Properties

Label 2304.2.l.a.575.3
Level $2304$
Weight $2$
Character 2304.575
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
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] = [2304,2,Mod(575,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.l (of order \(4\), degree \(2\), 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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 575.3
Root \(-0.581861 + 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 2304.575
Dual form 2304.2.l.a.1727.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.16372 - 1.16372i) q^{5} +3.74166 q^{7} +O(q^{10})\) \(q+(1.16372 - 1.16372i) q^{5} +3.74166 q^{7} +(1.64575 + 1.64575i) q^{11} +(0.645751 - 0.645751i) q^{13} -6.57008i q^{17} +(1.41421 + 1.41421i) q^{19} -6.00000i q^{23} +2.29150i q^{25} +(5.40636 + 5.40636i) q^{29} -0.913230i q^{31} +(4.35425 - 4.35425i) q^{35} +(-1.00000 - 1.00000i) q^{37} -6.57008 q^{41} +(3.74166 - 3.74166i) q^{43} +6.00000 q^{47} +7.00000 q^{49} +(-7.73381 + 7.73381i) q^{53} +3.83039 q^{55} +(-9.29150 - 9.29150i) q^{59} +(-10.2915 + 10.2915i) q^{61} -1.50295i q^{65} +(10.8127 + 10.8127i) q^{67} -2.70850i q^{71} -12.5830i q^{73} +(6.15784 + 6.15784i) q^{77} +14.0534i q^{79} +(10.9373 - 10.9373i) q^{83} +(-7.64575 - 7.64575i) q^{85} +0.412247 q^{89} +(2.41618 - 2.41618i) q^{91} +3.29150 q^{95} +2.70850 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{11} - 16 q^{13} + 56 q^{35} - 8 q^{37} + 48 q^{47} + 56 q^{49} - 32 q^{59} - 40 q^{61} + 24 q^{83} - 40 q^{85} - 16 q^{95} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) 1.16372 1.16372i 0.520432 0.520432i −0.397270 0.917702i \(-0.630042\pi\)
0.917702 + 0.397270i \(0.130042\pi\)
\(6\) 0 0
\(7\) 3.74166 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.64575 + 1.64575i 0.496213 + 0.496213i 0.910257 0.414044i \(-0.135884\pi\)
−0.414044 + 0.910257i \(0.635884\pi\)
\(12\) 0 0
\(13\) 0.645751 0.645751i 0.179099 0.179099i −0.611864 0.790963i \(-0.709580\pi\)
0.790963 + 0.611864i \(0.209580\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.57008i 1.59348i −0.604323 0.796740i \(-0.706556\pi\)
0.604323 0.796740i \(-0.293444\pi\)
\(18\) 0 0
\(19\) 1.41421 + 1.41421i 0.324443 + 0.324443i 0.850469 0.526026i \(-0.176318\pi\)
−0.526026 + 0.850469i \(0.676318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 2.29150i 0.458301i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.40636 + 5.40636i 1.00394 + 1.00394i 0.999992 + 0.00394411i \(0.00125545\pi\)
0.00394411 + 0.999992i \(0.498745\pi\)
\(30\) 0 0
\(31\) 0.913230i 0.164021i −0.996631 0.0820105i \(-0.973866\pi\)
0.996631 0.0820105i \(-0.0261341\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.35425 4.35425i 0.736002 0.736002i
\(36\) 0 0
\(37\) −1.00000 1.00000i −0.164399 0.164399i 0.620113 0.784512i \(-0.287087\pi\)
−0.784512 + 0.620113i \(0.787087\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.57008 −1.02607 −0.513037 0.858366i \(-0.671480\pi\)
−0.513037 + 0.858366i \(0.671480\pi\)
\(42\) 0 0
\(43\) 3.74166 3.74166i 0.570597 0.570597i −0.361698 0.932295i \(-0.617803\pi\)
0.932295 + 0.361698i \(0.117803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.73381 + 7.73381i −1.06232 + 1.06232i −0.0643956 + 0.997924i \(0.520512\pi\)
−0.997924 + 0.0643956i \(0.979488\pi\)
\(54\) 0 0
\(55\) 3.83039 0.516490
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.29150 9.29150i −1.20965 1.20965i −0.971140 0.238511i \(-0.923341\pi\)
−0.238511 0.971140i \(-0.576659\pi\)
\(60\) 0 0
\(61\) −10.2915 + 10.2915i −1.31769 + 1.31769i −0.402093 + 0.915599i \(0.631717\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.50295i 0.186418i
\(66\) 0 0
\(67\) 10.8127 + 10.8127i 1.32098 + 1.32098i 0.912980 + 0.408005i \(0.133775\pi\)
0.408005 + 0.912980i \(0.366225\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.70850i 0.321440i −0.987000 0.160720i \(-0.948618\pi\)
0.987000 0.160720i \(-0.0513815\pi\)
\(72\) 0 0
\(73\) 12.5830i 1.47273i −0.676585 0.736365i \(-0.736541\pi\)
0.676585 0.736365i \(-0.263459\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.15784 + 6.15784i 0.701751 + 0.701751i
\(78\) 0 0
\(79\) 14.0534i 1.58113i 0.612378 + 0.790565i \(0.290213\pi\)
−0.612378 + 0.790565i \(0.709787\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.9373 10.9373i 1.20052 1.20052i 0.226511 0.974009i \(-0.427268\pi\)
0.974009 0.226511i \(-0.0727319\pi\)
\(84\) 0 0
\(85\) −7.64575 7.64575i −0.829298 0.829298i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.412247 0.0436981 0.0218490 0.999761i \(-0.493045\pi\)
0.0218490 + 0.999761i \(0.493045\pi\)
\(90\) 0 0
\(91\) 2.41618 2.41618i 0.253285 0.253285i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.29150 0.337701
\(96\) 0 0
\(97\) 2.70850 0.275006 0.137503 0.990501i \(-0.456092\pi\)
0.137503 + 0.990501i \(0.456092\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.07892 + 3.07892i −0.306364 + 0.306364i −0.843497 0.537133i \(-0.819507\pi\)
0.537133 + 0.843497i \(0.319507\pi\)
\(102\) 0 0
\(103\) −0.913230 −0.0899833 −0.0449916 0.998987i \(-0.514326\pi\)
−0.0449916 + 0.998987i \(0.514326\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.29150 3.29150i −0.318202 0.318202i 0.529874 0.848076i \(-0.322239\pi\)
−0.848076 + 0.529874i \(0.822239\pi\)
\(108\) 0 0
\(109\) −6.64575 + 6.64575i −0.636548 + 0.636548i −0.949702 0.313155i \(-0.898614\pi\)
0.313155 + 0.949702i \(0.398614\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) −6.98233 6.98233i −0.651106 0.651106i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.5830i 2.25352i
\(120\) 0 0
\(121\) 5.58301i 0.507546i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.48528 + 8.48528i 0.758947 + 0.758947i
\(126\) 0 0
\(127\) 9.39851i 0.833983i −0.908910 0.416992i \(-0.863084\pi\)
0.908910 0.416992i \(-0.136916\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 12.0000i 1.04844 1.04844i 0.0496797 0.998765i \(-0.484180\pi\)
0.998765 0.0496797i \(-0.0158200\pi\)
\(132\) 0 0
\(133\) 5.29150 + 5.29150i 0.458831 + 0.458831i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.91520 −0.163626 −0.0818132 0.996648i \(-0.526071\pi\)
−0.0818132 + 0.996648i \(0.526071\pi\)
\(138\) 0 0
\(139\) −10.8127 + 10.8127i −0.917123 + 0.917123i −0.996819 0.0796958i \(-0.974605\pi\)
0.0796958 + 0.996819i \(0.474605\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.12549 0.177743
\(144\) 0 0
\(145\) 12.5830 1.04496
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.49117 3.49117i 0.286007 0.286007i −0.549492 0.835499i \(-0.685179\pi\)
0.835499 + 0.549492i \(0.185179\pi\)
\(150\) 0 0
\(151\) 17.8838 1.45536 0.727681 0.685915i \(-0.240598\pi\)
0.727681 + 0.685915i \(0.240598\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.06275 1.06275i −0.0853618 0.0853618i
\(156\) 0 0
\(157\) 4.29150 4.29150i 0.342499 0.342499i −0.514807 0.857306i \(-0.672136\pi\)
0.857306 + 0.514807i \(0.172136\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 22.4499i 1.76930i
\(162\) 0 0
\(163\) 9.39851 + 9.39851i 0.736148 + 0.736148i 0.971830 0.235682i \(-0.0757324\pi\)
−0.235682 + 0.971830i \(0.575732\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.29150i 0.254704i −0.991858 0.127352i \(-0.959352\pi\)
0.991858 0.127352i \(-0.0406478\pi\)
\(168\) 0 0
\(169\) 12.1660i 0.935847i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.16372 1.16372i −0.0884761 0.0884761i 0.661484 0.749960i \(-0.269927\pi\)
−0.749960 + 0.661484i \(0.769927\pi\)
\(174\) 0 0
\(175\) 8.57402i 0.648135i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.8745 + 15.8745i −1.18652 + 1.18652i −0.208492 + 0.978024i \(0.566856\pi\)
−0.978024 + 0.208492i \(0.933144\pi\)
\(180\) 0 0
\(181\) 6.64575 + 6.64575i 0.493975 + 0.493975i 0.909556 0.415581i \(-0.136422\pi\)
−0.415581 + 0.909556i \(0.636422\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.32744 −0.171117
\(186\) 0 0
\(187\) 10.8127 10.8127i 0.790705 0.790705i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.70850 −0.630125 −0.315062 0.949071i \(-0.602025\pi\)
−0.315062 + 0.949071i \(0.602025\pi\)
\(192\) 0 0
\(193\) 16.5830 1.19367 0.596835 0.802364i \(-0.296425\pi\)
0.596835 + 0.802364i \(0.296425\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.751475 0.751475i 0.0535404 0.0535404i −0.679830 0.733370i \(-0.737946\pi\)
0.733370 + 0.679830i \(0.237946\pi\)
\(198\) 0 0
\(199\) 5.56812 0.394713 0.197357 0.980332i \(-0.436764\pi\)
0.197357 + 0.980332i \(0.436764\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.2288 + 20.2288i 1.41978 + 1.41978i
\(204\) 0 0
\(205\) −7.64575 + 7.64575i −0.534002 + 0.534002i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.65489i 0.321985i
\(210\) 0 0
\(211\) 6.98233 + 6.98233i 0.480684 + 0.480684i 0.905350 0.424666i \(-0.139609\pi\)
−0.424666 + 0.905350i \(0.639609\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.70850i 0.593915i
\(216\) 0 0
\(217\) 3.41699i 0.231961i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.24264 4.24264i −0.285391 0.285391i
\(222\) 0 0
\(223\) 7.57205i 0.507062i −0.967327 0.253531i \(-0.918408\pi\)
0.967327 0.253531i \(-0.0815920\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.64575 7.64575i 0.507466 0.507466i −0.406282 0.913748i \(-0.633175\pi\)
0.913748 + 0.406282i \(0.133175\pi\)
\(228\) 0 0
\(229\) −14.6458 14.6458i −0.967818 0.967818i 0.0316796 0.999498i \(-0.489914\pi\)
−0.999498 + 0.0316796i \(0.989914\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.412247 −0.0270072 −0.0135036 0.999909i \(-0.504298\pi\)
−0.0135036 + 0.999909i \(0.504298\pi\)
\(234\) 0 0
\(235\) 6.98233 6.98233i 0.455477 0.455477i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.8745 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(240\) 0 0
\(241\) −17.8745 −1.15140 −0.575699 0.817662i \(-0.695270\pi\)
−0.575699 + 0.817662i \(0.695270\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.14605 8.14605i 0.520432 0.520432i
\(246\) 0 0
\(247\) 1.82646 0.116215
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.6458 13.6458i −0.861312 0.861312i 0.130178 0.991491i \(-0.458445\pi\)
−0.991491 + 0.130178i \(0.958445\pi\)
\(252\) 0 0
\(253\) 9.87451 9.87451i 0.620805 0.620805i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.89753i 0.555013i 0.960724 + 0.277506i \(0.0895079\pi\)
−0.960724 + 0.277506i \(0.910492\pi\)
\(258\) 0 0
\(259\) −3.74166 3.74166i −0.232495 0.232495i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.5830i 1.88583i 0.333035 + 0.942914i \(0.391927\pi\)
−0.333035 + 0.942914i \(0.608073\pi\)
\(264\) 0 0
\(265\) 18.0000i 1.10573i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.7161 + 14.7161i 0.897259 + 0.897259i 0.995193 0.0979341i \(-0.0312235\pi\)
−0.0979341 + 0.995193i \(0.531223\pi\)
\(270\) 0 0
\(271\) 9.39851i 0.570919i −0.958391 0.285459i \(-0.907854\pi\)
0.958391 0.285459i \(-0.0921462\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.77124 + 3.77124i −0.227415 + 0.227415i
\(276\) 0 0
\(277\) 5.35425 + 5.35425i 0.321706 + 0.321706i 0.849421 0.527716i \(-0.176951\pi\)
−0.527716 + 0.849421i \(0.676951\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.2073 −1.08616 −0.543078 0.839682i \(-0.682741\pi\)
−0.543078 + 0.839682i \(0.682741\pi\)
\(282\) 0 0
\(283\) −3.83039 + 3.83039i −0.227693 + 0.227693i −0.811728 0.584035i \(-0.801473\pi\)
0.584035 + 0.811728i \(0.301473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.5830 −1.45109
\(288\) 0 0
\(289\) −26.1660 −1.53918
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −18.5465 + 18.5465i −1.08350 + 1.08350i −0.0873196 + 0.996180i \(0.527830\pi\)
−0.996180 + 0.0873196i \(0.972170\pi\)
\(294\) 0 0
\(295\) −21.6255 −1.25908
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.87451 3.87451i −0.224069 0.224069i
\(300\) 0 0
\(301\) 14.0000 14.0000i 0.806947 0.806947i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 23.9529i 1.37154i
\(306\) 0 0
\(307\) −14.6431 14.6431i −0.835727 0.835727i 0.152566 0.988293i \(-0.451246\pi\)
−0.988293 + 0.152566i \(0.951246\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.1660i 0.746576i 0.927716 + 0.373288i \(0.121770\pi\)
−0.927716 + 0.373288i \(0.878230\pi\)
\(312\) 0 0
\(313\) 15.1660i 0.857234i −0.903486 0.428617i \(-0.859001\pi\)
0.903486 0.428617i \(-0.140999\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.73381 7.73381i −0.434374 0.434374i 0.455739 0.890113i \(-0.349375\pi\)
−0.890113 + 0.455739i \(0.849375\pi\)
\(318\) 0 0
\(319\) 17.7951i 0.996332i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.29150 9.29150i 0.516993 0.516993i
\(324\) 0 0
\(325\) 1.47974 + 1.47974i 0.0820813 + 0.0820813i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 22.4499 1.23771
\(330\) 0 0
\(331\) −6.98233 + 6.98233i −0.383784 + 0.383784i −0.872463 0.488680i \(-0.837479\pi\)
0.488680 + 0.872463i \(0.337479\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 25.1660 1.37497
\(336\) 0 0
\(337\) −12.5830 −0.685440 −0.342720 0.939438i \(-0.611348\pi\)
−0.342720 + 0.939438i \(0.611348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.50295 1.50295i 0.0813893 0.0813893i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.64575 7.64575i −0.410445 0.410445i 0.471448 0.881894i \(-0.343731\pi\)
−0.881894 + 0.471448i \(0.843731\pi\)
\(348\) 0 0
\(349\) 0.416995 0.416995i 0.0223212 0.0223212i −0.695858 0.718179i \(-0.744976\pi\)
0.718179 + 0.695858i \(0.244976\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.07303i 0.429684i 0.976649 + 0.214842i \(0.0689237\pi\)
−0.976649 + 0.214842i \(0.931076\pi\)
\(354\) 0 0
\(355\) −3.15194 3.15194i −0.167287 0.167287i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.7085i 1.40962i 0.709396 + 0.704810i \(0.248968\pi\)
−0.709396 + 0.704810i \(0.751032\pi\)
\(360\) 0 0
\(361\) 15.0000i 0.789474i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.6431 14.6431i −0.766456 0.766456i
\(366\) 0 0
\(367\) 3.74166i 0.195313i −0.995220 0.0976565i \(-0.968865\pi\)
0.995220 0.0976565i \(-0.0311346\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −28.9373 + 28.9373i −1.50235 + 1.50235i
\(372\) 0 0
\(373\) −13.7085 13.7085i −0.709799 0.709799i 0.256694 0.966493i \(-0.417367\pi\)
−0.966493 + 0.256694i \(0.917367\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.98233 0.359608
\(378\) 0 0
\(379\) 20.2112 20.2112i 1.03818 1.03818i 0.0389399 0.999242i \(-0.487602\pi\)
0.999242 0.0389399i \(-0.0123981\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.4575 −1.45411 −0.727055 0.686579i \(-0.759112\pi\)
−0.727055 + 0.686579i \(0.759112\pi\)
\(384\) 0 0
\(385\) 14.3320 0.730427
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.8068 15.8068i 0.801439 0.801439i −0.181882 0.983320i \(-0.558219\pi\)
0.983320 + 0.181882i \(0.0582188\pi\)
\(390\) 0 0
\(391\) −39.4205 −1.99358
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.3542 + 16.3542i 0.822872 + 0.822872i
\(396\) 0 0
\(397\) −13.5830 + 13.5830i −0.681711 + 0.681711i −0.960386 0.278674i \(-0.910105\pi\)
0.278674 + 0.960386i \(0.410105\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.5406i 1.17556i 0.809019 + 0.587782i \(0.199999\pi\)
−0.809019 + 0.587782i \(0.800001\pi\)
\(402\) 0 0
\(403\) −0.589720 0.589720i −0.0293760 0.0293760i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.29150i 0.163154i
\(408\) 0 0
\(409\) 33.2915i 1.64616i 0.567926 + 0.823079i \(0.307746\pi\)
−0.567926 + 0.823079i \(0.692254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −34.7656 34.7656i −1.71070 1.71070i
\(414\) 0 0
\(415\) 25.4558i 1.24958i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.9373 + 10.9373i −0.534320 + 0.534320i −0.921855 0.387535i \(-0.873327\pi\)
0.387535 + 0.921855i \(0.373327\pi\)
\(420\) 0 0
\(421\) 9.93725 + 9.93725i 0.484312 + 0.484312i 0.906506 0.422194i \(-0.138740\pi\)
−0.422194 + 0.906506i \(0.638740\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.0554 0.730293
\(426\) 0 0
\(427\) −38.5073 + 38.5073i −1.86350 + 1.86350i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.5830 −0.606102 −0.303051 0.952974i \(-0.598005\pi\)
−0.303051 + 0.952974i \(0.598005\pi\)
\(432\) 0 0
\(433\) −5.41699 −0.260324 −0.130162 0.991493i \(-0.541550\pi\)
−0.130162 + 0.991493i \(0.541550\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.48528 8.48528i 0.405906 0.405906i
\(438\) 0 0
\(439\) 7.57205 0.361394 0.180697 0.983539i \(-0.442165\pi\)
0.180697 + 0.983539i \(0.442165\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.64575 + 7.64575i 0.363261 + 0.363261i 0.865012 0.501751i \(-0.167311\pi\)
−0.501751 + 0.865012i \(0.667311\pi\)
\(444\) 0 0
\(445\) 0.479741 0.479741i 0.0227419 0.0227419i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.0554i 0.710507i 0.934770 + 0.355253i \(0.115605\pi\)
−0.934770 + 0.355253i \(0.884395\pi\)
\(450\) 0 0
\(451\) −10.8127 10.8127i −0.509151 0.509151i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.62352i 0.263635i
\(456\) 0 0
\(457\) 22.4575i 1.05052i −0.850942 0.525259i \(-0.823968\pi\)
0.850942 0.525259i \(-0.176032\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −10.4735 10.4735i −0.487799 0.487799i 0.419812 0.907611i \(-0.362096\pi\)
−0.907611 + 0.419812i \(0.862096\pi\)
\(462\) 0 0
\(463\) 30.1995i 1.40349i 0.712429 + 0.701744i \(0.247595\pi\)
−0.712429 + 0.701744i \(0.752405\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.6458 + 19.6458i −0.909097 + 0.909097i −0.996199 0.0871024i \(-0.972239\pi\)
0.0871024 + 0.996199i \(0.472239\pi\)
\(468\) 0 0
\(469\) 40.4575 + 40.4575i 1.86815 + 1.86815i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.3157 0.566275
\(474\) 0 0
\(475\) −3.24067 + 3.24067i −0.148692 + 0.148692i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.7085 −0.672048 −0.336024 0.941853i \(-0.609082\pi\)
−0.336024 + 0.941853i \(0.609082\pi\)
\(480\) 0 0
\(481\) −1.29150 −0.0588875
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.15194 3.15194i 0.143122 0.143122i
\(486\) 0 0
\(487\) −12.2269 −0.554055 −0.277028 0.960862i \(-0.589349\pi\)
−0.277028 + 0.960862i \(0.589349\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.29150 + 9.29150i 0.419320 + 0.419320i 0.884969 0.465650i \(-0.154179\pi\)
−0.465650 + 0.884969i \(0.654179\pi\)
\(492\) 0 0
\(493\) 35.5203 35.5203i 1.59975 1.59975i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.1343i 0.454584i
\(498\) 0 0
\(499\) −21.6255 21.6255i −0.968088 0.968088i 0.0314182 0.999506i \(-0.489998\pi\)
−0.999506 + 0.0314182i \(0.989998\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.87451i 0.172756i −0.996262 0.0863779i \(-0.972471\pi\)
0.996262 0.0863779i \(-0.0275292\pi\)
\(504\) 0 0
\(505\) 7.16601i 0.318883i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.73381 + 7.73381i 0.342795 + 0.342795i 0.857417 0.514622i \(-0.172068\pi\)
−0.514622 + 0.857417i \(0.672068\pi\)
\(510\) 0 0
\(511\) 47.0813i 2.08275i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.06275 + 1.06275i −0.0468302 + 0.0468302i
\(516\) 0 0
\(517\) 9.87451 + 9.87451i 0.434280 + 0.434280i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.91520 −0.0839063 −0.0419531 0.999120i \(-0.513358\pi\)
−0.0419531 + 0.999120i \(0.513358\pi\)
\(522\) 0 0
\(523\) −16.3808 + 16.3808i −0.716284 + 0.716284i −0.967842 0.251558i \(-0.919057\pi\)
0.251558 + 0.967842i \(0.419057\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.24264 + 4.24264i −0.183769 + 0.183769i
\(534\) 0 0
\(535\) −7.66079 −0.331205
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.5203 + 11.5203i 0.496213 + 0.496213i
\(540\) 0 0
\(541\) 8.64575 8.64575i 0.371710 0.371710i −0.496390 0.868100i \(-0.665341\pi\)
0.868100 + 0.496390i \(0.165341\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.4676i 0.662560i
\(546\) 0 0
\(547\) −0.0887363 0.0887363i −0.00379409 0.00379409i 0.705207 0.709001i \(-0.250854\pi\)
−0.709001 + 0.705207i \(0.750854\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.2915i 0.651440i
\(552\) 0 0
\(553\) 52.5830i 2.23606i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.6196 26.6196i −1.12791 1.12791i −0.990517 0.137390i \(-0.956129\pi\)
−0.137390 0.990517i \(-0.543871\pi\)
\(558\) 0 0
\(559\) 4.83236i 0.204387i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3542 + 10.3542i −0.436380 + 0.436380i −0.890792 0.454412i \(-0.849849\pi\)
0.454412 + 0.890792i \(0.349849\pi\)
\(564\) 0 0
\(565\) −14.8118 14.8118i −0.623136 0.623136i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.0200 1.21658 0.608291 0.793714i \(-0.291855\pi\)
0.608291 + 0.793714i \(0.291855\pi\)
\(570\) 0 0
\(571\) −3.83039 + 3.83039i −0.160297 + 0.160297i −0.782698 0.622401i \(-0.786157\pi\)
0.622401 + 0.782698i \(0.286157\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.7490 0.573374
\(576\) 0 0
\(577\) −30.5830 −1.27319 −0.636594 0.771199i \(-0.719657\pi\)
−0.636594 + 0.771199i \(0.719657\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.9235 40.9235i 1.69779 1.69779i
\(582\) 0 0
\(583\) −25.4558 −1.05427
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.70850 + 2.70850i 0.111792 + 0.111792i 0.760790 0.648998i \(-0.224812\pi\)
−0.648998 + 0.760790i \(0.724812\pi\)
\(588\) 0 0
\(589\) 1.29150 1.29150i 0.0532154 0.0532154i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.24264i 0.174224i −0.996199 0.0871122i \(-0.972236\pi\)
0.996199 0.0871122i \(-0.0277639\pi\)
\(594\) 0 0
\(595\) −28.6078 28.6078i −1.17280 1.17280i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.1660i 0.783102i 0.920156 + 0.391551i \(0.128061\pi\)
−0.920156 + 0.391551i \(0.871939\pi\)
\(600\) 0 0
\(601\) 30.5830i 1.24751i −0.781621 0.623753i \(-0.785607\pi\)
0.781621 0.623753i \(-0.214393\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.49707 6.49707i −0.264143 0.264143i
\(606\) 0 0
\(607\) 38.5073i 1.56296i 0.623929 + 0.781481i \(0.285535\pi\)
−0.623929 + 0.781481i \(0.714465\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.87451 3.87451i 0.156746 0.156746i
\(612\) 0 0
\(613\) −13.5830 13.5830i −0.548612 0.548612i 0.377427 0.926039i \(-0.376809\pi\)
−0.926039 + 0.377427i \(0.876809\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.3475 −1.26200 −0.631001 0.775782i \(-0.717356\pi\)
−0.631001 + 0.775782i \(0.717356\pi\)
\(618\) 0 0
\(619\) 17.7951 17.7951i 0.715244 0.715244i −0.252384 0.967627i \(-0.581214\pi\)
0.967627 + 0.252384i \(0.0812145\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.54249 0.0617984
\(624\) 0 0
\(625\) 8.29150 0.331660
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.57008 + 6.57008i −0.261966 + 0.261966i
\(630\) 0 0
\(631\) 26.1916 1.04267 0.521336 0.853352i \(-0.325434\pi\)
0.521336 + 0.853352i \(0.325434\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −10.9373 10.9373i −0.434032 0.434032i
\(636\) 0 0
\(637\) 4.52026 4.52026i 0.179099 0.179099i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 32.0259i 1.26495i 0.774582 + 0.632474i \(0.217961\pi\)
−0.774582 + 0.632474i \(0.782039\pi\)
\(642\) 0 0
\(643\) 20.7122 + 20.7122i 0.816810 + 0.816810i 0.985644 0.168834i \(-0.0540003\pi\)
−0.168834 + 0.985644i \(0.554000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.3320i 1.74287i 0.490510 + 0.871436i \(0.336811\pi\)
−0.490510 + 0.871436i \(0.663189\pi\)
\(648\) 0 0
\(649\) 30.5830i 1.20049i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.16372 + 1.16372i 0.0455400 + 0.0455400i 0.729510 0.683970i \(-0.239748\pi\)
−0.683970 + 0.729510i \(0.739748\pi\)
\(654\) 0 0
\(655\) 27.9293i 1.09129i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19.1660 + 19.1660i −0.746602 + 0.746602i −0.973839 0.227238i \(-0.927031\pi\)
0.227238 + 0.973839i \(0.427031\pi\)
\(660\) 0 0
\(661\) −29.4575 29.4575i −1.14576 1.14576i −0.987377 0.158387i \(-0.949371\pi\)
−0.158387 0.987377i \(-0.550629\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.3157 0.477581
\(666\) 0 0
\(667\) 32.4382 32.4382i 1.25601 1.25601i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33.8745 −1.30771
\(672\) 0 0
\(673\) −19.7490 −0.761269 −0.380634 0.924726i \(-0.624294\pi\)
−0.380634 + 0.924726i \(0.624294\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −27.4441 + 27.4441i −1.05476 + 1.05476i −0.0563498 + 0.998411i \(0.517946\pi\)
−0.998411 + 0.0563498i \(0.982054\pi\)
\(678\) 0 0
\(679\) 10.1343 0.388918
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.3542 + 16.3542i 0.625778 + 0.625778i 0.947003 0.321225i \(-0.104095\pi\)
−0.321225 + 0.947003i \(0.604095\pi\)
\(684\) 0 0
\(685\) −2.22876 + 2.22876i −0.0851564 + 0.0851564i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.98823i 0.380521i
\(690\) 0 0
\(691\) 21.5367 + 21.5367i 0.819295 + 0.819295i 0.986006 0.166711i \(-0.0533146\pi\)
−0.166711 + 0.986006i \(0.553315\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.1660i 0.954601i
\(696\) 0 0
\(697\) 43.1660i 1.63503i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.6867 + 31.6867i 1.19679 + 1.19679i 0.975122 + 0.221668i \(0.0711501\pi\)
0.221668 + 0.975122i \(0.428850\pi\)
\(702\) 0 0
\(703\) 2.82843i 0.106676i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.5203 + 11.5203i −0.433264 + 0.433264i
\(708\) 0 0
\(709\) 23.3542 + 23.3542i 0.877087 + 0.877087i 0.993232 0.116145i \(-0.0370537\pi\)
−0.116145 + 0.993232i \(0.537054\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.47938 −0.205204
\(714\) 0 0
\(715\) 2.47348 2.47348i 0.0925030 0.0925030i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 34.4575 1.28505 0.642524 0.766265i \(-0.277887\pi\)
0.642524 + 0.766265i \(0.277887\pi\)
\(720\) 0 0
\(721\) −3.41699 −0.127256
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.3887 + 12.3887i −0.460105 + 0.460105i
\(726\) 0 0
\(727\) −6.39261 −0.237089 −0.118544 0.992949i \(-0.537823\pi\)
−0.118544 + 0.992949i \(0.537823\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.5830 24.5830i −0.909235 0.909235i
\(732\) 0 0
\(733\) 6.06275 6.06275i 0.223933 0.223933i −0.586220 0.810152i \(-0.699384\pi\)
0.810152 + 0.586220i \(0.199384\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 35.5901i 1.31098i
\(738\) 0 0
\(739\) −3.83039 3.83039i −0.140903 0.140903i 0.633137 0.774040i \(-0.281767\pi\)
−0.774040 + 0.633137i \(0.781767\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35.0405i 1.28551i 0.766071 + 0.642756i \(0.222209\pi\)
−0.766071 + 0.642756i \(0.777791\pi\)
\(744\) 0 0
\(745\) 8.12549i 0.297695i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.3157 12.3157i −0.450005 0.450005i
\(750\) 0 0
\(751\) 34.6769i 1.26538i 0.774406 + 0.632689i \(0.218049\pi\)
−0.774406 + 0.632689i \(0.781951\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.8118 20.8118i 0.757418 0.757418i
\(756\) 0 0
\(757\) 9.35425 + 9.35425i 0.339986 + 0.339986i 0.856362 0.516376i \(-0.172719\pi\)
−0.516376 + 0.856362i \(0.672719\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.7162 −0.823460 −0.411730 0.911306i \(-0.635075\pi\)
−0.411730 + 0.911306i \(0.635075\pi\)
\(762\) 0 0
\(763\) −24.8661 + 24.8661i −0.900214 + 0.900214i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −11.1660 −0.402657 −0.201328 0.979524i \(-0.564526\pi\)
−0.201328 + 0.979524i \(0.564526\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.73381 7.73381i 0.278166 0.278166i −0.554211 0.832376i \(-0.686980\pi\)
0.832376 + 0.554211i \(0.186980\pi\)
\(774\) 0 0
\(775\) 2.09267 0.0751709
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.29150 9.29150i −0.332903 0.332903i
\(780\) 0 0
\(781\) 4.45751 4.45751i 0.159502 0.159502i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.98823i 0.356495i
\(786\) 0 0
\(787\) −6.06910 6.06910i −0.216340 0.216340i 0.590614 0.806954i \(-0.298886\pi\)
−0.806954 + 0.590614i \(0.798886\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 47.6235i 1.69330i
\(792\) 0 0
\(793\) 13.2915i 0.471995i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.81861 + 5.81861i 0.206106 + 0.206106i 0.802610 0.596504i \(-0.203444\pi\)
−0.596504 + 0.802610i \(0.703444\pi\)
\(798\) 0 0
\(799\) 39.4205i 1.39460i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.7085 20.7085i 0.730787 0.730787i
\(804\) 0 0
\(805\) −26.1255 26.1255i −0.920803 0.920803i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0495 −0.423637 −0.211818 0.977309i \(-0.567939\pi\)
−0.211818 + 0.977309i \(0.567939\pi\)
\(810\) 0 0
\(811\) 24.0416 24.0416i 0.844216 0.844216i −0.145188 0.989404i \(-0.546379\pi\)
0.989404 + 0.145188i \(0.0463788\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.8745 0.766231
\(816\) 0 0
\(817\) 10.5830 0.370252
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.55830 + 8.55830i −0.298687 + 0.298687i −0.840499 0.541813i \(-0.817738\pi\)
0.541813 + 0.840499i \(0.317738\pi\)
\(822\) 0 0
\(823\) −44.1641 −1.53946 −0.769732 0.638367i \(-0.779610\pi\)
−0.769732 + 0.638367i \(0.779610\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.12549 2.12549i −0.0739106 0.0739106i 0.669185 0.743096i \(-0.266643\pi\)
−0.743096 + 0.669185i \(0.766643\pi\)
\(828\) 0 0
\(829\) −23.9373 + 23.9373i −0.831375 + 0.831375i −0.987705 0.156330i \(-0.950034\pi\)
0.156330 + 0.987705i \(0.450034\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 45.9906i 1.59348i
\(834\) 0 0
\(835\) −3.83039 3.83039i −0.132556 0.132556i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27.8745i 0.962335i −0.876629 0.481167i \(-0.840213\pi\)
0.876629 0.481167i \(-0.159787\pi\)
\(840\) 0 0
\(841\) 29.4575i 1.01578i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.1579 + 14.1579i 0.487045 + 0.487045i
\(846\) 0 0
\(847\) 20.8897i 0.717778i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.00000 + 6.00000i −0.205677 + 0.205677i
\(852\) 0 0
\(853\) −7.70850 7.70850i −0.263934 0.263934i 0.562716 0.826650i \(-0.309756\pi\)
−0.826650 + 0.562716i \(0.809756\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.74559 −0.196266 −0.0981328 0.995173i \(-0.531287\pi\)
−0.0981328 + 0.995173i \(0.531287\pi\)
\(858\) 0 0
\(859\) −18.7083 + 18.7083i −0.638319 + 0.638319i −0.950141 0.311822i \(-0.899061\pi\)
0.311822 + 0.950141i \(0.399061\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 47.0405 1.60128 0.800639 0.599147i \(-0.204494\pi\)
0.800639 + 0.599147i \(0.204494\pi\)
\(864\) 0 0
\(865\) −2.70850 −0.0920917
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −23.1284 + 23.1284i −0.784577 + 0.784577i
\(870\) 0 0
\(871\) 13.9647 0.473175
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 31.7490 + 31.7490i 1.07331 + 1.07331i
\(876\) 0 0
\(877\) 12.4170 12.4170i 0.419292 0.419292i −0.465668 0.884960i \(-0.654186\pi\)
0.884960 + 0.465668i \(0.154186\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.24854i 0.244210i −0.992517 0.122105i \(-0.961036\pi\)
0.992517 0.122105i \(-0.0389644\pi\)
\(882\) 0 0
\(883\) 12.2269 + 12.2269i 0.411469 + 0.411469i 0.882250 0.470781i \(-0.156028\pi\)
−0.470781 + 0.882250i \(0.656028\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.1660i 1.65083i −0.564524 0.825417i \(-0.690940\pi\)
0.564524 0.825417i \(-0.309060\pi\)
\(888\) 0 0
\(889\) 35.1660i 1.17943i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.48528 + 8.48528i 0.283949 + 0.283949i
\(894\) 0 0
\(895\) 36.9470i 1.23500i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.93725 4.93725i 0.164667 0.164667i
\(900\) 0 0
\(901\) 50.8118 + 50.8118i 1.69279 + 1.69279i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.4676 0.514161
\(906\) 0 0
\(907\) 21.7142 21.7142i 0.721008 0.721008i −0.247803 0.968811i \(-0.579708\pi\)
0.968811 + 0.247803i \(0.0797085\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −53.6235 −1.77663 −0.888313 0.459238i \(-0.848123\pi\)
−0.888313 + 0.459238i \(0.848123\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 44.8999 44.8999i 1.48272 1.48272i
\(918\) 0 0
\(919\) 2.09267 0.0690308 0.0345154 0.999404i \(-0.489011\pi\)
0.0345154 + 0.999404i \(0.489011\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.74902 1.74902i −0.0575696 0.0575696i
\(924\) 0 0
\(925\) 2.29150 2.29150i 0.0753441 0.0753441i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.1661i 1.48185i −0.671587 0.740926i \(-0.734387\pi\)
0.671587 0.740926i \(-0.265613\pi\)
\(930\) 0 0
\(931\) 9.89949 + 9.89949i 0.324443 + 0.324443i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 25.1660i 0.823017i
\(936\) 0 0
\(937\) 52.3320i 1.70961i −0.518947 0.854806i \(-0.673676\pi\)
0.518947 0.854806i \(-0.326324\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.9588 18.9588i −0.618039 0.618039i 0.326989 0.945028i \(-0.393966\pi\)
−0.945028 + 0.326989i \(0.893966\pi\)
\(942\) 0 0
\(943\) 39.4205i 1.28371i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.5830 12.5830i 0.408893 0.408893i −0.472460 0.881352i \(-0.656634\pi\)
0.881352 + 0.472460i \(0.156634\pi\)
\(948\) 0 0
\(949\) −8.12549 8.12549i −0.263765 0.263765i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.8858 0.611770 0.305885 0.952069i \(-0.401048\pi\)
0.305885 + 0.952069i \(0.401048\pi\)
\(954\) 0 0
\(955\) −10.1343 + 10.1343i −0.327937 + 0.327937i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.16601 −0.231403
\(960\) 0 0
\(961\) 30.1660 0.973097
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 19.2980 19.2980i 0.621225 0.621225i
\(966\) 0 0
\(967\) 56.3023 1.81056 0.905280 0.424814i \(-0.139661\pi\)
0.905280 + 0.424814i \(0.139661\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34.9373 34.9373i −1.12119 1.12119i −0.991563 0.129627i \(-0.958622\pi\)
−0.129627 0.991563i \(-0.541378\pi\)
\(972\) 0 0
\(973\) −40.4575 + 40.4575i −1.29701 + 1.29701i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.5347i 0.656965i −0.944510 0.328482i \(-0.893463\pi\)
0.944510 0.328482i \(-0.106537\pi\)
\(978\) 0 0
\(979\) 0.678456 + 0.678456i 0.0216835 + 0.0216835i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.58301i 0.209965i −0.994474 0.104983i \(-0.966521\pi\)
0.994474 0.104983i \(-0.0334787\pi\)
\(984\) 0 0
\(985\) 1.74902i 0.0557283i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.4499 22.4499i −0.713867 0.713867i
\(990\) 0 0
\(991\) 16.0573i 0.510078i −0.966931 0.255039i \(-0.917912\pi\)
0.966931 0.255039i \(-0.0820883\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.47974 6.47974i 0.205422 0.205422i
\(996\) 0 0
\(997\) −10.2915 10.2915i −0.325935 0.325935i 0.525103 0.851038i \(-0.324027\pi\)
−0.851038 + 0.525103i \(0.824027\pi\)
\(998\) 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 2304.2.l.a.575.3 yes 8
3.2 odd 2 2304.2.l.e.575.2 yes 8
4.3 odd 2 2304.2.l.e.575.3 yes 8
8.3 odd 2 2304.2.l.d.575.2 yes 8
8.5 even 2 2304.2.l.h.575.2 yes 8
12.11 even 2 inner 2304.2.l.a.575.2 8
16.3 odd 4 2304.2.l.e.1727.2 yes 8
16.5 even 4 2304.2.l.h.1727.3 yes 8
16.11 odd 4 2304.2.l.d.1727.3 yes 8
16.13 even 4 inner 2304.2.l.a.1727.2 yes 8
24.5 odd 2 2304.2.l.d.575.3 yes 8
24.11 even 2 2304.2.l.h.575.3 yes 8
48.5 odd 4 2304.2.l.d.1727.2 yes 8
48.11 even 4 2304.2.l.h.1727.2 yes 8
48.29 odd 4 2304.2.l.e.1727.3 yes 8
48.35 even 4 inner 2304.2.l.a.1727.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2304.2.l.a.575.2 8 12.11 even 2 inner
2304.2.l.a.575.3 yes 8 1.1 even 1 trivial
2304.2.l.a.1727.2 yes 8 16.13 even 4 inner
2304.2.l.a.1727.3 yes 8 48.35 even 4 inner
2304.2.l.d.575.2 yes 8 8.3 odd 2
2304.2.l.d.575.3 yes 8 24.5 odd 2
2304.2.l.d.1727.2 yes 8 48.5 odd 4
2304.2.l.d.1727.3 yes 8 16.11 odd 4
2304.2.l.e.575.2 yes 8 3.2 odd 2
2304.2.l.e.575.3 yes 8 4.3 odd 2
2304.2.l.e.1727.2 yes 8 16.3 odd 4
2304.2.l.e.1727.3 yes 8 48.29 odd 4
2304.2.l.h.575.2 yes 8 8.5 even 2
2304.2.l.h.575.3 yes 8 24.11 even 2
2304.2.l.h.1727.2 yes 8 48.11 even 4
2304.2.l.h.1727.3 yes 8 16.5 even 4