107249
\(316\) 0 0
\(317\) 22.8016 1.28066 0.640332 0.768098i \(-0.278797\pi\)
0.640332 + 0.768098i \(0.278797\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −26.4159 −1.47439
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −0.439374 −0.0243721
\(326\) 0 0
\(327\) −1.51991 −0.0840512
\(328\) 0 0
\(329\) −2.07490 −0.114393
\(330\) 0 0
\(331\) 3.95293 0.217273 0.108636 0.994082i \(-0.465352\pi\)
0.108636 + 0.994082i \(0.465352\pi\)
\(332\) 0 0
\(333\) −0.386099 −0.0211581
\(334\) 0 0
\(335\) 5.67933 0.310295
\(336\) 0 0
\(337\) 13.5112 0.736000 0.368000 0.929826i \(-0.380043\pi\)
0.368000 + 0.929826i \(0.380043\pi\)
\(338\) 0 0
\(339\) −0.764837 −0.0415402
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.6577 −1.11541
\(344\) 0 0
\(345\) −6.24632 −0.336291
\(346\) 0 0
\(347\) −11.6203 −0.623808 −0.311904 0.950114i \(-0.600967\pi\)
−0.311904 + 0.950114i \(0.600967\pi\)
\(348\) 0 0
\(349\) −15.9228 −0.852329 −0.426164 0.904646i \(-0.640136\pi\)
−0.426164 + 0.904646i \(0.640136\pi\)
\(350\) 0 0
\(351\) 2.41817 0.129072
\(352\) 0 0
\(353\) 1.76640 0.0940161 0.0470081 0.998895i \(-0.485031\pi\)
0.0470081 + 0.998895i \(0.485031\pi\)
\(354\) 0 0
\(355\) 16.0321 0.850894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.4329 −1.50063 −0.750314 0.661081i \(-0.770098\pi\)
−0.750314 + 0.661081i \(0.770098\pi\)
\(360\) 0 0
\(361\) 31.4649 1.65605
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 7.60020 0.396727 0.198363 0.980129i \(-0.436437\pi\)
0.198363 + 0.980129i \(0.436437\pi\)
\(368\) 0 0
\(369\) −4.76102 −0.247849
\(370\) 0 0
\(371\) 7.10387 0.368814
\(372\) 0 0
\(373\) −17.9529 −0.929568 −0.464784 0.885424i \(-0.653868\pi\)
−0.464784 + 0.885424i \(0.653868\pi\)
\(374\) 0 0
\(375\) −1.60020 −0.0826338
\(376\) 0 0
\(377\) −1.75749 −0.0905156
\(378\) 0 0
\(379\) 7.80891 0.401117 0.200558 0.979682i \(-0.435724\pi\)
0.200558 + 0.979682i \(0.435724\pi\)
\(380\) 0 0
\(381\) −7.72823 −0.395929
\(382\) 0 0
\(383\) −33.3907 −1.70619 −0.853094 0.521758i \(-0.825276\pi\)
−0.853094 + 0.521758i \(0.825276\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.57449 −0.232534
\(388\) 0 0
\(389\) −5.28305 −0.267861 −0.133931 0.990991i \(-0.542760\pi\)
−0.133931 + 0.990991i \(0.542760\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −27.4487 −1.38460
\(394\) 0 0
\(395\) −3.78575 −0.190482
\(396\) 0 0
\(397\) −35.6139 −1.78741 −0.893705 0.448655i \(-0.851903\pi\)
−0.893705 + 0.448655i \(0.851903\pi\)
\(398\) 0 0
\(399\) −49.2471 −2.46544
\(400\) 0 0
\(401\) −27.5591 −1.37623 −0.688117 0.725600i \(-0.741562\pi\)
−0.688117 + 0.725600i \(0.741562\pi\)
\(402\) 0 0
\(403\) 3.61390 0.180021
\(404\) 0 0
\(405\) 7.48883 0.372123
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 24.5162 1.21225 0.606125 0.795370i \(-0.292723\pi\)
0.606125 + 0.795370i \(0.292723\pi\)
\(410\) 0 0
\(411\) 13.7104 0.676285
\(412\) 0 0
\(413\) −11.7104 −0.576232
\(414\) 0 0
\(415\) 3.50269 0.171940
\(416\) 0 0
\(417\) −13.1166 −0.642322
\(418\) 0 0
\(419\) 24.4625 1.19507 0.597537 0.801842i \(-0.296146\pi\)
0.597537 + 0.801842i \(0.296146\pi\)
\(420\) 0 0
\(421\) −2.00352 −0.0976457 −0.0488229 0.998807i \(-0.515547\pi\)
−0.0488229 + 0.998807i \(0.515547\pi\)
\(422\) 0 0
\(423\) −0.210435 −0.0102317
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 52.1443 2.52344
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.1919 1.40612 0.703062 0.711128i \(-0.251815\pi\)
0.703062 + 0.711128i \(0.251815\pi\)
\(432\) 0 0
\(433\) 30.6491 1.47290 0.736450 0.676492i \(-0.236501\pi\)
0.736450 + 0.676492i \(0.236501\pi\)
\(434\) 0 0
\(435\) −6.40078 −0.306894
\(436\) 0 0
\(437\) 27.7298 1.32649
\(438\) 0 0
\(439\) 16.6857 0.796365 0.398182 0.917306i \(-0.369641\pi\)
0.398182 + 0.917306i \(0.369641\pi\)
\(440\) 0 0
\(441\) −5.17071 −0.246224
\(442\) 0 0
\(443\) −9.69672 −0.460705 −0.230353 0.973107i \(-0.573988\pi\)
−0.230353 + 0.973107i \(0.573988\pi\)
\(444\) 0 0
\(445\) 11.2077 0.531298
\(446\) 0 0
\(447\) −29.1940 −1.38083
\(448\) 0 0
\(449\) 18.8772 0.890869 0.445435 0.895314i \(-0.353049\pi\)
0.445435 + 0.895314i \(0.353049\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 11.0247 0.517987
\(454\) 0 0
\(455\) −1.90348 −0.0892363
\(456\) 0 0
\(457\) 40.0487 1.87340 0.936700 0.350133i \(-0.113864\pi\)
0.936700 + 0.350133i \(0.113864\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.58112 −0.120215 −0.0601073 0.998192i \(-0.519144\pi\)
−0.0601073 + 0.998192i \(0.519144\pi\)
\(462\) 0 0
\(463\) −31.7759 −1.47675 −0.738375 0.674391i \(-0.764406\pi\)
−0.738375 + 0.674391i \(0.764406\pi\)
\(464\) 0 0
\(465\) 13.1618 0.610364
\(466\) 0 0
\(467\) 10.5779 0.489485 0.244742 0.969588i \(-0.421297\pi\)
0.244742 + 0.969588i \(0.421297\pi\)
\(468\) 0 0
\(469\) 24.6043 1.13612
\(470\) 0 0
\(471\) −13.0548 −0.601535
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.10387 0.325948
\(476\) 0 0
\(477\) 0.720470 0.0329880
\(478\) 0 0
\(479\) −22.5209 −1.02901 −0.514503 0.857489i \(-0.672024\pi\)
−0.514503 + 0.857489i \(0.672024\pi\)
\(480\) 0 0
\(481\) −0.386099 −0.0176046
\(482\) 0 0
\(483\) −27.0606 −1.23130
\(484\) 0 0
\(485\) 14.9688 0.679696
\(486\) 0 0
\(487\) −2.43794 −0.110474 −0.0552368 0.998473i \(-0.517591\pi\)
−0.0552368 + 0.998473i \(0.517591\pi\)
\(488\) 0 0
\(489\) 15.0811 0.681991
\(490\) 0 0
\(491\) 21.1919 0.956378 0.478189 0.878257i \(-0.341293\pi\)
0.478189 + 0.878257i \(0.341293\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 69.4549 3.11548
\(498\) 0 0
\(499\) −2.20578 −0.0987441 −0.0493721 0.998780i \(-0.515722\pi\)
−0.0493721 + 0.998780i \(0.515722\pi\)
\(500\) 0 0
\(501\) 11.6181 0.519060
\(502\) 0 0
\(503\) −15.9932 −0.713102 −0.356551 0.934276i \(-0.616047\pi\)
−0.356551 + 0.934276i \(0.616047\pi\)
\(504\) 0 0
\(505\) 3.46834 0.154339
\(506\) 0 0
\(507\) −20.4936 −0.910154
\(508\) 0 0
\(509\) −10.4706 −0.464102 −0.232051 0.972704i \(-0.574544\pi\)
−0.232051 + 0.972704i \(0.574544\pi\)
\(510\) 0 0
\(511\) −34.6580 −1.53318
\(512\) 0 0
\(513\) −39.0973 −1.72619
\(514\) 0 0
\(515\) 1.85402 0.0816979
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 22.1556 0.972522
\(520\) 0 0
\(521\) 31.5313 1.38141 0.690706 0.723135i \(-0.257300\pi\)
0.690706 + 0.723135i \(0.257300\pi\)
\(522\) 0 0
\(523\) 33.1058 1.44762 0.723808 0.690001i \(-0.242390\pi\)
0.723808 + 0.690001i \(0.242390\pi\)
\(524\) 0 0
\(525\) −6.93244 −0.302557
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.76288 −0.337517
\(530\) 0 0
\(531\) −1.18766 −0.0515402
\(532\) 0 0
\(533\) −4.76102 −0.206223
\(534\) 0 0
\(535\) 16.5079 0.713699
\(536\) 0 0
\(537\) −30.4155 −1.31252
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22.8166 −0.980961 −0.490481 0.871452i \(-0.663179\pi\)
−0.490481 + 0.871452i \(0.663179\pi\)
\(542\) 0 0
\(543\) 33.5801 1.44106
\(544\) 0 0
\(545\) 0.949828 0.0406862
\(546\) 0 0
\(547\) 4.78927 0.204774 0.102387 0.994745i \(-0.467352\pi\)
0.102387 + 0.994745i \(0.467352\pi\)
\(548\) 0 0
\(549\) 5.28844 0.225705
\(550\) 0 0
\(551\) 28.4155 1.21054
\(552\) 0 0
\(553\) −16.4008 −0.697432
\(554\) 0 0
\(555\) −1.40617 −0.0596885
\(556\) 0 0
\(557\) 21.9158 0.928601 0.464301 0.885678i \(-0.346306\pi\)
0.464301 + 0.885678i \(0.346306\pi\)
\(558\) 0 0
\(559\) −4.57449 −0.193480
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −30.5349 −1.28689 −0.643446 0.765491i \(-0.722496\pi\)
−0.643446 + 0.765491i \(0.722496\pi\)
\(564\) 0 0
\(565\) 0.477965 0.0201081
\(566\) 0 0
\(567\) 32.4435 1.36250
\(568\) 0 0
\(569\) 12.4351 0.521308 0.260654 0.965432i \(-0.416062\pi\)
0.260654 + 0.965432i \(0.416062\pi\)
\(570\) 0 0
\(571\) −24.9217 −1.04294 −0.521470 0.853270i \(-0.674616\pi\)
−0.521470 + 0.853270i \(0.674616\pi\)
\(572\) 0 0
\(573\) −4.76102 −0.198894
\(574\) 0 0
\(575\) 3.90348 0.162786
\(576\) 0 0
\(577\) −37.9464 −1.57973 −0.789865 0.613281i \(-0.789849\pi\)
−0.789865 + 0.613281i \(0.789849\pi\)
\(578\) 0 0
\(579\) 36.0853 1.49966
\(580\) 0 0
\(581\) 15.1745 0.629545
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −0.193049 −0.00798161
\(586\) 0 0
\(587\) 12.4469 0.513737 0.256869 0.966446i \(-0.417309\pi\)
0.256869 + 0.966446i \(0.417309\pi\)
\(588\) 0 0
\(589\) −58.4302 −2.40757
\(590\) 0 0
\(591\) 16.3479 0.672464
\(592\) 0 0
\(593\) −17.4981 −0.718562 −0.359281 0.933229i \(-0.616978\pi\)
−0.359281 + 0.933229i \(0.616978\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.65406 0.354187
\(598\) 0 0
\(599\) −10.1389 −0.414266 −0.207133 0.978313i \(-0.566413\pi\)
−0.207133 + 0.978313i \(0.566413\pi\)
\(600\) 0 0
\(601\) −13.0718 −0.533210 −0.266605 0.963806i \(-0.585902\pi\)
−0.266605 + 0.963806i \(0.585902\pi\)
\(602\) 0 0
\(603\) 2.49535 0.101619
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.95293 0.322800 0.161400 0.986889i \(-0.448399\pi\)
0.161400 + 0.986889i \(0.448399\pi\)
\(608\) 0 0
\(609\) −27.7298 −1.12367
\(610\) 0 0
\(611\) −0.210435 −0.00851329
\(612\) 0 0
\(613\) 20.5576 0.830315 0.415157 0.909750i \(-0.363727\pi\)
0.415157 + 0.909750i \(0.363727\pi\)
\(614\) 0 0
\(615\) −17.3396 −0.699200
\(616\) 0 0
\(617\) 24.4244 0.983288 0.491644 0.870796i \(-0.336396\pi\)
0.491644 + 0.870796i \(0.336396\pi\)
\(618\) 0 0
\(619\) −25.8564 −1.03926 −0.519628 0.854392i \(-0.673930\pi\)
−0.519628 + 0.854392i \(0.673930\pi\)
\(620\) 0 0
\(621\) −21.4834 −0.862101
\(622\) 0 0
\(623\) 48.5547 1.94530
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −38.6580 −1.53895 −0.769475 0.638677i \(-0.779482\pi\)
−0.769475 + 0.638677i \(0.779482\pi\)
\(632\) 0 0
\(633\) −44.0024 −1.74894
\(634\) 0 0
\(635\) 4.82955 0.191655
\(636\) 0 0
\(637\) −5.17071 −0.204871
\(638\) 0 0
\(639\) 7.04407 0.278659
\(640\) 0 0
\(641\) −22.4333 −0.886061 −0.443031 0.896506i \(-0.646097\pi\)
−0.443031 + 0.896506i \(0.646097\pi\)
\(642\) 0 0
\(643\) −22.3052 −0.879633 −0.439816 0.898088i \(-0.644956\pi\)
−0.439816 + 0.898088i \(0.644956\pi\)
\(644\) 0 0
\(645\) −16.6603 −0.655997
\(646\) 0 0
\(647\) 45.5401 1.79037 0.895183 0.445699i \(-0.147045\pi\)
0.895183 + 0.445699i \(0.147045\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 57.0202 2.23480
\(652\) 0 0
\(653\) 3.13594 0.122719 0.0613593 0.998116i \(-0.480456\pi\)
0.0613593 + 0.998116i \(0.480456\pi\)
\(654\) 0 0
\(655\) 17.1533 0.670236
\(656\) 0 0
\(657\) −3.51499 −0.137133
\(658\) 0 0
\(659\) −44.0853 −1.71732 −0.858661 0.512545i \(-0.828703\pi\)
−0.858661 + 0.512545i \(0.828703\pi\)
\(660\) 0 0
\(661\) −30.0182 −1.16757 −0.583786 0.811907i \(-0.698429\pi\)
−0.583786 + 0.811907i \(0.698429\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30.7757 1.19343
\(666\) 0 0
\(667\) 15.6139 0.604573
\(668\) 0 0
\(669\) −43.8290 −1.69453
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 14.3814 0.554364 0.277182 0.960817i \(-0.410600\pi\)
0.277182 + 0.960817i \(0.410600\pi\)
\(674\) 0 0
\(675\) −5.50367 −0.211836
\(676\) 0 0
\(677\) 27.9059 1.07251 0.536255 0.844056i \(-0.319839\pi\)
0.536255 + 0.844056i \(0.319839\pi\)
\(678\) 0 0
\(679\) 64.8483 2.48865
\(680\) 0 0
\(681\) −24.2139 −0.927880
\(682\) 0 0
\(683\) 19.8055 0.757838 0.378919 0.925430i \(-0.376296\pi\)
0.378919 + 0.925430i \(0.376296\pi\)
\(684\) 0 0
\(685\) −8.56797 −0.327365
\(686\) 0 0
\(687\) 24.4620 0.933283
\(688\) 0 0
\(689\) 0.720470 0.0274477
\(690\) 0 0
\(691\) −6.00239 −0.228342 −0.114171 0.993461i \(-0.536421\pi\)
−0.114171 + 0.993461i \(0.536421\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.19687 0.310925
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −21.0027 −0.794394
\(700\) 0 0
\(701\) −9.91577 −0.374514 −0.187257 0.982311i \(-0.559960\pi\)
−0.187257 + 0.982311i \(0.559960\pi\)
\(702\) 0 0
\(703\) 6.24251 0.235441
\(704\) 0 0
\(705\) −0.766403 −0.0288644
\(706\) 0 0
\(707\) 15.0257 0.565100
\(708\) 0 0
\(709\) −47.9132 −1.79942 −0.899709 0.436490i \(-0.856221\pi\)
−0.899709 + 0.436490i \(0.856221\pi\)
\(710\) 0 0
\(711\) −1.66336 −0.0623808
\(712\) 0 0
\(713\) −32.1066 −1.20240
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −11.8000 −0.440679
\(718\) 0 0
\(719\) −20.1884 −0.752900 −0.376450 0.926437i \(-0.622855\pi\)
−0.376450 + 0.926437i \(0.622855\pi\)
\(720\) 0 0
\(721\) 8.03207 0.299130
\(722\) 0 0
\(723\) −24.5843 −0.914302
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 29.5570 1.09621 0.548105 0.836410i \(-0.315349\pi\)
0.548105 + 0.836410i \(0.315349\pi\)
\(728\) 0 0
\(729\) 29.7112 1.10042
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 47.0734 1.73870 0.869349 0.494199i \(-0.164539\pi\)
0.869349 + 0.494199i \(0.164539\pi\)
\(734\) 0 0
\(735\) −18.8317 −0.694617
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.9810 0.624658 0.312329 0.949974i \(-0.398891\pi\)
0.312329 + 0.949974i \(0.398891\pi\)
\(740\) 0 0
\(741\) −4.99461 −0.183482
\(742\) 0 0
\(743\) −4.34115 −0.159262 −0.0796308 0.996824i \(-0.525374\pi\)
−0.0796308 + 0.996824i \(0.525374\pi\)
\(744\) 0 0
\(745\) 18.2440 0.668410
\(746\) 0 0
\(747\) 1.53899 0.0563087
\(748\) 0 0
\(749\) 71.5163 2.61315
\(750\) 0 0
\(751\) 43.5217 1.58813 0.794065 0.607833i \(-0.207961\pi\)
0.794065 + 0.607833i \(0.207961\pi\)
\(752\) 0 0
\(753\) −27.7811 −1.01240
\(754\) 0 0
\(755\) −6.88961 −0.250739
\(756\) 0 0
\(757\) −44.8082 −1.62858 −0.814291 0.580457i \(-0.802874\pi\)
−0.814291 + 0.580457i \(0.802874\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.2007 −1.31227 −0.656137 0.754642i \(-0.727811\pi\)
−0.656137 + 0.754642i \(0.727811\pi\)
\(762\) 0 0
\(763\) 4.11489 0.148969
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.18766 −0.0428841
\(768\) 0 0
\(769\) −15.9652 −0.575721 −0.287860 0.957672i \(-0.592944\pi\)
−0.287860 + 0.957672i \(0.592944\pi\)
\(770\) 0 0
\(771\) 47.7618 1.72010
\(772\) 0 0
\(773\) −17.4190 −0.626518 −0.313259 0.949668i \(-0.601421\pi\)
−0.313259 + 0.949668i \(0.601421\pi\)
\(774\) 0 0
\(775\) −8.22512 −0.295455
\(776\) 0 0
\(777\) −6.09187 −0.218544
\(778\) 0 0
\(779\) 76.9769 2.75798
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −22.0147 −0.786741
\(784\) 0 0
\(785\) 8.15828 0.291181
\(786\) 0 0
\(787\) 42.4844 1.51441 0.757203 0.653179i \(-0.226565\pi\)
0.757203 + 0.653179i \(0.226565\pi\)
\(788\) 0 0
\(789\) −18.2831 −0.650894
\(790\) 0 0
\(791\) 2.07066 0.0736242
\(792\) 0 0
\(793\) 5.28844 0.187798
\(794\) 0 0
\(795\) 2.62395 0.0930617
\(796\) 0 0
\(797\) −9.33789 −0.330765 −0.165383 0.986229i \(-0.552886\pi\)
−0.165383 + 0.986229i \(0.552886\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.92438 0.173995
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 16.9108 0.596028
\(806\) 0 0
\(807\) −37.3253 −1.31391
\(808\) 0 0
\(809\) −19.1854 −0.674522 −0.337261 0.941411i \(-0.609501\pi\)
−0.337261 + 0.941411i \(0.609501\pi\)
\(810\) 0 0
\(811\) 11.7466 0.412480 0.206240 0.978501i \(-0.433877\pi\)
0.206240 + 0.978501i \(0.433877\pi\)
\(812\) 0 0
\(813\) −3.27953 −0.115018
\(814\) 0 0
\(815\) −9.42453 −0.330127
\(816\) 0 0
\(817\) 73.9611 2.58757
\(818\) 0 0
\(819\) −0.836337 −0.0292240
\(820\) 0 0
\(821\) −9.45552 −0.330000 −0.165000 0.986294i \(-0.552762\pi\)
−0.165000 + 0.986294i \(0.552762\pi\)
\(822\) 0 0
\(823\) −21.1203 −0.736206 −0.368103 0.929785i \(-0.619993\pi\)
−0.368103 + 0.929785i \(0.619993\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.2860 0.948827 0.474414 0.880302i \(-0.342660\pi\)
0.474414 + 0.880302i \(0.342660\pi\)
\(828\) 0 0
\(829\) 53.4113 1.85505 0.927526 0.373758i \(-0.121931\pi\)
0.927526 + 0.373758i \(0.121931\pi\)
\(830\) 0 0
\(831\) 0.604267 0.0209618
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −7.26045 −0.251258
\(836\) 0 0
\(837\) 45.2684 1.56470
\(838\) 0 0
\(839\) 13.5070 0.466315 0.233157 0.972439i \(-0.425094\pi\)
0.233157 + 0.972439i \(0.425094\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 30.8510 1.06257
\(844\) 0 0
\(845\) 12.8070 0.440572
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 16.9278 0.580960
\(850\) 0 0
\(851\) 3.43017 0.117585
\(852\) 0 0
\(853\) −14.3236 −0.490431 −0.245215 0.969469i \(-0.578859\pi\)
−0.245215 + 0.969469i \(0.578859\pi\)
\(854\) 0 0
\(855\) 3.12125 0.106745
\(856\) 0 0
\(857\) −22.8078 −0.779099 −0.389549 0.921006i \(-0.627369\pi\)
−0.389549 + 0.921006i \(0.627369\pi\)
\(858\) 0 0
\(859\) 0.556674 0.0189935 0.00949673 0.999955i \(-0.496977\pi\)
0.00949673 + 0.999955i \(0.496977\pi\)
\(860\) 0 0
\(861\) −75.1194 −2.56006
\(862\) 0 0
\(863\) −51.7257 −1.76076 −0.880381 0.474267i \(-0.842713\pi\)
−0.880381 + 0.474267i \(0.842713\pi\)
\(864\) 0 0
\(865\) −13.8455 −0.470763
\(866\) 0 0
\(867\) −27.2033 −0.923873
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.49535 0.0845517
\(872\) 0 0
\(873\) 6.57688 0.222594
\(874\) 0 0
\(875\) 4.33225 0.146457
\(876\) 0 0
\(877\) 1.30964 0.0442235 0.0221118 0.999756i \(-0.492961\pi\)
0.0221118 + 0.999756i \(0.492961\pi\)
\(878\) 0 0
\(879\) 36.7884 1.24084
\(880\) 0 0
\(881\) −14.8772 −0.501225 −0.250612 0.968087i \(-0.580632\pi\)
−0.250612 + 0.968087i \(0.580632\pi\)
\(882\) 0 0
\(883\) −21.0742 −0.709203 −0.354601 0.935018i \(-0.615383\pi\)
−0.354601 + 0.935018i \(0.615383\pi\)
\(884\) 0 0
\(885\) −4.32546 −0.145399
\(886\) 0 0
\(887\) −46.9521 −1.57650 −0.788248 0.615358i \(-0.789012\pi\)
−0.788248 + 0.615358i \(0.789012\pi\)
\(888\) 0 0
\(889\) 20.9228 0.701729
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.40235 0.113855
\(894\) 0 0
\(895\) 19.0073 0.635345
\(896\) 0 0
\(897\) −2.74447 −0.0916352
\(898\) 0 0
\(899\) −32.9005 −1.09729
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −72.1763 −2.40188
\(904\) 0 0
\(905\) −20.9850 −0.697565
\(906\) 0 0
\(907\) 19.9242 0.661573 0.330787 0.943706i \(-0.392686\pi\)
0.330787 + 0.943706i \(0.392686\pi\)
\(908\) 0 0
\(909\) 1.52390 0.0505445
\(910\) 0 0
\(911\) 19.6506 0.651054 0.325527 0.945533i \(-0.394458\pi\)
0.325527 + 0.945533i \(0.394458\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 19.2604 0.636731
\(916\) 0 0
\(917\) 74.3124 2.45401
\(918\) 0 0
\(919\) −12.0468 −0.397386 −0.198693 0.980062i \(-0.563670\pi\)
−0.198693 + 0.980062i \(0.563670\pi\)
\(920\) 0 0
\(921\) −48.1395 −1.58625
\(922\) 0 0
\(923\) 7.04407 0.231858
\(924\) 0 0
\(925\) 0.878747 0.0288930
\(926\) 0 0
\(927\) 0.814608 0.0267552
\(928\) 0 0
\(929\) 47.4525 1.55687 0.778433 0.627728i \(-0.216015\pi\)
0.778433 + 0.627728i \(0.216015\pi\)
\(930\) 0 0
\(931\) 83.6009 2.73991
\(932\) 0 0
\(933\) 27.6544 0.905366
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.00847793 0.000276962 0 0.000138481 1.00000i \(-0.499956\pi\)
0.000138481 1.00000i \(0.499956\pi\)
\(938\) 0 0
\(939\) −10.7398 −0.350481
\(940\) 0 0
\(941\) 31.9695 1.04217 0.521087 0.853503i \(-0.325527\pi\)
0.521087 + 0.853503i \(0.325527\pi\)
\(942\) 0 0
\(943\) 42.2977 1.37740
\(944\) 0 0
\(945\) −23.8433 −0.775621
\(946\) 0 0
\(947\) −18.8534 −0.612653 −0.306327 0.951926i \(-0.599100\pi\)
−0.306327 + 0.951926i \(0.599100\pi\)
\(948\) 0 0
\(949\) −3.51499 −0.114101
\(950\) 0 0
\(951\) 36.4870 1.18317
\(952\) 0 0
\(953\) −1.07180 −0.0347189 −0.0173595 0.999849i \(-0.505526\pi\)
−0.0173595 + 0.999849i \(0.505526\pi\)
\(954\) 0 0
\(955\) 2.97527 0.0962775
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −37.1186 −1.19862
\(960\) 0 0
\(961\) 36.6526 1.18234
\(962\) 0 0
\(963\) 7.25314 0.233729
\(964\) 0 0
\(965\) −22.5506 −0.725929
\(966\) 0 0
\(967\) −43.2749 −1.39163 −0.695813 0.718223i \(-0.744956\pi\)
−0.695813 + 0.718223i \(0.744956\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.0048 0.898716 0.449358 0.893352i \(-0.351653\pi\)
0.449358 + 0.893352i \(0.351653\pi\)
\(972\) 0 0
\(973\) 35.5109 1.13843
\(974\) 0 0
\(975\) −0.703084 −0.0225167
\(976\) 0 0
\(977\) −18.8341 −0.602555 −0.301278 0.953536i \(-0.597413\pi\)
−0.301278 + 0.953536i \(0.597413\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.417329 0.0133243
\(982\) 0 0
\(983\) −60.5281 −1.93055 −0.965273 0.261243i \(-0.915868\pi\)
−0.965273 + 0.261243i \(0.915868\pi\)
\(984\) 0 0
\(985\) −10.2162 −0.325516
\(986\) 0 0
\(987\) −3.32025 −0.105685
\(988\) 0 0
\(989\) 40.6406 1.29230
\(990\) 0 0
\(991\) −54.5931 −1.73421 −0.867103 0.498128i \(-0.834021\pi\)
−0.867103 + 0.498128i \(0.834021\pi\)
\(992\) 0 0
\(993\) 6.32546 0.200733
\(994\) 0 0
\(995\) −5.40812 −0.171449
\(996\) 0 0
\(997\) −17.9445 −0.568307 −0.284153 0.958779i \(-0.591712\pi\)
−0.284153 + 0.958779i \(0.591712\pi\)
\(998\) 0 0
\(999\) −4.83634 −0.153015
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 9680.2.a.cq.1.3 4
4.3 odd 2 4840.2.a.x.1.2 yes 4
11.10 odd 2 9680.2.a.cr.1.3 4
44.43 even 2 4840.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.2 4 44.43 even 2
4840.2.a.x.1.2 yes 4 4.3 odd 2
9680.2.a.cq.1.3 4 1.1 even 1 trivial
9680.2.a.cr.1.3 4 11.10 odd 2