Properties

Label 504.2.bl.a.17.4
Level $504$
Weight $2$
Character 504.17
Analytic conductor $4.024$
Analytic rank $0$
Dimension $16$
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] = [504,2,Mod(17,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bl (of order \(6\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.4
Root \(0.144868 + 1.25092i\) of defining polynomial
Character \(\chi\) \(=\) 504.17
Dual form 504.2.bl.a.89.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.144868 - 0.250919i) q^{5} +(-2.26907 - 1.36064i) q^{7} +O(q^{10})\) \(q+(-0.144868 - 0.250919i) q^{5} +(-2.26907 - 1.36064i) q^{7} +(5.23077 + 3.01999i) q^{11} -5.46050i q^{13} +(2.22666 - 3.85669i) q^{17} +(3.51739 - 2.03076i) q^{19} +(-1.11743 + 0.645146i) q^{23} +(2.45803 - 4.25743i) q^{25} -0.377918i q^{29} +(-3.09749 - 1.78834i) q^{31} +(-0.0126942 + 0.766467i) q^{35} +(1.01555 + 1.75898i) q^{37} +5.50384 q^{41} -6.45419 q^{43} +(5.38833 + 9.33287i) q^{47} +(3.29733 + 6.17475i) q^{49} +(-9.77422 - 5.64315i) q^{53} -1.75000i q^{55} +(0.790140 - 1.36856i) q^{59} +(9.54984 - 5.51360i) q^{61} +(-1.37015 + 0.791054i) q^{65} +(-2.04381 + 3.53999i) q^{67} -0.410536i q^{71} +(-11.1149 - 6.41718i) q^{73} +(-7.75986 - 13.9697i) q^{77} +(6.01355 + 10.4158i) q^{79} -0.155917 q^{83} -1.29029 q^{85} +(3.34409 + 5.79213i) q^{89} +(-7.42976 + 12.3902i) q^{91} +(-1.01912 - 0.588387i) q^{95} +16.5090i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} + 12 q^{19} + 12 q^{25} + 24 q^{31} + 4 q^{37} + 8 q^{43} + 32 q^{49} - 28 q^{67} - 60 q^{73} - 32 q^{79} - 32 q^{85} - 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(-1\)

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
\(5\) −0.144868 0.250919i −0.0647871 0.112215i 0.831812 0.555057i \(-0.187304\pi\)
−0.896600 + 0.442842i \(0.853970\pi\)
\(6\) 0 0
\(7\) −2.26907 1.36064i −0.857627 0.514273i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.23077 + 3.01999i 1.57714 + 0.910560i 0.995256 + 0.0972858i \(0.0310161\pi\)
0.581880 + 0.813275i \(0.302317\pi\)
\(12\) 0 0
\(13\) 5.46050i 1.51447i −0.653142 0.757235i \(-0.726550\pi\)
0.653142 0.757235i \(-0.273450\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.22666 3.85669i 0.540045 0.935385i −0.458856 0.888511i \(-0.651741\pi\)
0.998901 0.0468746i \(-0.0149261\pi\)
\(18\) 0 0
\(19\) 3.51739 2.03076i 0.806944 0.465889i −0.0389497 0.999241i \(-0.512401\pi\)
0.845893 + 0.533352i \(0.179068\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.11743 + 0.645146i −0.232999 + 0.134522i −0.611955 0.790893i \(-0.709617\pi\)
0.378956 + 0.925415i \(0.376283\pi\)
\(24\) 0 0
\(25\) 2.45803 4.25743i 0.491605 0.851485i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.377918i 0.0701776i −0.999384 0.0350888i \(-0.988829\pi\)
0.999384 0.0350888i \(-0.0111714\pi\)
\(30\) 0 0
\(31\) −3.09749 1.78834i −0.556326 0.321195i 0.195343 0.980735i \(-0.437418\pi\)
−0.751670 + 0.659540i \(0.770751\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.0126942 + 0.766467i −0.00214570 + 0.129557i
\(36\) 0 0
\(37\) 1.01555 + 1.75898i 0.166955 + 0.289174i 0.937348 0.348395i \(-0.113273\pi\)
−0.770393 + 0.637569i \(0.779940\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.50384 0.859556 0.429778 0.902935i \(-0.358592\pi\)
0.429778 + 0.902935i \(0.358592\pi\)
\(42\) 0 0
\(43\) −6.45419 −0.984254 −0.492127 0.870523i \(-0.663780\pi\)
−0.492127 + 0.870523i \(0.663780\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.38833 + 9.33287i 0.785969 + 1.36134i 0.928418 + 0.371536i \(0.121169\pi\)
−0.142449 + 0.989802i \(0.545498\pi\)
\(48\) 0 0
\(49\) 3.29733 + 6.17475i 0.471048 + 0.882108i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.77422 5.64315i −1.34259 0.775146i −0.355405 0.934713i \(-0.615657\pi\)
−0.987187 + 0.159567i \(0.948990\pi\)
\(54\) 0 0
\(55\) 1.75000i 0.235970i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.790140 1.36856i 0.102867 0.178172i −0.809998 0.586433i \(-0.800532\pi\)
0.912865 + 0.408262i \(0.133865\pi\)
\(60\) 0 0
\(61\) 9.54984 5.51360i 1.22273 0.705945i 0.257233 0.966350i \(-0.417189\pi\)
0.965500 + 0.260405i \(0.0838560\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.37015 + 0.791054i −0.169946 + 0.0981182i
\(66\) 0 0
\(67\) −2.04381 + 3.53999i −0.249691 + 0.432478i −0.963440 0.267924i \(-0.913663\pi\)
0.713749 + 0.700402i \(0.246996\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.410536i 0.0487216i −0.999703 0.0243608i \(-0.992245\pi\)
0.999703 0.0243608i \(-0.00775505\pi\)
\(72\) 0 0
\(73\) −11.1149 6.41718i −1.30090 0.751074i −0.320340 0.947303i \(-0.603797\pi\)
−0.980558 + 0.196229i \(0.937131\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.75986 13.9697i −0.884319 1.59200i
\(78\) 0 0
\(79\) 6.01355 + 10.4158i 0.676577 + 1.17187i 0.976005 + 0.217747i \(0.0698707\pi\)
−0.299428 + 0.954119i \(0.596796\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.155917 −0.0171142 −0.00855708 0.999963i \(-0.502724\pi\)
−0.00855708 + 0.999963i \(0.502724\pi\)
\(84\) 0 0
\(85\) −1.29029 −0.139952
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.34409 + 5.79213i 0.354473 + 0.613965i 0.987028 0.160551i \(-0.0513271\pi\)
−0.632555 + 0.774516i \(0.717994\pi\)
\(90\) 0 0
\(91\) −7.42976 + 12.3902i −0.778851 + 1.29885i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.01912 0.588387i −0.104559 0.0603672i
\(96\) 0 0
\(97\) 16.5090i 1.67623i 0.545494 + 0.838115i \(0.316342\pi\)
−0.545494 + 0.838115i \(0.683658\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.99552 + 10.3845i −0.596577 + 1.03330i 0.396746 + 0.917929i \(0.370140\pi\)
−0.993322 + 0.115373i \(0.963194\pi\)
\(102\) 0 0
\(103\) −0.100344 + 0.0579336i −0.00988718 + 0.00570837i −0.504935 0.863157i \(-0.668484\pi\)
0.495048 + 0.868865i \(0.335150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.1501 + 8.16957i −1.36794 + 0.789782i −0.990665 0.136318i \(-0.956473\pi\)
−0.377278 + 0.926100i \(0.623140\pi\)
\(108\) 0 0
\(109\) 0.454348 0.786954i 0.0435187 0.0753766i −0.843446 0.537215i \(-0.819477\pi\)
0.886964 + 0.461838i \(0.152810\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.37176i 0.787549i −0.919207 0.393775i \(-0.871169\pi\)
0.919207 0.393775i \(-0.128831\pi\)
\(114\) 0 0
\(115\) 0.323760 + 0.186923i 0.0301907 + 0.0174306i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.3000 + 5.72142i −0.944200 + 0.524481i
\(120\) 0 0
\(121\) 12.7406 + 22.0674i 1.15824 + 2.00613i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.87305 −0.256973
\(126\) 0 0
\(127\) 6.13280 0.544198 0.272099 0.962269i \(-0.412282\pi\)
0.272099 + 0.962269i \(0.412282\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.36514 12.7568i −0.643495 1.11457i −0.984647 0.174558i \(-0.944150\pi\)
0.341152 0.940008i \(-0.389183\pi\)
\(132\) 0 0
\(133\) −10.7443 0.177947i −0.931651 0.0154299i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.7797 + 6.80101i 1.00641 + 0.581050i 0.910138 0.414306i \(-0.135976\pi\)
0.0962696 + 0.995355i \(0.469309\pi\)
\(138\) 0 0
\(139\) 17.3580i 1.47229i −0.676827 0.736143i \(-0.736645\pi\)
0.676827 0.736143i \(-0.263355\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.4906 28.5626i 1.37902 2.38853i
\(144\) 0 0
\(145\) −0.0948270 + 0.0547484i −0.00787495 + 0.00454661i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.3600 + 7.71339i −1.09449 + 0.631905i −0.934769 0.355257i \(-0.884393\pi\)
−0.159723 + 0.987162i \(0.551060\pi\)
\(150\) 0 0
\(151\) −2.68712 + 4.65423i −0.218675 + 0.378756i −0.954403 0.298521i \(-0.903507\pi\)
0.735728 + 0.677277i \(0.236840\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.03630i 0.0832373i
\(156\) 0 0
\(157\) −6.03109 3.48205i −0.481334 0.277898i 0.239638 0.970862i \(-0.422971\pi\)
−0.720972 + 0.692964i \(0.756304\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.41332 + 0.0565312i 0.269008 + 0.00445528i
\(162\) 0 0
\(163\) 5.00803 + 8.67416i 0.392259 + 0.679413i 0.992747 0.120221i \(-0.0383603\pi\)
−0.600488 + 0.799634i \(0.705027\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.03254 −0.466812 −0.233406 0.972379i \(-0.574987\pi\)
−0.233406 + 0.972379i \(0.574987\pi\)
\(168\) 0 0
\(169\) −16.8171 −1.29362
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.18258 + 15.9047i 0.698139 + 1.20921i 0.969111 + 0.246625i \(0.0793214\pi\)
−0.270972 + 0.962587i \(0.587345\pi\)
\(174\) 0 0
\(175\) −11.3702 + 6.31591i −0.859509 + 0.477438i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.83604 + 5.67884i 0.735180 + 0.424457i 0.820314 0.571913i \(-0.193799\pi\)
−0.0851340 + 0.996370i \(0.527132\pi\)
\(180\) 0 0
\(181\) 11.8647i 0.881893i −0.897533 0.440947i \(-0.854643\pi\)
0.897533 0.440947i \(-0.145357\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.294241 0.509641i 0.0216331 0.0374696i
\(186\) 0 0
\(187\) 23.2943 13.4490i 1.70345 0.983487i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.6310 + 6.71514i −0.841587 + 0.485890i −0.857803 0.513978i \(-0.828171\pi\)
0.0162166 + 0.999869i \(0.494838\pi\)
\(192\) 0 0
\(193\) 2.87792 4.98470i 0.207157 0.358807i −0.743661 0.668557i \(-0.766912\pi\)
0.950818 + 0.309751i \(0.100246\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.66053i 0.189555i 0.995498 + 0.0947775i \(0.0302140\pi\)
−0.995498 + 0.0947775i \(0.969786\pi\)
\(198\) 0 0
\(199\) 7.25552 + 4.18898i 0.514330 + 0.296949i 0.734612 0.678488i \(-0.237364\pi\)
−0.220282 + 0.975436i \(0.570698\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.514209 + 0.857521i −0.0360904 + 0.0601862i
\(204\) 0 0
\(205\) −0.797333 1.38102i −0.0556881 0.0964547i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.5315 1.69688
\(210\) 0 0
\(211\) 7.12240 0.490326 0.245163 0.969482i \(-0.421159\pi\)
0.245163 + 0.969482i \(0.421159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.935008 + 1.61948i 0.0637670 + 0.110448i
\(216\) 0 0
\(217\) 4.59514 + 8.27243i 0.311939 + 0.561569i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −21.0595 12.1587i −1.41661 0.817882i
\(222\) 0 0
\(223\) 13.6407i 0.913449i −0.889608 0.456725i \(-0.849023\pi\)
0.889608 0.456725i \(-0.150977\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.59561 9.69188i 0.371393 0.643272i −0.618387 0.785874i \(-0.712213\pi\)
0.989780 + 0.142602i \(0.0455468\pi\)
\(228\) 0 0
\(229\) 7.40517 4.27538i 0.489348 0.282525i −0.234956 0.972006i \(-0.575495\pi\)
0.724304 + 0.689481i \(0.242161\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.8863 6.28522i 0.713186 0.411758i −0.0990536 0.995082i \(-0.531582\pi\)
0.812240 + 0.583324i \(0.198248\pi\)
\(234\) 0 0
\(235\) 1.56120 2.70408i 0.101841 0.176394i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.3263i 1.37948i −0.724056 0.689741i \(-0.757725\pi\)
0.724056 0.689741i \(-0.242275\pi\)
\(240\) 0 0
\(241\) −17.7751 10.2625i −1.14499 0.661063i −0.197332 0.980337i \(-0.563228\pi\)
−0.947663 + 0.319274i \(0.896561\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.07169 1.72189i 0.0684676 0.110008i
\(246\) 0 0
\(247\) −11.0890 19.2067i −0.705575 1.22209i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −11.9577 −0.754764 −0.377382 0.926058i \(-0.623176\pi\)
−0.377382 + 0.926058i \(0.623176\pi\)
\(252\) 0 0
\(253\) −7.79333 −0.489963
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.8581 + 27.4670i 0.989199 + 1.71334i 0.621541 + 0.783381i \(0.286507\pi\)
0.367658 + 0.929961i \(0.380160\pi\)
\(258\) 0 0
\(259\) 0.0889877 5.37303i 0.00552943 0.333864i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.8678 + 12.0480i 1.28676 + 0.742912i 0.978075 0.208252i \(-0.0667775\pi\)
0.308686 + 0.951164i \(0.400111\pi\)
\(264\) 0 0
\(265\) 3.27006i 0.200878i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.31897 + 7.48068i −0.263332 + 0.456105i −0.967125 0.254300i \(-0.918155\pi\)
0.703793 + 0.710405i \(0.251488\pi\)
\(270\) 0 0
\(271\) 21.1099 12.1878i 1.28233 0.740356i 0.305060 0.952333i \(-0.401324\pi\)
0.977275 + 0.211977i \(0.0679902\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 25.7147 14.8464i 1.55066 0.895272i
\(276\) 0 0
\(277\) 12.9122 22.3646i 0.775820 1.34376i −0.158513 0.987357i \(-0.550670\pi\)
0.934333 0.356402i \(-0.115997\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.9846i 1.78873i 0.447334 + 0.894367i \(0.352374\pi\)
−0.447334 + 0.894367i \(0.647626\pi\)
\(282\) 0 0
\(283\) −19.8489 11.4597i −1.17989 0.681211i −0.223902 0.974612i \(-0.571880\pi\)
−0.955990 + 0.293401i \(0.905213\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.4886 7.48873i −0.737178 0.442046i
\(288\) 0 0
\(289\) −1.41605 2.45267i −0.0832972 0.144275i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.4782 1.66372 0.831858 0.554988i \(-0.187277\pi\)
0.831858 + 0.554988i \(0.187277\pi\)
\(294\) 0 0
\(295\) −0.457865 −0.0266579
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.52282 + 6.10171i 0.203730 + 0.352871i
\(300\) 0 0
\(301\) 14.6450 + 8.78181i 0.844123 + 0.506175i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.76694 1.59749i −0.158435 0.0914723i
\(306\) 0 0
\(307\) 5.73643i 0.327395i 0.986511 + 0.163698i \(0.0523422\pi\)
−0.986511 + 0.163698i \(0.947658\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.71985 + 8.17502i −0.267638 + 0.463563i −0.968251 0.249978i \(-0.919577\pi\)
0.700613 + 0.713541i \(0.252910\pi\)
\(312\) 0 0
\(313\) −10.0870 + 5.82373i −0.570150 + 0.329176i −0.757209 0.653172i \(-0.773438\pi\)
0.187059 + 0.982349i \(0.440104\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.90757 + 1.10133i −0.107140 + 0.0618571i −0.552612 0.833438i \(-0.686369\pi\)
0.445473 + 0.895296i \(0.353036\pi\)
\(318\) 0 0
\(319\) 1.14131 1.97680i 0.0639009 0.110680i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.0873i 1.00640i
\(324\) 0 0
\(325\) −23.2477 13.4221i −1.28955 0.744522i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.472155 28.5085i 0.0260307 1.57172i
\(330\) 0 0
\(331\) 12.5017 + 21.6535i 0.687154 + 1.19019i 0.972755 + 0.231837i \(0.0744737\pi\)
−0.285600 + 0.958349i \(0.592193\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.18434 0.0647072
\(336\) 0 0
\(337\) 11.8462 0.645303 0.322651 0.946518i \(-0.395426\pi\)
0.322651 + 0.946518i \(0.395426\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.8015 18.7088i −0.584935 1.01314i
\(342\) 0 0
\(343\) 0.919731 18.4974i 0.0496608 0.998766i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.2907 7.67341i −0.713484 0.411930i 0.0988655 0.995101i \(-0.468479\pi\)
−0.812350 + 0.583170i \(0.801812\pi\)
\(348\) 0 0
\(349\) 0.666155i 0.0356585i 0.999841 + 0.0178292i \(0.00567552\pi\)
−0.999841 + 0.0178292i \(0.994324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.18680 + 7.25175i −0.222841 + 0.385972i −0.955669 0.294442i \(-0.904866\pi\)
0.732829 + 0.680413i \(0.238200\pi\)
\(354\) 0 0
\(355\) −0.103011 + 0.0594736i −0.00546728 + 0.00315653i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.9438 + 14.4013i −1.31649 + 0.760073i −0.983161 0.182739i \(-0.941504\pi\)
−0.333324 + 0.942812i \(0.608170\pi\)
\(360\) 0 0
\(361\) −1.25200 + 2.16852i −0.0658946 + 0.114133i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.71859i 0.194640i
\(366\) 0 0
\(367\) 24.5554 + 14.1770i 1.28178 + 0.740036i 0.977173 0.212443i \(-0.0681419\pi\)
0.304606 + 0.952478i \(0.401475\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.5001 + 26.1038i 0.752807 + 1.35524i
\(372\) 0 0
\(373\) 3.87744 + 6.71593i 0.200766 + 0.347737i 0.948776 0.315951i \(-0.102323\pi\)
−0.748009 + 0.663688i \(0.768990\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.06362 −0.106282
\(378\) 0 0
\(379\) −17.3391 −0.890652 −0.445326 0.895368i \(-0.646912\pi\)
−0.445326 + 0.895368i \(0.646912\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.930503 + 1.61168i 0.0475465 + 0.0823530i 0.888819 0.458258i \(-0.151526\pi\)
−0.841273 + 0.540611i \(0.818193\pi\)
\(384\) 0 0
\(385\) −2.38112 + 3.97087i −0.121353 + 0.202375i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.34272 2.50727i −0.220184 0.127124i 0.385851 0.922561i \(-0.373908\pi\)
−0.606036 + 0.795437i \(0.707241\pi\)
\(390\) 0 0
\(391\) 5.74609i 0.290592i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.74235 3.01783i 0.0876670 0.151844i
\(396\) 0 0
\(397\) −20.0838 + 11.5954i −1.00798 + 0.581955i −0.910599 0.413292i \(-0.864379\pi\)
−0.0973778 + 0.995247i \(0.531046\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.51104 0.872401i 0.0754579 0.0435656i −0.461796 0.886986i \(-0.652795\pi\)
0.537254 + 0.843420i \(0.319462\pi\)
\(402\) 0 0
\(403\) −9.76523 + 16.9139i −0.486441 + 0.842540i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.2678i 0.608090i
\(408\) 0 0
\(409\) 27.7362 + 16.0135i 1.37147 + 0.791816i 0.991113 0.133026i \(-0.0424694\pi\)
0.380352 + 0.924842i \(0.375803\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.65500 + 2.03027i −0.179851 + 0.0999028i
\(414\) 0 0
\(415\) 0.0225875 + 0.0391227i 0.00110878 + 0.00192046i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.5506 −0.808549 −0.404274 0.914638i \(-0.632476\pi\)
−0.404274 + 0.914638i \(0.632476\pi\)
\(420\) 0 0
\(421\) 19.2673 0.939029 0.469514 0.882925i \(-0.344429\pi\)
0.469514 + 0.882925i \(0.344429\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −10.9464 18.9597i −0.530978 0.919681i
\(426\) 0 0
\(427\) −29.1713 0.483132i −1.41170 0.0233804i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.7026 + 17.7262i 1.47889 + 0.853840i 0.999715 0.0238778i \(-0.00760125\pi\)
0.479179 + 0.877717i \(0.340935\pi\)
\(432\) 0 0
\(433\) 10.7841i 0.518251i −0.965844 0.259125i \(-0.916566\pi\)
0.965844 0.259125i \(-0.0834343\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.62028 + 4.53846i −0.125345 + 0.217104i
\(438\) 0 0
\(439\) −35.4313 + 20.4563i −1.69104 + 0.976324i −0.737364 + 0.675495i \(0.763930\pi\)
−0.953678 + 0.300829i \(0.902737\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.84672 + 2.22090i −0.182763 + 0.105518i −0.588590 0.808432i \(-0.700317\pi\)
0.405827 + 0.913950i \(0.366983\pi\)
\(444\) 0 0
\(445\) 0.968906 1.67819i 0.0459305 0.0795540i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.3226i 0.959082i −0.877519 0.479541i \(-0.840803\pi\)
0.877519 0.479541i \(-0.159197\pi\)
\(450\) 0 0
\(451\) 28.7893 + 16.6215i 1.35564 + 0.782677i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.18529 + 0.0693165i 0.196209 + 0.00324961i
\(456\) 0 0
\(457\) −0.324749 0.562482i −0.0151911 0.0263118i 0.858330 0.513098i \(-0.171502\pi\)
−0.873521 + 0.486786i \(0.838169\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −35.7882 −1.66682 −0.833412 0.552652i \(-0.813616\pi\)
−0.833412 + 0.552652i \(0.813616\pi\)
\(462\) 0 0
\(463\) −16.8281 −0.782068 −0.391034 0.920376i \(-0.627883\pi\)
−0.391034 + 0.920376i \(0.627883\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.6807 20.2315i −0.540517 0.936203i −0.998874 0.0474348i \(-0.984895\pi\)
0.458357 0.888768i \(-0.348438\pi\)
\(468\) 0 0
\(469\) 9.45419 5.25158i 0.436554 0.242496i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.7604 19.4916i −1.55230 0.896223i
\(474\) 0 0
\(475\) 19.9667i 0.916134i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.9020 32.7393i 0.863657 1.49590i −0.00471828 0.999989i \(-0.501502\pi\)
0.868375 0.495908i \(-0.165165\pi\)
\(480\) 0 0
\(481\) 9.60491 5.54540i 0.437946 0.252848i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.14242 2.39163i 0.188098 0.108598i
\(486\) 0 0
\(487\) −15.1049 + 26.1624i −0.684466 + 1.18553i 0.289138 + 0.957287i \(0.406631\pi\)
−0.973604 + 0.228243i \(0.926702\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 38.8318i 1.75245i 0.481899 + 0.876227i \(0.339947\pi\)
−0.481899 + 0.876227i \(0.660053\pi\)
\(492\) 0 0
\(493\) −1.45751 0.841496i −0.0656431 0.0378991i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.558590 + 0.931533i −0.0250562 + 0.0417850i
\(498\) 0 0
\(499\) −3.52542 6.10620i −0.157819 0.273351i 0.776263 0.630410i \(-0.217113\pi\)
−0.934082 + 0.357058i \(0.883780\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.9561 −0.577683 −0.288842 0.957377i \(-0.593270\pi\)
−0.288842 + 0.957377i \(0.593270\pi\)
\(504\) 0 0
\(505\) 3.47425 0.154602
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.964368 1.67033i −0.0427449 0.0740363i 0.843861 0.536561i \(-0.180277\pi\)
−0.886606 + 0.462525i \(0.846944\pi\)
\(510\) 0 0
\(511\) 16.4890 + 29.6843i 0.729429 + 1.31316i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.0290733 + 0.0167855i 0.00128112 + 0.000739657i
\(516\) 0 0
\(517\) 65.0908i 2.86269i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.43874 + 16.3484i −0.413519 + 0.716235i −0.995272 0.0971301i \(-0.969034\pi\)
0.581753 + 0.813366i \(0.302367\pi\)
\(522\) 0 0
\(523\) −3.87627 + 2.23797i −0.169497 + 0.0978594i −0.582349 0.812939i \(-0.697866\pi\)
0.412851 + 0.910798i \(0.364533\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.7941 + 7.96406i −0.600883 + 0.346920i
\(528\) 0 0
\(529\) −10.6676 + 18.4768i −0.463808 + 0.803338i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.0537i 1.30177i
\(534\) 0 0
\(535\) 4.09981 + 2.36703i 0.177250 + 0.102335i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.40009 + 42.2566i −0.0603060 + 1.82012i
\(540\) 0 0
\(541\) −0.637629 1.10441i −0.0274138 0.0474821i 0.851993 0.523553i \(-0.175394\pi\)
−0.879407 + 0.476071i \(0.842061\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.263283 −0.0112778
\(546\) 0 0
\(547\) 30.8770 1.32020 0.660102 0.751176i \(-0.270513\pi\)
0.660102 + 0.751176i \(0.270513\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.767462 1.32928i −0.0326950 0.0566294i
\(552\) 0 0
\(553\) 0.526940 31.8163i 0.0224078 1.35297i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.0859 + 16.7927i 1.23241 + 0.711531i 0.967531 0.252751i \(-0.0813355\pi\)
0.264876 + 0.964282i \(0.414669\pi\)
\(558\) 0 0
\(559\) 35.2431i 1.49062i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.4046 + 23.2175i −0.564937 + 0.978499i 0.432119 + 0.901817i \(0.357766\pi\)
−0.997056 + 0.0766823i \(0.975567\pi\)
\(564\) 0 0
\(565\) −2.10064 + 1.21280i −0.0883745 + 0.0510231i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.7065 7.91345i 0.574606 0.331749i −0.184381 0.982855i \(-0.559028\pi\)
0.758987 + 0.651106i \(0.225695\pi\)
\(570\) 0 0
\(571\) −3.77658 + 6.54124i −0.158045 + 0.273742i −0.934164 0.356845i \(-0.883852\pi\)
0.776119 + 0.630587i \(0.217186\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.34315i 0.264527i
\(576\) 0 0
\(577\) 11.7609 + 6.79013i 0.489611 + 0.282677i 0.724413 0.689366i \(-0.242111\pi\)
−0.234802 + 0.972043i \(0.575444\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.353787 + 0.212147i 0.0146776 + 0.00880134i
\(582\) 0 0
\(583\) −34.0845 59.0360i −1.41163 2.44502i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.7243 −1.06176 −0.530878 0.847449i \(-0.678138\pi\)
−0.530878 + 0.847449i \(0.678138\pi\)
\(588\) 0 0
\(589\) −14.5268 −0.598565
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.59806 11.4282i −0.270950 0.469299i 0.698156 0.715946i \(-0.254004\pi\)
−0.969105 + 0.246647i \(0.920671\pi\)
\(594\) 0 0
\(595\) 2.92776 + 1.75562i 0.120026 + 0.0719734i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.8018 8.54581i −0.604784 0.349172i 0.166137 0.986103i \(-0.446871\pi\)
−0.770921 + 0.636930i \(0.780204\pi\)
\(600\) 0 0
\(601\) 10.7137i 0.437019i −0.975835 0.218510i \(-0.929881\pi\)
0.975835 0.218510i \(-0.0701195\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.69143 6.39375i 0.150078 0.259943i
\(606\) 0 0
\(607\) −14.0938 + 8.13707i −0.572050 + 0.330273i −0.757968 0.652292i \(-0.773808\pi\)
0.185917 + 0.982565i \(0.440474\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.9621 29.4230i 2.06171 1.19033i
\(612\) 0 0
\(613\) 7.76754 13.4538i 0.313728 0.543393i −0.665438 0.746453i \(-0.731755\pi\)
0.979166 + 0.203060i \(0.0650886\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.18413i 0.128188i −0.997944 0.0640941i \(-0.979584\pi\)
0.997944 0.0640941i \(-0.0204158\pi\)
\(618\) 0 0
\(619\) −23.3325 13.4710i −0.937811 0.541446i −0.0485379 0.998821i \(-0.515456\pi\)
−0.889274 + 0.457376i \(0.848789\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.293027 17.6928i 0.0117399 0.708848i
\(624\) 0 0
\(625\) −11.8739 20.5662i −0.474957 0.822649i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.04512 0.360653
\(630\) 0 0
\(631\) 19.7789 0.787387 0.393693 0.919242i \(-0.371197\pi\)
0.393693 + 0.919242i \(0.371197\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.888449 1.53884i −0.0352570 0.0610670i
\(636\) 0 0
\(637\) 33.7173 18.0051i 1.33593 0.713388i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −39.5884 22.8564i −1.56365 0.902772i −0.996883 0.0788925i \(-0.974862\pi\)
−0.566764 0.823880i \(-0.691805\pi\)
\(642\) 0 0
\(643\) 11.1743i 0.440672i −0.975424 0.220336i \(-0.929285\pi\)
0.975424 0.220336i \(-0.0707155\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.35600 + 14.4730i −0.328508 + 0.568993i −0.982216 0.187754i \(-0.939879\pi\)
0.653708 + 0.756747i \(0.273213\pi\)
\(648\) 0 0
\(649\) 8.26608 4.77242i 0.324472 0.187334i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.0979 21.4185i 1.45175 0.838170i 0.453173 0.891423i \(-0.350292\pi\)
0.998581 + 0.0532524i \(0.0169588\pi\)
\(654\) 0 0
\(655\) −2.13395 + 3.69611i −0.0833804 + 0.144419i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.00672i 0.117125i −0.998284 0.0585627i \(-0.981348\pi\)
0.998284 0.0585627i \(-0.0186518\pi\)
\(660\) 0 0
\(661\) 13.0503 + 7.53461i 0.507599 + 0.293062i 0.731846 0.681470i \(-0.238659\pi\)
−0.224247 + 0.974532i \(0.571992\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.51186 + 2.72174i 0.0586275 + 0.105544i
\(666\) 0 0
\(667\) 0.243812 + 0.422295i 0.00944045 + 0.0163513i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 66.6040 2.57122
\(672\) 0 0
\(673\) −6.01408 −0.231826 −0.115913 0.993259i \(-0.536979\pi\)
−0.115913 + 0.993259i \(0.536979\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.93572 + 6.81687i 0.151262 + 0.261994i 0.931692 0.363250i \(-0.118333\pi\)
−0.780430 + 0.625244i \(0.785000\pi\)
\(678\) 0 0
\(679\) 22.4627 37.4599i 0.862039 1.43758i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13.4955 + 7.79165i 0.516393 + 0.298139i 0.735457 0.677571i \(-0.236967\pi\)
−0.219065 + 0.975710i \(0.570301\pi\)
\(684\) 0 0
\(685\) 3.94101i 0.150578i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −30.8144 + 53.3721i −1.17394 + 2.03332i
\(690\) 0 0
\(691\) −24.4535 + 14.1183i −0.930256 + 0.537084i −0.886893 0.461976i \(-0.847141\pi\)
−0.0433635 + 0.999059i \(0.513807\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.35546 + 2.51462i −0.165212 + 0.0953851i
\(696\) 0 0
\(697\) 12.2552 21.2266i 0.464199 0.804016i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.7937i 0.634288i 0.948377 + 0.317144i \(0.102724\pi\)
−0.948377 + 0.317144i \(0.897276\pi\)
\(702\) 0 0
\(703\) 7.14414 + 4.12467i 0.269446 + 0.155565i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.7338 15.4055i 1.04304 0.579384i
\(708\) 0 0
\(709\) −3.57291 6.18846i −0.134183 0.232412i 0.791102 0.611684i \(-0.209508\pi\)
−0.925285 + 0.379272i \(0.876174\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.61496 0.172832
\(714\) 0 0
\(715\) −9.55589 −0.357370
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.03223 + 12.1802i 0.262258 + 0.454244i 0.966842 0.255377i \(-0.0821995\pi\)
−0.704584 + 0.709621i \(0.748866\pi\)
\(720\) 0 0
\(721\) 0.306514 + 0.00507646i 0.0114152 + 0.000189057i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.60896 0.928932i −0.0597552 0.0344997i
\(726\) 0 0
\(727\) 12.0721i 0.447728i −0.974620 0.223864i \(-0.928133\pi\)
0.974620 0.223864i \(-0.0718672\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −14.3713 + 24.8918i −0.531542 + 0.920657i
\(732\) 0 0
\(733\) −21.2754 + 12.2834i −0.785827 + 0.453697i −0.838491 0.544915i \(-0.816562\pi\)
0.0526646 + 0.998612i \(0.483229\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.3814 + 12.3446i −0.787595 + 0.454718i
\(738\) 0 0
\(739\) 19.0963 33.0758i 0.702470 1.21671i −0.265126 0.964214i \(-0.585414\pi\)
0.967597 0.252501i \(-0.0812530\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.94867i 0.364981i 0.983208 + 0.182491i \(0.0584159\pi\)
−0.983208 + 0.182491i \(0.941584\pi\)
\(744\) 0 0
\(745\) 3.87088 + 2.23485i 0.141818 + 0.0818786i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 43.2234 + 0.715862i 1.57935 + 0.0261570i
\(750\) 0 0
\(751\) −1.96469 3.40295i −0.0716927 0.124175i 0.827951 0.560801i \(-0.189507\pi\)
−0.899643 + 0.436626i \(0.856173\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.55712 0.0566692
\(756\) 0 0
\(757\) −10.7492 −0.390686 −0.195343 0.980735i \(-0.562582\pi\)
−0.195343 + 0.980735i \(0.562582\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.24304 3.88506i −0.0813102 0.140833i 0.822503 0.568761i \(-0.192577\pi\)
−0.903813 + 0.427928i \(0.859244\pi\)
\(762\) 0 0
\(763\) −2.10171 + 1.16745i −0.0760869 + 0.0422645i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.47303 4.31456i −0.269836 0.155790i
\(768\) 0 0
\(769\) 9.74739i 0.351500i −0.984435 0.175750i \(-0.943765\pi\)
0.984435 0.175750i \(-0.0562350\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.4609 31.9752i 0.663992 1.15007i −0.315565 0.948904i \(-0.602194\pi\)
0.979557 0.201165i \(-0.0644726\pi\)
\(774\) 0 0
\(775\) −15.2274 + 8.79157i −0.546986 + 0.315803i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.3591 11.1770i 0.693613 0.400458i
\(780\) 0 0
\(781\) 1.23981 2.14742i 0.0443640 0.0768406i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.01776i 0.0720169i
\(786\) 0 0
\(787\) 16.7796 + 9.68771i 0.598128 + 0.345330i 0.768305 0.640084i \(-0.221100\pi\)
−0.170177 + 0.985414i \(0.554434\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11.3909 + 18.9961i −0.405015 + 0.675423i
\(792\) 0 0
\(793\) −30.1070 52.1469i −1.06913 1.85179i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.8237 0.843880 0.421940 0.906624i \(-0.361349\pi\)
0.421940 + 0.906624i \(0.361349\pi\)
\(798\) 0 0
\(799\) 47.9920 1.69783
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −38.7596 67.1336i −1.36780 2.36909i
\(804\) 0 0
\(805\) −0.480298 0.864659i −0.0169283 0.0304752i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.6571 + 10.7717i 0.655948 + 0.378712i 0.790731 0.612163i \(-0.209701\pi\)
−0.134783 + 0.990875i \(0.543034\pi\)
\(810\) 0 0
\(811\) 6.36360i 0.223456i 0.993739 + 0.111728i \(0.0356386\pi\)
−0.993739 + 0.111728i \(0.964361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.45101 2.51322i 0.0508267 0.0880344i
\(816\) 0 0
\(817\) −22.7019 + 13.1069i −0.794238 + 0.458553i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0269 5.78905i 0.349942 0.202039i −0.314717 0.949185i \(-0.601910\pi\)
0.664660 + 0.747146i \(0.268576\pi\)
\(822\) 0 0
\(823\) 7.91237 13.7046i 0.275808 0.477714i −0.694531 0.719463i \(-0.744388\pi\)
0.970339 + 0.241750i \(0.0777213\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.6537i 0.440011i −0.975499 0.220006i \(-0.929392\pi\)
0.975499 0.220006i \(-0.0706076\pi\)
\(828\) 0 0
\(829\) 7.47709 + 4.31690i 0.259690 + 0.149932i 0.624193 0.781270i \(-0.285428\pi\)
−0.364503 + 0.931202i \(0.618761\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 31.1562 + 1.03230i 1.07950 + 0.0357669i
\(834\) 0 0
\(835\) 0.873924 + 1.51368i 0.0302434 + 0.0523831i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −33.7820 −1.16628 −0.583142 0.812371i \(-0.698177\pi\)
−0.583142 + 0.812371i \(0.698177\pi\)
\(840\) 0 0
\(841\) 28.8572 0.995075
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.43626 + 4.21973i 0.0838100 + 0.145163i
\(846\) 0 0
\(847\) 1.11640 67.4079i 0.0383601 2.31616i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.26960 1.31035i −0.0778008 0.0449183i
\(852\) 0 0
\(853\) 1.23446i 0.0422671i 0.999777 + 0.0211336i \(0.00672752\pi\)
−0.999777 + 0.0211336i \(0.993272\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.5237 30.3519i 0.598597 1.03680i −0.394431 0.918925i \(-0.629058\pi\)
0.993028 0.117875i \(-0.0376082\pi\)
\(858\) 0 0
\(859\) 18.5187 10.6918i 0.631852 0.364800i −0.149617 0.988744i \(-0.547804\pi\)
0.781469 + 0.623944i \(0.214471\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.61471 2.66430i 0.157087 0.0906940i −0.419396 0.907803i \(-0.637758\pi\)
0.576483 + 0.817109i \(0.304425\pi\)
\(864\) 0 0
\(865\) 2.66053 4.60818i 0.0904608 0.156683i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 72.6433i 2.46426i
\(870\) 0 0
\(871\) 19.3301 + 11.1602i 0.654976 + 0.378150i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.51913 + 3.90917i 0.220387 + 0.132154i
\(876\) 0 0
\(877\) −1.76822 3.06264i −0.0597085 0.103418i 0.834626 0.550817i \(-0.185684\pi\)
−0.894335 + 0.447399i \(0.852350\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −49.3358 −1.66217 −0.831083 0.556148i \(-0.812279\pi\)
−0.831083 + 0.556148i \(0.812279\pi\)
\(882\) 0 0
\(883\) −33.9893 −1.14383 −0.571915 0.820313i \(-0.693799\pi\)
−0.571915 + 0.820313i \(0.693799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.8153 + 32.5891i 0.631757 + 1.09423i 0.987192 + 0.159534i \(0.0509991\pi\)
−0.355436 + 0.934701i \(0.615668\pi\)
\(888\) 0 0
\(889\) −13.9157 8.34452i −0.466719 0.279866i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 37.9057 + 21.8849i 1.26847 + 0.732349i
\(894\) 0 0
\(895\) 3.29074i 0.109997i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.675845 + 1.17060i −0.0225407 + 0.0390417i
\(900\) 0 0
\(901\) −43.5278 + 25.1308i −1.45012 + 0.837227i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.97707 + 1.71881i −0.0989613 + 0.0571353i
\(906\) 0 0
\(907\) 5.75787 9.97293i 0.191187 0.331146i −0.754457 0.656350i \(-0.772100\pi\)
0.945644 + 0.325204i \(0.105433\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.3469i 1.70120i −0.525815 0.850599i \(-0.676240\pi\)
0.525815 0.850599i \(-0.323760\pi\)
\(912\) 0 0
\(913\) −0.815568 0.470869i −0.0269914 0.0155835i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.645373 + 38.9673i −0.0213121 + 1.28681i
\(918\) 0 0
\(919\) −1.49832 2.59517i −0.0494251 0.0856068i 0.840254 0.542192i \(-0.182406\pi\)
−0.889680 + 0.456586i \(0.849072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.24173 −0.0737874
\(924\) 0 0
\(925\) 9.98497 0.328304
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18.7345 32.4491i −0.614659 1.06462i −0.990444 0.137914i \(-0.955960\pi\)
0.375785 0.926707i \(-0.377373\pi\)
\(930\) 0 0
\(931\) 24.1375 + 15.0229i 0.791073 + 0.492355i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.74922 3.89667i −0.220723 0.127435i
\(936\) 0 0
\(937\) 4.97716i 0.162597i 0.996690 + 0.0812983i \(0.0259067\pi\)
−0.996690 + 0.0812983i \(0.974093\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.0429 + 29.5192i −0.555584 + 0.962299i 0.442274 + 0.896880i \(0.354172\pi\)
−0.997858 + 0.0654193i \(0.979162\pi\)
\(942\) 0 0
\(943\) −6.15014 + 3.55078i −0.200276 + 0.115629i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.1735 15.1113i 0.850524 0.491050i −0.0103036 0.999947i \(-0.503280\pi\)
0.860828 + 0.508897i \(0.169946\pi\)
\(948\) 0 0
\(949\) −35.0410 + 60.6928i −1.13748 + 1.97017i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20.5265i 0.664917i 0.943118 + 0.332459i \(0.107878\pi\)
−0.943118 + 0.332459i \(0.892122\pi\)
\(954\) 0 0
\(955\) 3.36992 + 1.94562i 0.109048 + 0.0629589i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −17.4752 31.4598i −0.564304 1.01589i
\(960\) 0 0
\(961\) −9.10369 15.7680i −0.293667 0.508647i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.66768 −0.0536845
\(966\) 0 0
\(967\) −21.8044 −0.701181 −0.350591 0.936529i \(-0.614019\pi\)
−0.350591 + 0.936529i \(0.614019\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −17.7817 30.7988i −0.570642 0.988382i −0.996500 0.0835915i \(-0.973361\pi\)
0.425858 0.904790i \(-0.359972\pi\)
\(972\) 0 0
\(973\) −23.6179 + 39.3864i −0.757156 + 1.26267i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.92632 + 5.15362i 0.285578 + 0.164879i 0.635946 0.771733i \(-0.280610\pi\)
−0.350368 + 0.936612i \(0.613943\pi\)
\(978\) 0 0
\(979\) 40.3964i 1.29108i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.1069 24.4340i 0.449942 0.779322i −0.548440 0.836190i \(-0.684778\pi\)
0.998382 + 0.0568679i \(0.0181114\pi\)
\(984\) 0 0
\(985\) 0.667579 0.385427i 0.0212708 0.0122807i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.21208 4.16389i 0.229331 0.132404i
\(990\) 0 0
\(991\) −17.2085 + 29.8061i −0.546647 + 0.946821i 0.451854 + 0.892092i \(0.350763\pi\)
−0.998501 + 0.0547288i \(0.982571\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.42740i 0.0769538i
\(996\) 0 0
\(997\) 41.0269 + 23.6869i 1.29934 + 0.750172i 0.980289 0.197567i \(-0.0633040\pi\)
0.319047 + 0.947739i \(0.396637\pi\)
\(998\) 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 504.2.bl.a.17.4 16
3.2 odd 2 inner 504.2.bl.a.17.5 yes 16
4.3 odd 2 1008.2.bt.d.17.4 16
7.2 even 3 3528.2.bl.a.1097.4 16
7.3 odd 6 3528.2.k.b.881.7 16
7.4 even 3 3528.2.k.b.881.9 16
7.5 odd 6 inner 504.2.bl.a.89.5 yes 16
7.6 odd 2 3528.2.bl.a.521.5 16
12.11 even 2 1008.2.bt.d.17.5 16
21.2 odd 6 3528.2.bl.a.1097.5 16
21.5 even 6 inner 504.2.bl.a.89.4 yes 16
21.11 odd 6 3528.2.k.b.881.8 16
21.17 even 6 3528.2.k.b.881.10 16
21.20 even 2 3528.2.bl.a.521.4 16
28.3 even 6 7056.2.k.h.881.8 16
28.11 odd 6 7056.2.k.h.881.10 16
28.19 even 6 1008.2.bt.d.593.5 16
84.11 even 6 7056.2.k.h.881.7 16
84.47 odd 6 1008.2.bt.d.593.4 16
84.59 odd 6 7056.2.k.h.881.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bl.a.17.4 16 1.1 even 1 trivial
504.2.bl.a.17.5 yes 16 3.2 odd 2 inner
504.2.bl.a.89.4 yes 16 21.5 even 6 inner
504.2.bl.a.89.5 yes 16 7.5 odd 6 inner
1008.2.bt.d.17.4 16 4.3 odd 2
1008.2.bt.d.17.5 16 12.11 even 2
1008.2.bt.d.593.4 16 84.47 odd 6
1008.2.bt.d.593.5 16 28.19 even 6
3528.2.k.b.881.7 16 7.3 odd 6
3528.2.k.b.881.8 16 21.11 odd 6
3528.2.k.b.881.9 16 7.4 even 3
3528.2.k.b.881.10 16 21.17 even 6
3528.2.bl.a.521.4 16 21.20 even 2
3528.2.bl.a.521.5 16 7.6 odd 2
3528.2.bl.a.1097.4 16 7.2 even 3
3528.2.bl.a.1097.5 16 21.2 odd 6
7056.2.k.h.881.7 16 84.11 even 6
7056.2.k.h.881.8 16 28.3 even 6
7056.2.k.h.881.9 16 84.59 odd 6
7056.2.k.h.881.10 16 28.11 odd 6