Properties

Label 2880.2.bl.b.431.1
Level $2880$
Weight $2$
Character 2880.431
Analytic conductor $22.997$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,2,Mod(431,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.bl (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 720)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 431.1
Character \(\chi\) \(=\) 2880.431
Dual form 2880.2.bl.b.1871.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +O(q^{10})\) \(q+(-0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +(4.37014 - 4.37014i) q^{11} +(1.44518 + 1.44518i) q^{13} +6.69649i q^{17} +(1.91147 - 1.91147i) q^{19} +1.18773i q^{23} +1.00000i q^{25} +(2.16029 - 2.16029i) q^{29} -4.28465i q^{31} +(3.17676 + 3.17676i) q^{35} +(0.00767006 - 0.00767006i) q^{37} +4.93930 q^{41} +(-2.61608 - 2.61608i) q^{43} -12.9356 q^{47} +13.1836 q^{49} +(-6.04040 - 6.04040i) q^{53} -6.18031 q^{55} +(-6.51088 + 6.51088i) q^{59} +(1.14755 + 1.14755i) q^{61} -2.04379i q^{65} +(5.87653 - 5.87653i) q^{67} -8.93153i q^{71} -13.7131i q^{73} +(-19.6333 + 19.6333i) q^{77} -4.10759i q^{79} +(-4.52750 - 4.52750i) q^{83} +(4.73514 - 4.73514i) q^{85} -12.3387 q^{89} +(-6.49261 - 6.49261i) q^{91} -2.70322 q^{95} -8.38827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0
\(4\) 0 0
\(5\) −0.707107 0.707107i −0.316228 0.316228i
\(6\) 0 0
\(7\) −4.49261 −1.69805 −0.849024 0.528355i \(-0.822809\pi\)
−0.849024 + 0.528355i \(0.822809\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.37014 4.37014i 1.31765 1.31765i 0.402012 0.915634i \(-0.368311\pi\)
0.915634 0.402012i \(-0.131689\pi\)
\(12\) 0 0
\(13\) 1.44518 + 1.44518i 0.400820 + 0.400820i 0.878522 0.477702i \(-0.158530\pi\)
−0.477702 + 0.878522i \(0.658530\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.69649i 1.62414i 0.583561 + 0.812069i \(0.301659\pi\)
−0.583561 + 0.812069i \(0.698341\pi\)
\(18\) 0 0
\(19\) 1.91147 1.91147i 0.438521 0.438521i −0.452993 0.891514i \(-0.649644\pi\)
0.891514 + 0.452993i \(0.149644\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.18773i 0.247659i 0.992303 + 0.123830i \(0.0395177\pi\)
−0.992303 + 0.123830i \(0.960482\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.16029 2.16029i 0.401155 0.401155i −0.477485 0.878640i \(-0.658451\pi\)
0.878640 + 0.477485i \(0.158451\pi\)
\(30\) 0 0
\(31\) 4.28465i 0.769546i −0.923011 0.384773i \(-0.874280\pi\)
0.923011 0.384773i \(-0.125720\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.17676 + 3.17676i 0.536970 + 0.536970i
\(36\) 0 0
\(37\) 0.00767006 0.00767006i 0.00126095 0.00126095i −0.706476 0.707737i \(-0.749716\pi\)
0.707737 + 0.706476i \(0.249716\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.93930 0.771389 0.385694 0.922627i \(-0.373962\pi\)
0.385694 + 0.922627i \(0.373962\pi\)
\(42\) 0 0
\(43\) −2.61608 2.61608i −0.398949 0.398949i 0.478913 0.877862i \(-0.341031\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.9356 −1.88686 −0.943429 0.331574i \(-0.892420\pi\)
−0.943429 + 0.331574i \(0.892420\pi\)
\(48\) 0 0
\(49\) 13.1836 1.88337
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.04040 6.04040i −0.829712 0.829712i 0.157765 0.987477i \(-0.449571\pi\)
−0.987477 + 0.157765i \(0.949571\pi\)
\(54\) 0 0
\(55\) −6.18031 −0.833353
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.51088 + 6.51088i −0.847644 + 0.847644i −0.989839 0.142195i \(-0.954584\pi\)
0.142195 + 0.989839i \(0.454584\pi\)
\(60\) 0 0
\(61\) 1.14755 + 1.14755i 0.146928 + 0.146928i 0.776744 0.629816i \(-0.216870\pi\)
−0.629816 + 0.776744i \(0.716870\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.04379i 0.253501i
\(66\) 0 0
\(67\) 5.87653 5.87653i 0.717932 0.717932i −0.250249 0.968181i \(-0.580513\pi\)
0.968181 + 0.250249i \(0.0805126\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.93153i 1.05998i −0.848005 0.529989i \(-0.822196\pi\)
0.848005 0.529989i \(-0.177804\pi\)
\(72\) 0 0
\(73\) 13.7131i 1.60500i −0.596654 0.802498i \(-0.703504\pi\)
0.596654 0.802498i \(-0.296496\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19.6333 + 19.6333i −2.23743 + 2.23743i
\(78\) 0 0
\(79\) 4.10759i 0.462140i −0.972937 0.231070i \(-0.925777\pi\)
0.972937 0.231070i \(-0.0742226\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.52750 4.52750i −0.496958 0.496958i 0.413532 0.910490i \(-0.364295\pi\)
−0.910490 + 0.413532i \(0.864295\pi\)
\(84\) 0 0
\(85\) 4.73514 4.73514i 0.513598 0.513598i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.3387 −1.30790 −0.653950 0.756538i \(-0.726889\pi\)
−0.653950 + 0.756538i \(0.726889\pi\)
\(90\) 0 0
\(91\) −6.49261 6.49261i −0.680611 0.680611i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.70322 −0.277345
\(96\) 0 0
\(97\) −8.38827 −0.851700 −0.425850 0.904794i \(-0.640025\pi\)
−0.425850 + 0.904794i \(0.640025\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.91015 2.91015i −0.289571 0.289571i 0.547339 0.836911i \(-0.315641\pi\)
−0.836911 + 0.547339i \(0.815641\pi\)
\(102\) 0 0
\(103\) 11.5754 1.14055 0.570277 0.821452i \(-0.306836\pi\)
0.570277 + 0.821452i \(0.306836\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.53540 + 3.53540i −0.341780 + 0.341780i −0.857036 0.515256i \(-0.827697\pi\)
0.515256 + 0.857036i \(0.327697\pi\)
\(108\) 0 0
\(109\) −6.33976 6.33976i −0.607239 0.607239i 0.334985 0.942224i \(-0.391269\pi\)
−0.942224 + 0.334985i \(0.891269\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.54008i 0.427095i −0.976933 0.213547i \(-0.931498\pi\)
0.976933 0.213547i \(-0.0685018\pi\)
\(114\) 0 0
\(115\) 0.839854 0.839854i 0.0783168 0.0783168i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 30.0847i 2.75786i
\(120\) 0 0
\(121\) 27.1962i 2.47239i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.707107 0.707107i 0.0632456 0.0632456i
\(126\) 0 0
\(127\) 6.51781i 0.578362i −0.957274 0.289181i \(-0.906617\pi\)
0.957274 0.289181i \(-0.0933830\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.77886 2.77886i −0.242790 0.242790i 0.575213 0.818003i \(-0.304919\pi\)
−0.818003 + 0.575213i \(0.804919\pi\)
\(132\) 0 0
\(133\) −8.58748 + 8.58748i −0.744629 + 0.744629i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.63676 0.567017 0.283508 0.958970i \(-0.408502\pi\)
0.283508 + 0.958970i \(0.408502\pi\)
\(138\) 0 0
\(139\) −7.05388 7.05388i −0.598302 0.598302i 0.341558 0.939861i \(-0.389045\pi\)
−0.939861 + 0.341558i \(0.889045\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.6312 1.05628
\(144\) 0 0
\(145\) −3.05511 −0.253713
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.52545 + 9.52545i 0.780355 + 0.780355i 0.979891 0.199535i \(-0.0639432\pi\)
−0.199535 + 0.979891i \(0.563943\pi\)
\(150\) 0 0
\(151\) −23.5203 −1.91405 −0.957027 0.290000i \(-0.906345\pi\)
−0.957027 + 0.290000i \(0.906345\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.02971 + 3.02971i −0.243352 + 0.243352i
\(156\) 0 0
\(157\) 10.1912 + 10.1912i 0.813348 + 0.813348i 0.985134 0.171786i \(-0.0549537\pi\)
−0.171786 + 0.985134i \(0.554954\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.33602i 0.420537i
\(162\) 0 0
\(163\) 16.0686 16.0686i 1.25859 1.25859i 0.306823 0.951766i \(-0.400734\pi\)
0.951766 0.306823i \(-0.0992661\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.149541i 0.0115718i 0.999983 + 0.00578592i \(0.00184173\pi\)
−0.999983 + 0.00578592i \(0.998158\pi\)
\(168\) 0 0
\(169\) 8.82294i 0.678687i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.93140 + 5.93140i −0.450956 + 0.450956i −0.895672 0.444716i \(-0.853305\pi\)
0.444716 + 0.895672i \(0.353305\pi\)
\(174\) 0 0
\(175\) 4.49261i 0.339609i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.5103 + 13.5103i 1.00981 + 1.00981i 0.999951 + 0.00985834i \(0.00313806\pi\)
0.00985834 + 0.999951i \(0.496862\pi\)
\(180\) 0 0
\(181\) 11.5235 11.5235i 0.856536 0.856536i −0.134393 0.990928i \(-0.542908\pi\)
0.990928 + 0.134393i \(0.0429083\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.0108471 −0.000797495
\(186\) 0 0
\(187\) 29.2646 + 29.2646i 2.14004 + 2.14004i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.81786 0.565680 0.282840 0.959167i \(-0.408723\pi\)
0.282840 + 0.959167i \(0.408723\pi\)
\(192\) 0 0
\(193\) −23.4854 −1.69052 −0.845258 0.534358i \(-0.820553\pi\)
−0.845258 + 0.534358i \(0.820553\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.3998 12.3998i −0.883453 0.883453i 0.110431 0.993884i \(-0.464777\pi\)
−0.993884 + 0.110431i \(0.964777\pi\)
\(198\) 0 0
\(199\) 1.42807 0.101233 0.0506167 0.998718i \(-0.483881\pi\)
0.0506167 + 0.998718i \(0.483881\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.70533 + 9.70533i −0.681180 + 0.681180i
\(204\) 0 0
\(205\) −3.49261 3.49261i −0.243935 0.243935i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.7068i 1.15563i
\(210\) 0 0
\(211\) −9.18871 + 9.18871i −0.632577 + 0.632577i −0.948714 0.316137i \(-0.897614\pi\)
0.316137 + 0.948714i \(0.397614\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.69970i 0.252318i
\(216\) 0 0
\(217\) 19.2493i 1.30673i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.67761 + 9.67761i −0.650986 + 0.650986i
\(222\) 0 0
\(223\) 1.56615i 0.104877i 0.998624 + 0.0524385i \(0.0166994\pi\)
−0.998624 + 0.0524385i \(0.983301\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.36008 8.36008i −0.554878 0.554878i 0.372967 0.927845i \(-0.378341\pi\)
−0.927845 + 0.372967i \(0.878341\pi\)
\(228\) 0 0
\(229\) −12.1157 + 12.1157i −0.800630 + 0.800630i −0.983194 0.182564i \(-0.941560\pi\)
0.182564 + 0.983194i \(0.441560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.7468 1.22815 0.614073 0.789249i \(-0.289530\pi\)
0.614073 + 0.789249i \(0.289530\pi\)
\(234\) 0 0
\(235\) 9.14688 + 9.14688i 0.596677 + 0.596677i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.624861 0.0404189 0.0202095 0.999796i \(-0.493567\pi\)
0.0202095 + 0.999796i \(0.493567\pi\)
\(240\) 0 0
\(241\) 17.9025 1.15320 0.576600 0.817027i \(-0.304379\pi\)
0.576600 + 0.817027i \(0.304379\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.32218 9.32218i −0.595572 0.595572i
\(246\) 0 0
\(247\) 5.52481 0.351535
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.7919 + 15.7919i −0.996779 + 0.996779i −0.999995 0.00321589i \(-0.998976\pi\)
0.00321589 + 0.999995i \(0.498976\pi\)
\(252\) 0 0
\(253\) 5.19056 + 5.19056i 0.326328 + 0.326328i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.2199i 1.01177i 0.862601 + 0.505885i \(0.168834\pi\)
−0.862601 + 0.505885i \(0.831166\pi\)
\(258\) 0 0
\(259\) −0.0344586 + 0.0344586i −0.00214115 + 0.00214115i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.316457i 0.0195136i −0.999952 0.00975679i \(-0.996894\pi\)
0.999952 0.00975679i \(-0.00310573\pi\)
\(264\) 0 0
\(265\) 8.54241i 0.524756i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.71152 + 6.71152i −0.409208 + 0.409208i −0.881462 0.472254i \(-0.843441\pi\)
0.472254 + 0.881462i \(0.343441\pi\)
\(270\) 0 0
\(271\) 11.3985i 0.692411i 0.938159 + 0.346205i \(0.112530\pi\)
−0.938159 + 0.346205i \(0.887470\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.37014 + 4.37014i 0.263529 + 0.263529i
\(276\) 0 0
\(277\) 17.2648 17.2648i 1.03734 1.03734i 0.0380679 0.999275i \(-0.487880\pi\)
0.999275 0.0380679i \(-0.0121203\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.69745 0.160916 0.0804582 0.996758i \(-0.474362\pi\)
0.0804582 + 0.996758i \(0.474362\pi\)
\(282\) 0 0
\(283\) −5.38979 5.38979i −0.320389 0.320389i 0.528527 0.848916i \(-0.322745\pi\)
−0.848916 + 0.528527i \(0.822745\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −22.1903 −1.30985
\(288\) 0 0
\(289\) −27.8430 −1.63782
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.09213 3.09213i −0.180644 0.180644i 0.610992 0.791636i \(-0.290771\pi\)
−0.791636 + 0.610992i \(0.790771\pi\)
\(294\) 0 0
\(295\) 9.20777 0.536097
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.71648 + 1.71648i −0.0992667 + 0.0992667i
\(300\) 0 0
\(301\) 11.7531 + 11.7531i 0.677435 + 0.677435i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.62287i 0.0929255i
\(306\) 0 0
\(307\) 17.8156 17.8156i 1.01679 1.01679i 0.0169305 0.999857i \(-0.494611\pi\)
0.999857 0.0169305i \(-0.00538941\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.3422i 0.926683i 0.886180 + 0.463342i \(0.153350\pi\)
−0.886180 + 0.463342i \(0.846650\pi\)
\(312\) 0 0
\(313\) 24.0707i 1.36056i −0.732954 0.680278i \(-0.761859\pi\)
0.732954 0.680278i \(-0.238141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.4498 13.4498i 0.755414 0.755414i −0.220070 0.975484i \(-0.570629\pi\)
0.975484 + 0.220070i \(0.0706287\pi\)
\(318\) 0 0
\(319\) 18.8815i 1.05716i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.8001 + 12.8001i 0.712218 + 0.712218i
\(324\) 0 0
\(325\) −1.44518 + 1.44518i −0.0801639 + 0.0801639i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 58.1148 3.20397
\(330\) 0 0
\(331\) −3.82091 3.82091i −0.210016 0.210016i 0.594258 0.804274i \(-0.297446\pi\)
−0.804274 + 0.594258i \(0.797446\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.31066 −0.454060
\(336\) 0 0
\(337\) −1.44837 −0.0788978 −0.0394489 0.999222i \(-0.512560\pi\)
−0.0394489 + 0.999222i \(0.512560\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.7245 18.7245i −1.01399 1.01399i
\(342\) 0 0
\(343\) −27.7803 −1.50000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.90479 + 2.90479i −0.155937 + 0.155937i −0.780764 0.624826i \(-0.785170\pi\)
0.624826 + 0.780764i \(0.285170\pi\)
\(348\) 0 0
\(349\) −22.0266 22.0266i −1.17906 1.17906i −0.979985 0.199072i \(-0.936207\pi\)
−0.199072 0.979985i \(-0.563793\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.17990i 0.382147i −0.981576 0.191074i \(-0.938803\pi\)
0.981576 0.191074i \(-0.0611969\pi\)
\(354\) 0 0
\(355\) −6.31555 + 6.31555i −0.335194 + 0.335194i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.4618i 1.23826i −0.785287 0.619132i \(-0.787484\pi\)
0.785287 0.619132i \(-0.212516\pi\)
\(360\) 0 0
\(361\) 11.6926i 0.615399i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.69662 + 9.69662i −0.507544 + 0.507544i
\(366\) 0 0
\(367\) 2.96970i 0.155017i 0.996992 + 0.0775086i \(0.0246965\pi\)
−0.996992 + 0.0775086i \(0.975303\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 27.1372 + 27.1372i 1.40889 + 1.40889i
\(372\) 0 0
\(373\) −5.38231 + 5.38231i −0.278686 + 0.278686i −0.832584 0.553899i \(-0.813140\pi\)
0.553899 + 0.832584i \(0.313140\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.24398 0.321582
\(378\) 0 0
\(379\) −8.79894 8.79894i −0.451971 0.451971i 0.444037 0.896008i \(-0.353546\pi\)
−0.896008 + 0.444037i \(0.853546\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.1823 −0.775780 −0.387890 0.921706i \(-0.626796\pi\)
−0.387890 + 0.921706i \(0.626796\pi\)
\(384\) 0 0
\(385\) 27.7657 1.41507
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.38360 5.38360i −0.272959 0.272959i 0.557331 0.830290i \(-0.311825\pi\)
−0.830290 + 0.557331i \(0.811825\pi\)
\(390\) 0 0
\(391\) −7.95364 −0.402233
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.90450 + 2.90450i −0.146141 + 0.146141i
\(396\) 0 0
\(397\) −4.98305 4.98305i −0.250092 0.250092i 0.570916 0.821008i \(-0.306588\pi\)
−0.821008 + 0.570916i \(0.806588\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.2573i 0.612099i −0.952016 0.306050i \(-0.900993\pi\)
0.952016 0.306050i \(-0.0990074\pi\)
\(402\) 0 0
\(403\) 6.19207 6.19207i 0.308449 0.308449i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.0670385i 0.00332297i
\(408\) 0 0
\(409\) 28.4064i 1.40461i −0.711877 0.702304i \(-0.752155\pi\)
0.711877 0.702304i \(-0.247845\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29.2508 29.2508i 1.43934 1.43934i
\(414\) 0 0
\(415\) 6.40286i 0.314304i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.3131 + 11.3131i 0.552681 + 0.552681i 0.927214 0.374533i \(-0.122197\pi\)
−0.374533 + 0.927214i \(0.622197\pi\)
\(420\) 0 0
\(421\) −1.71988 + 1.71988i −0.0838219 + 0.0838219i −0.747775 0.663953i \(-0.768878\pi\)
0.663953 + 0.747775i \(0.268878\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.69649 −0.324828
\(426\) 0 0
\(427\) −5.15548 5.15548i −0.249491 0.249491i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.03049 −0.194142 −0.0970709 0.995277i \(-0.530947\pi\)
−0.0970709 + 0.995277i \(0.530947\pi\)
\(432\) 0 0
\(433\) 6.01068 0.288855 0.144427 0.989515i \(-0.453866\pi\)
0.144427 + 0.989515i \(0.453866\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.27031 + 2.27031i 0.108604 + 0.108604i
\(438\) 0 0
\(439\) 13.1277 0.626549 0.313275 0.949663i \(-0.398574\pi\)
0.313275 + 0.949663i \(0.398574\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.57959 1.57959i 0.0750488 0.0750488i −0.668586 0.743635i \(-0.733100\pi\)
0.743635 + 0.668586i \(0.233100\pi\)
\(444\) 0 0
\(445\) 8.72478 + 8.72478i 0.413594 + 0.413594i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.4970i 1.06170i −0.847467 0.530849i \(-0.821873\pi\)
0.847467 0.530849i \(-0.178127\pi\)
\(450\) 0 0
\(451\) 21.5854 21.5854i 1.01642 1.01642i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.18194i 0.430456i
\(456\) 0 0
\(457\) 5.39811i 0.252513i −0.991998 0.126256i \(-0.959704\pi\)
0.991998 0.126256i \(-0.0402962\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.3939 21.3939i 0.996412 0.996412i −0.00358156 0.999994i \(-0.501140\pi\)
0.999994 + 0.00358156i \(0.00114005\pi\)
\(462\) 0 0
\(463\) 24.9382i 1.15898i 0.814980 + 0.579489i \(0.196748\pi\)
−0.814980 + 0.579489i \(0.803252\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.82512 3.82512i −0.177006 0.177006i 0.613044 0.790049i \(-0.289945\pi\)
−0.790049 + 0.613044i \(0.789945\pi\)
\(468\) 0 0
\(469\) −26.4009 + 26.4009i −1.21908 + 1.21908i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −22.8653 −1.05135
\(474\) 0 0
\(475\) 1.91147 + 1.91147i 0.0877042 + 0.0877042i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.0760 −0.460383 −0.230192 0.973145i \(-0.573935\pi\)
−0.230192 + 0.973145i \(0.573935\pi\)
\(480\) 0 0
\(481\) 0.0221692 0.00101083
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.93140 + 5.93140i 0.269331 + 0.269331i
\(486\) 0 0
\(487\) 11.1510 0.505302 0.252651 0.967557i \(-0.418698\pi\)
0.252651 + 0.967557i \(0.418698\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.7851 + 10.7851i −0.486725 + 0.486725i −0.907271 0.420546i \(-0.861839\pi\)
0.420546 + 0.907271i \(0.361839\pi\)
\(492\) 0 0
\(493\) 14.4663 + 14.4663i 0.651531 + 0.651531i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 40.1259i 1.79989i
\(498\) 0 0
\(499\) −10.6160 + 10.6160i −0.475238 + 0.475238i −0.903605 0.428367i \(-0.859089\pi\)
0.428367 + 0.903605i \(0.359089\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.3508i 0.818224i −0.912484 0.409112i \(-0.865839\pi\)
0.912484 0.409112i \(-0.134161\pi\)
\(504\) 0 0
\(505\) 4.11558i 0.183141i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.9467 + 15.9467i −0.706824 + 0.706824i −0.965866 0.259042i \(-0.916593\pi\)
0.259042 + 0.965866i \(0.416593\pi\)
\(510\) 0 0
\(511\) 61.6076i 2.72536i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.18502 8.18502i −0.360675 0.360675i
\(516\) 0 0
\(517\) −56.5306 + 56.5306i −2.48621 + 2.48621i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −22.6636 −0.992908 −0.496454 0.868063i \(-0.665365\pi\)
−0.496454 + 0.868063i \(0.665365\pi\)
\(522\) 0 0
\(523\) 1.12095 + 1.12095i 0.0490158 + 0.0490158i 0.731190 0.682174i \(-0.238965\pi\)
−0.682174 + 0.731190i \(0.738965\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.6921 1.24985
\(528\) 0 0
\(529\) 21.5893 0.938665
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.13815 + 7.13815i 0.309188 + 0.309188i
\(534\) 0 0
\(535\) 4.99981 0.216161
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 57.6140 57.6140i 2.48161 2.48161i
\(540\) 0 0
\(541\) 32.3559 + 32.3559i 1.39109 + 1.39109i 0.822894 + 0.568195i \(0.192358\pi\)
0.568195 + 0.822894i \(0.307642\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.96577i 0.384051i
\(546\) 0 0
\(547\) −19.5398 + 19.5398i −0.835463 + 0.835463i −0.988258 0.152795i \(-0.951172\pi\)
0.152795 + 0.988258i \(0.451172\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.25864i 0.351830i
\(552\) 0 0
\(553\) 18.4538i 0.784735i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.8255 + 25.8255i −1.09426 + 1.09426i −0.0991915 + 0.995068i \(0.531626\pi\)
−0.995068 + 0.0991915i \(0.968374\pi\)
\(558\) 0 0
\(559\) 7.56140i 0.319813i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.3693 + 10.3693i 0.437013 + 0.437013i 0.891006 0.453992i \(-0.150001\pi\)
−0.453992 + 0.891006i \(0.650001\pi\)
\(564\) 0 0
\(565\) −3.21032 + 3.21032i −0.135059 + 0.135059i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 44.4509 1.86348 0.931738 0.363130i \(-0.118292\pi\)
0.931738 + 0.363130i \(0.118292\pi\)
\(570\) 0 0
\(571\) 23.7391 + 23.7391i 0.993450 + 0.993450i 0.999979 0.00652855i \(-0.00207812\pi\)
−0.00652855 + 0.999979i \(0.502078\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.18773 −0.0495319
\(576\) 0 0
\(577\) −2.99852 −0.124830 −0.0624150 0.998050i \(-0.519880\pi\)
−0.0624150 + 0.998050i \(0.519880\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.3403 + 20.3403i 0.843858 + 0.843858i
\(582\) 0 0
\(583\) −52.7947 −2.18653
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.4168 + 20.4168i −0.842691 + 0.842691i −0.989208 0.146517i \(-0.953194\pi\)
0.146517 + 0.989208i \(0.453194\pi\)
\(588\) 0 0
\(589\) −8.18998 8.18998i −0.337462 0.337462i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.16808i 0.0890322i 0.999009 + 0.0445161i \(0.0141746\pi\)
−0.999009 + 0.0445161i \(0.985825\pi\)
\(594\) 0 0
\(595\) −21.2731 + 21.2731i −0.872113 + 0.872113i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.37935i 0.301512i 0.988571 + 0.150756i \(0.0481708\pi\)
−0.988571 + 0.150756i \(0.951829\pi\)
\(600\) 0 0
\(601\) 2.20199i 0.0898209i −0.998991 0.0449105i \(-0.985700\pi\)
0.998991 0.0449105i \(-0.0143003\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −19.2306 + 19.2306i −0.781837 + 0.781837i
\(606\) 0 0
\(607\) 1.10100i 0.0446880i 0.999750 + 0.0223440i \(0.00711291\pi\)
−0.999750 + 0.0223440i \(0.992887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.6943 18.6943i −0.756290 0.756290i
\(612\) 0 0
\(613\) 25.9748 25.9748i 1.04911 1.04911i 0.0503809 0.998730i \(-0.483956\pi\)
0.998730 0.0503809i \(-0.0160435\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.4011 −1.26416 −0.632081 0.774903i \(-0.717799\pi\)
−0.632081 + 0.774903i \(0.717799\pi\)
\(618\) 0 0
\(619\) 10.6007 + 10.6007i 0.426078 + 0.426078i 0.887290 0.461212i \(-0.152585\pi\)
−0.461212 + 0.887290i \(0.652585\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 55.4330 2.22088
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.0513625 + 0.0513625i 0.00204796 + 0.00204796i
\(630\) 0 0
\(631\) −13.5474 −0.539313 −0.269657 0.962957i \(-0.586910\pi\)
−0.269657 + 0.962957i \(0.586910\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.60879 + 4.60879i −0.182894 + 0.182894i
\(636\) 0 0
\(637\) 19.0526 + 19.0526i 0.754890 + 0.754890i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 40.4032i 1.59583i 0.602770 + 0.797915i \(0.294063\pi\)
−0.602770 + 0.797915i \(0.705937\pi\)
\(642\) 0 0
\(643\) 2.22995 2.22995i 0.0879407 0.0879407i −0.661768 0.749709i \(-0.730194\pi\)
0.749709 + 0.661768i \(0.230194\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.10482i 0.0827490i −0.999144 0.0413745i \(-0.986826\pi\)
0.999144 0.0413745i \(-0.0131737\pi\)
\(648\) 0 0
\(649\) 56.9069i 2.23379i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.4398 24.4398i 0.956402 0.956402i −0.0426867 0.999089i \(-0.513592\pi\)
0.999089 + 0.0426867i \(0.0135917\pi\)
\(654\) 0 0
\(655\) 3.92990i 0.153554i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.67541 5.67541i −0.221082 0.221082i 0.587872 0.808954i \(-0.299966\pi\)
−0.808954 + 0.587872i \(0.799966\pi\)
\(660\) 0 0
\(661\) −9.18787 + 9.18787i −0.357367 + 0.357367i −0.862841 0.505475i \(-0.831317\pi\)
0.505475 + 0.862841i \(0.331317\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.1445 0.470945
\(666\) 0 0
\(667\) 2.56584 + 2.56584i 0.0993498 + 0.0993498i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.0299 0.387199
\(672\) 0 0
\(673\) −5.85854 −0.225830 −0.112915 0.993605i \(-0.536019\pi\)
−0.112915 + 0.993605i \(0.536019\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.6489 + 29.6489i 1.13950 + 1.13950i 0.988540 + 0.150958i \(0.0482357\pi\)
0.150958 + 0.988540i \(0.451764\pi\)
\(678\) 0 0
\(679\) 37.6852 1.44623
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.4194 29.4194i 1.12570 1.12570i 0.134834 0.990868i \(-0.456950\pi\)
0.990868 0.134834i \(-0.0430501\pi\)
\(684\) 0 0
\(685\) −4.69290 4.69290i −0.179307 0.179307i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 17.4589i 0.665130i
\(690\) 0 0
\(691\) −20.6070 + 20.6070i −0.783926 + 0.783926i −0.980491 0.196565i \(-0.937021\pi\)
0.196565 + 0.980491i \(0.437021\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.97570i 0.378400i
\(696\) 0 0
\(697\) 33.0760i 1.25284i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.37293 + 3.37293i −0.127394 + 0.127394i −0.767929 0.640535i \(-0.778713\pi\)
0.640535 + 0.767929i \(0.278713\pi\)
\(702\) 0 0
\(703\) 0.0293221i 0.00110591i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0742 + 13.0742i 0.491706 + 0.491706i
\(708\) 0 0
\(709\) 34.1260 34.1260i 1.28163 1.28163i 0.341886 0.939741i \(-0.388934\pi\)
0.939741 0.341886i \(-0.111066\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.08902 0.190585
\(714\) 0 0
\(715\) −8.93163 8.93163i −0.334024 0.334024i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.4927 1.54742 0.773708 0.633542i \(-0.218400\pi\)
0.773708 + 0.633542i \(0.218400\pi\)
\(720\) 0 0
\(721\) −52.0036 −1.93672
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.16029 + 2.16029i 0.0802310 + 0.0802310i
\(726\) 0 0
\(727\) 15.1442 0.561669 0.280835 0.959756i \(-0.409389\pi\)
0.280835 + 0.959756i \(0.409389\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.5186 17.5186i 0.647949 0.647949i
\(732\) 0 0
\(733\) −1.53796 1.53796i −0.0568060 0.0568060i 0.678133 0.734939i \(-0.262789\pi\)
−0.734939 + 0.678133i \(0.762789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 51.3625i 1.89196i
\(738\) 0 0
\(739\) −35.3121 + 35.3121i −1.29898 + 1.29898i −0.369910 + 0.929068i \(0.620611\pi\)
−0.929068 + 0.369910i \(0.879389\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.6663i 1.41853i −0.704943 0.709264i \(-0.749028\pi\)
0.704943 0.709264i \(-0.250972\pi\)
\(744\) 0 0
\(745\) 13.4710i 0.493540i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 15.8832 15.8832i 0.580359 0.580359i
\(750\) 0 0
\(751\) 53.8013i 1.96324i −0.190853 0.981619i \(-0.561125\pi\)
0.190853 0.981619i \(-0.438875\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.6314 + 16.6314i 0.605277 + 0.605277i
\(756\) 0 0
\(757\) 10.7306 10.7306i 0.390010 0.390010i −0.484681 0.874691i \(-0.661064\pi\)
0.874691 + 0.484681i \(0.161064\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.3367 0.700954 0.350477 0.936571i \(-0.386019\pi\)
0.350477 + 0.936571i \(0.386019\pi\)
\(762\) 0 0
\(763\) 28.4821 + 28.4821i 1.03112 + 1.03112i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.8187 −0.679504
\(768\) 0 0
\(769\) 16.6094 0.598949 0.299474 0.954104i \(-0.403189\pi\)
0.299474 + 0.954104i \(0.403189\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.9353 + 27.9353i 1.00476 + 1.00476i 0.999989 + 0.00477424i \(0.00151969\pi\)
0.00477424 + 0.999989i \(0.498480\pi\)
\(774\) 0 0
\(775\) 4.28465 0.153909
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.44131 9.44131i 0.338270 0.338270i
\(780\) 0 0
\(781\) −39.0320 39.0320i −1.39668 1.39668i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.4126i 0.514407i
\(786\) 0 0
\(787\) 9.37684 9.37684i 0.334248 0.334248i −0.519949 0.854197i \(-0.674049\pi\)
0.854197 + 0.519949i \(0.174049\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.3968i 0.725227i
\(792\) 0 0
\(793\) 3.31681i 0.117783i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.3229 + 12.3229i −0.436498 + 0.436498i −0.890832 0.454334i \(-0.849877\pi\)
0.454334 + 0.890832i \(0.349877\pi\)
\(798\) 0 0
\(799\) 86.6235i 3.06452i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −59.9281 59.9281i −2.11482 2.11482i
\(804\) 0 0
\(805\) −3.77314 + 3.77314i −0.132986 + 0.132986i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.4755 0.684721 0.342360 0.939569i \(-0.388774\pi\)
0.342360 + 0.939569i \(0.388774\pi\)
\(810\) 0 0
\(811\) −5.32021 5.32021i −0.186818 0.186818i 0.607501 0.794319i \(-0.292172\pi\)
−0.794319 + 0.607501i \(0.792172\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −22.7244 −0.796002
\(816\) 0 0
\(817\) −10.0011 −0.349895
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.3478 21.3478i −0.745042 0.745042i 0.228502 0.973544i \(-0.426617\pi\)
−0.973544 + 0.228502i \(0.926617\pi\)
\(822\) 0 0
\(823\) −17.9419 −0.625414 −0.312707 0.949850i \(-0.601236\pi\)
−0.312707 + 0.949850i \(0.601236\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25.7459 + 25.7459i −0.895272 + 0.895272i −0.995013 0.0997415i \(-0.968198\pi\)
0.0997415 + 0.995013i \(0.468198\pi\)
\(828\) 0 0
\(829\) −17.3731 17.3731i −0.603394 0.603394i 0.337818 0.941211i \(-0.390311\pi\)
−0.941211 + 0.337818i \(0.890311\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 88.2836i 3.05885i
\(834\) 0 0
\(835\) 0.105742 0.105742i 0.00365934 0.00365934i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.5757i 0.675827i 0.941177 + 0.337913i \(0.109721\pi\)
−0.941177 + 0.337913i \(0.890279\pi\)
\(840\) 0 0
\(841\) 19.6663i 0.678149i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.23876 + 6.23876i −0.214620 + 0.214620i
\(846\) 0 0
\(847\) 122.182i 4.19823i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.00910998 + 0.00910998i 0.000312286 + 0.000312286i
\(852\) 0 0
\(853\) 6.26218 6.26218i 0.214413 0.214413i −0.591726 0.806139i \(-0.701553\pi\)
0.806139 + 0.591726i \(0.201553\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.9392 −1.39846 −0.699229 0.714898i \(-0.746473\pi\)
−0.699229 + 0.714898i \(0.746473\pi\)
\(858\) 0 0
\(859\) 24.9076 + 24.9076i 0.849837 + 0.849837i 0.990113 0.140275i \(-0.0447987\pi\)
−0.140275 + 0.990113i \(0.544799\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.89680 0.336891 0.168445 0.985711i \(-0.446125\pi\)
0.168445 + 0.985711i \(0.446125\pi\)
\(864\) 0 0
\(865\) 8.38827 0.285210
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −17.9507 17.9507i −0.608937 0.608937i
\(870\) 0 0
\(871\) 16.9852 0.575522
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.17676 + 3.17676i −0.107394 + 0.107394i
\(876\) 0 0
\(877\) 22.2824 + 22.2824i 0.752424 + 0.752424i 0.974931 0.222507i \(-0.0714240\pi\)
−0.222507 + 0.974931i \(0.571424\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47.9458i 1.61533i −0.589639 0.807667i \(-0.700730\pi\)
0.589639 0.807667i \(-0.299270\pi\)
\(882\) 0 0
\(883\) 23.2635 23.2635i 0.782880 0.782880i −0.197436 0.980316i \(-0.563261\pi\)
0.980316 + 0.197436i \(0.0632615\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.4020i 1.08795i 0.839100 + 0.543977i \(0.183082\pi\)
−0.839100 + 0.543977i \(0.816918\pi\)
\(888\) 0 0
\(889\) 29.2820i 0.982087i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −24.7261 + 24.7261i −0.827427 + 0.827427i
\(894\) 0 0
\(895\) 19.1065i 0.638660i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9.25608 9.25608i −0.308707 0.308707i
\(900\) 0 0
\(901\) 40.4495 40.4495i 1.34757 1.34757i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.2967 −0.541721
\(906\) 0 0
\(907\) −15.2567 15.2567i −0.506589 0.506589i 0.406888 0.913478i \(-0.366614\pi\)
−0.913478 + 0.406888i \(0.866614\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 45.6786 1.51340 0.756700 0.653762i \(-0.226810\pi\)
0.756700 + 0.653762i \(0.226810\pi\)
\(912\) 0 0
\(913\) −39.5716 −1.30963
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.4843 + 12.4843i 0.412269 + 0.412269i
\(918\) 0 0
\(919\) −48.1001 −1.58668 −0.793338 0.608782i \(-0.791658\pi\)
−0.793338 + 0.608782i \(0.791658\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.9076 12.9076i 0.424860 0.424860i
\(924\) 0 0
\(925\) 0.00767006 + 0.00767006i 0.000252190 + 0.000252190i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.8562i 0.356180i −0.984014 0.178090i \(-0.943008\pi\)
0.984014 0.178090i \(-0.0569918\pi\)
\(930\) 0 0
\(931\) 25.1999 25.1999i 0.825895 0.825895i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 41.3864i 1.35348i
\(936\) 0 0
\(937\) 52.5798i 1.71771i −0.512220 0.858854i \(-0.671177\pi\)
0.512220 0.858854i \(-0.328823\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 34.9046 34.9046i 1.13786 1.13786i 0.149023 0.988834i \(-0.452387\pi\)
0.988834 0.149023i \(-0.0476128\pi\)
\(942\) 0 0
\(943\) 5.86657i 0.191042i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.6778 + 15.6778i 0.509459 + 0.509459i 0.914360 0.404902i \(-0.132694\pi\)
−0.404902 + 0.914360i \(0.632694\pi\)
\(948\) 0 0
\(949\) 19.8178 19.8178i 0.643314 0.643314i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4.85833 −0.157377 −0.0786884 0.996899i \(-0.525073\pi\)
−0.0786884 + 0.996899i \(0.525073\pi\)
\(954\) 0 0
\(955\) −5.52806 5.52806i −0.178884 0.178884i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −29.8164 −0.962822
\(960\) 0 0
\(961\) 12.6418 0.407799
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16.6067 + 16.6067i 0.534588 + 0.534588i
\(966\) 0 0
\(967\) 24.0916 0.774733 0.387367 0.921926i \(-0.373385\pi\)
0.387367 + 0.921926i \(0.373385\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.3267 13.3267i 0.427673 0.427673i −0.460162 0.887835i \(-0.652209\pi\)
0.887835 + 0.460162i \(0.152209\pi\)
\(972\) 0 0
\(973\) 31.6904 + 31.6904i 1.01595 + 1.01595i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34.7266i 1.11100i 0.831516 + 0.555501i \(0.187474\pi\)
−0.831516 + 0.555501i \(0.812526\pi\)
\(978\) 0 0
\(979\) −53.9219 + 53.9219i −1.72335 + 1.72335i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 52.7711i 1.68314i 0.540152 + 0.841568i \(0.318367\pi\)
−0.540152 + 0.841568i \(0.681633\pi\)
\(984\) 0 0
\(985\) 17.5360i 0.558744i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.10721 3.10721i 0.0988035 0.0988035i
\(990\) 0 0
\(991\) 29.3280i 0.931634i −0.884881 0.465817i \(-0.845761\pi\)
0.884881 0.465817i \(-0.154239\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.00980 1.00980i −0.0320128 0.0320128i
\(996\) 0 0
\(997\) −17.1758 + 17.1758i −0.543964 + 0.543964i −0.924688 0.380725i \(-0.875675\pi\)
0.380725 + 0.924688i \(0.375675\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2880.2.bl.b.431.1 24
3.2 odd 2 inner 2880.2.bl.b.431.7 24
4.3 odd 2 720.2.bl.b.611.5 yes 24
12.11 even 2 720.2.bl.b.611.8 yes 24
16.5 even 4 720.2.bl.b.251.8 yes 24
16.11 odd 4 inner 2880.2.bl.b.1871.7 24
48.5 odd 4 720.2.bl.b.251.5 24
48.11 even 4 inner 2880.2.bl.b.1871.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.2.bl.b.251.5 24 48.5 odd 4
720.2.bl.b.251.8 yes 24 16.5 even 4
720.2.bl.b.611.5 yes 24 4.3 odd 2
720.2.bl.b.611.8 yes 24 12.11 even 2
2880.2.bl.b.431.1 24 1.1 even 1 trivial
2880.2.bl.b.431.7 24 3.2 odd 2 inner
2880.2.bl.b.1871.1 24 48.11 even 4 inner
2880.2.bl.b.1871.7 24 16.11 odd 4 inner