Properties

Label 2736.2.dc.e.1889.4
Level $2736$
Weight $2$
Character 2736.1889
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (of order \(6\), 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.4
Root \(-0.561107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1889
Dual form 2736.2.dc.e.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.524572 - 0.302862i) q^{5} -4.37361 q^{7} +O(q^{10})\) \(q+(-0.524572 - 0.302862i) q^{5} -4.37361 q^{7} -4.96560i q^{11} +(1.91196 - 1.10387i) q^{13} +(2.80061 + 1.61693i) q^{17} +(0.965553 + 4.25061i) q^{19} +(1.27590 - 0.736640i) q^{23} +(-2.31655 - 4.01238i) q^{25} +(1.05366 + 1.82499i) q^{29} -9.85749i q^{31} +(2.29427 + 1.32460i) q^{35} +5.03086i q^{37} +(-3.79948 + 6.58090i) q^{41} +(-1.50446 + 2.60580i) q^{43} +(-9.40077 + 5.42754i) q^{47} +12.1285 q^{49} +(-1.08442 - 1.87827i) q^{53} +(-1.50389 + 2.60481i) q^{55} +(-0.306301 + 0.530529i) q^{59} +(-4.45845 - 7.72227i) q^{61} -1.33728 q^{65} +(-11.7394 + 6.77776i) q^{67} +(-1.90222 + 3.29475i) q^{71} +(-1.75981 + 3.04808i) q^{73} +21.7176i q^{77} +(-13.7950 - 7.96456i) q^{79} +14.1763i q^{83} +(-0.979412 - 1.69639i) q^{85} +(6.78655 + 11.7546i) q^{89} +(-8.36216 + 4.82790i) q^{91} +(0.780846 - 2.52218i) q^{95} +(3.12989 + 1.80704i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 4 q^{19} + 14 q^{25} - 12 q^{35} - 8 q^{41} + 2 q^{43} - 36 q^{47} + 32 q^{49} + 8 q^{53} - 12 q^{55} + 8 q^{59} - 2 q^{61} - 8 q^{65} - 30 q^{67} - 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} + 32 q^{89} - 18 q^{91} - 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(1\)

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
\(5\) −0.524572 0.302862i −0.234596 0.135444i 0.378095 0.925767i \(-0.376579\pi\)
−0.612690 + 0.790323i \(0.709913\pi\)
\(6\) 0 0
\(7\) −4.37361 −1.65307 −0.826535 0.562885i \(-0.809691\pi\)
−0.826535 + 0.562885i \(0.809691\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.96560i 1.49718i −0.663031 0.748592i \(-0.730730\pi\)
0.663031 0.748592i \(-0.269270\pi\)
\(12\) 0 0
\(13\) 1.91196 1.10387i 0.530282 0.306158i −0.210850 0.977519i \(-0.567623\pi\)
0.741131 + 0.671360i \(0.234290\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.80061 + 1.61693i 0.679247 + 0.392163i 0.799571 0.600571i \(-0.205060\pi\)
−0.120325 + 0.992735i \(0.538394\pi\)
\(18\) 0 0
\(19\) 0.965553 + 4.25061i 0.221513 + 0.975157i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.27590 0.736640i 0.266043 0.153600i −0.361045 0.932548i \(-0.617580\pi\)
0.627088 + 0.778948i \(0.284247\pi\)
\(24\) 0 0
\(25\) −2.31655 4.01238i −0.463310 0.802476i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.05366 + 1.82499i 0.195660 + 0.338893i 0.947117 0.320889i \(-0.103982\pi\)
−0.751457 + 0.659782i \(0.770648\pi\)
\(30\) 0 0
\(31\) 9.85749i 1.77046i −0.465157 0.885228i \(-0.654002\pi\)
0.465157 0.885228i \(-0.345998\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.29427 + 1.32460i 0.387803 + 0.223898i
\(36\) 0 0
\(37\) 5.03086i 0.827069i 0.910488 + 0.413534i \(0.135706\pi\)
−0.910488 + 0.413534i \(0.864294\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.79948 + 6.58090i −0.593379 + 1.02776i 0.400394 + 0.916343i \(0.368873\pi\)
−0.993773 + 0.111420i \(0.964460\pi\)
\(42\) 0 0
\(43\) −1.50446 + 2.60580i −0.229428 + 0.397381i −0.957639 0.287972i \(-0.907019\pi\)
0.728211 + 0.685353i \(0.240352\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.40077 + 5.42754i −1.37124 + 0.791687i −0.991085 0.133234i \(-0.957464\pi\)
−0.380158 + 0.924921i \(0.624130\pi\)
\(48\) 0 0
\(49\) 12.1285 1.73264
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.08442 1.87827i −0.148957 0.258000i 0.781885 0.623422i \(-0.214258\pi\)
−0.930842 + 0.365422i \(0.880925\pi\)
\(54\) 0 0
\(55\) −1.50389 + 2.60481i −0.202784 + 0.351233i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.306301 + 0.530529i −0.0398770 + 0.0690690i −0.885275 0.465068i \(-0.846030\pi\)
0.845398 + 0.534137i \(0.179363\pi\)
\(60\) 0 0
\(61\) −4.45845 7.72227i −0.570846 0.988735i −0.996479 0.0838390i \(-0.973282\pi\)
0.425633 0.904896i \(-0.360051\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.33728 −0.165869
\(66\) 0 0
\(67\) −11.7394 + 6.77776i −1.43420 + 0.828035i −0.997438 0.0715426i \(-0.977208\pi\)
−0.436761 + 0.899578i \(0.643874\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.90222 + 3.29475i −0.225752 + 0.391014i −0.956545 0.291585i \(-0.905817\pi\)
0.730793 + 0.682600i \(0.239151\pi\)
\(72\) 0 0
\(73\) −1.75981 + 3.04808i −0.205970 + 0.356751i −0.950441 0.310904i \(-0.899368\pi\)
0.744471 + 0.667655i \(0.232702\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 21.7176i 2.47495i
\(78\) 0 0
\(79\) −13.7950 7.96456i −1.55206 0.896083i −0.997974 0.0636245i \(-0.979734\pi\)
−0.554087 0.832458i \(-0.686933\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.1763i 1.55606i 0.628230 + 0.778028i \(0.283780\pi\)
−0.628230 + 0.778028i \(0.716220\pi\)
\(84\) 0 0
\(85\) −0.979412 1.69639i −0.106232 0.183999i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.78655 + 11.7546i 0.719373 + 1.24599i 0.961249 + 0.275683i \(0.0889040\pi\)
−0.241876 + 0.970307i \(0.577763\pi\)
\(90\) 0 0
\(91\) −8.36216 + 4.82790i −0.876593 + 0.506101i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.780846 2.52218i 0.0801131 0.258770i
\(96\) 0 0
\(97\) 3.12989 + 1.80704i 0.317792 + 0.183478i 0.650408 0.759585i \(-0.274598\pi\)
−0.332616 + 0.943062i \(0.607931\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −16.6728 + 9.62603i −1.65900 + 0.957826i −0.685826 + 0.727765i \(0.740559\pi\)
−0.973176 + 0.230060i \(0.926108\pi\)
\(102\) 0 0
\(103\) 3.29310i 0.324479i −0.986751 0.162239i \(-0.948128\pi\)
0.986751 0.162239i \(-0.0518717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00308 0.193645 0.0968224 0.995302i \(-0.469132\pi\)
0.0968224 + 0.995302i \(0.469132\pi\)
\(108\) 0 0
\(109\) 0.514531 + 0.297065i 0.0492832 + 0.0284536i 0.524439 0.851448i \(-0.324275\pi\)
−0.475156 + 0.879902i \(0.657608\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.13975 0.671651 0.335825 0.941924i \(-0.390985\pi\)
0.335825 + 0.941924i \(0.390985\pi\)
\(114\) 0 0
\(115\) −0.892400 −0.0832167
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.2488 7.07183i −1.12284 0.648273i
\(120\) 0 0
\(121\) −13.6572 −1.24156
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.83499i 0.521898i
\(126\) 0 0
\(127\) −9.79592 + 5.65568i −0.869247 + 0.501860i −0.867098 0.498138i \(-0.834017\pi\)
−0.00214915 + 0.999998i \(0.500684\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.84637 2.79805i −0.423430 0.244467i 0.273114 0.961982i \(-0.411946\pi\)
−0.696544 + 0.717515i \(0.745280\pi\)
\(132\) 0 0
\(133\) −4.22295 18.5905i −0.366176 1.61200i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.97602 4.02761i 0.596002 0.344102i −0.171465 0.985190i \(-0.554850\pi\)
0.767467 + 0.641088i \(0.221517\pi\)
\(138\) 0 0
\(139\) −6.58012 11.3971i −0.558118 0.966689i −0.997654 0.0684640i \(-0.978190\pi\)
0.439535 0.898225i \(-0.355143\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.48137 9.49401i −0.458375 0.793929i
\(144\) 0 0
\(145\) 1.27645i 0.106004i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.6556 11.3482i −1.61025 0.929679i −0.989311 0.145820i \(-0.953418\pi\)
−0.620939 0.783859i \(-0.713249\pi\)
\(150\) 0 0
\(151\) 11.4226i 0.929556i −0.885427 0.464778i \(-0.846134\pi\)
0.885427 0.464778i \(-0.153866\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.98545 + 5.17096i −0.239797 + 0.415341i
\(156\) 0 0
\(157\) −4.15228 + 7.19197i −0.331388 + 0.573981i −0.982784 0.184757i \(-0.940850\pi\)
0.651396 + 0.758738i \(0.274184\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.58028 + 3.22178i −0.439788 + 0.253912i
\(162\) 0 0
\(163\) 13.1546 1.03035 0.515173 0.857086i \(-0.327728\pi\)
0.515173 + 0.857086i \(0.327728\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.1178 + 19.2567i 0.860324 + 1.49013i 0.871616 + 0.490189i \(0.163072\pi\)
−0.0112920 + 0.999936i \(0.503594\pi\)
\(168\) 0 0
\(169\) −4.06295 + 7.03723i −0.312534 + 0.541325i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.63651 9.76272i 0.428536 0.742246i −0.568208 0.822885i \(-0.692363\pi\)
0.996743 + 0.0806396i \(0.0256963\pi\)
\(174\) 0 0
\(175\) 10.1317 + 17.5486i 0.765884 + 1.32655i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.0361 1.87128 0.935642 0.352951i \(-0.114822\pi\)
0.935642 + 0.352951i \(0.114822\pi\)
\(180\) 0 0
\(181\) 6.39225 3.69057i 0.475132 0.274318i −0.243254 0.969963i \(-0.578215\pi\)
0.718386 + 0.695645i \(0.244881\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.52366 2.63905i 0.112021 0.194027i
\(186\) 0 0
\(187\) 8.02903 13.9067i 0.587141 1.01696i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.4138i 1.33237i 0.745785 + 0.666187i \(0.232075\pi\)
−0.745785 + 0.666187i \(0.767925\pi\)
\(192\) 0 0
\(193\) −12.0421 6.95252i −0.866811 0.500454i −0.000523893 1.00000i \(-0.500167\pi\)
−0.866287 + 0.499546i \(0.833500\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.26259i 0.659932i −0.943993 0.329966i \(-0.892963\pi\)
0.943993 0.329966i \(-0.107037\pi\)
\(198\) 0 0
\(199\) 6.97107 + 12.0743i 0.494166 + 0.855921i 0.999977 0.00672315i \(-0.00214006\pi\)
−0.505811 + 0.862644i \(0.668807\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.60830 7.98181i −0.323439 0.560214i
\(204\) 0 0
\(205\) 3.98620 2.30143i 0.278408 0.160739i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21.1068 4.79455i 1.45999 0.331646i
\(210\) 0 0
\(211\) 0.223008 + 0.128754i 0.0153525 + 0.00886378i 0.507657 0.861559i \(-0.330512\pi\)
−0.492304 + 0.870423i \(0.663845\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.57839 0.911286i 0.107646 0.0621492i
\(216\) 0 0
\(217\) 43.1128i 2.92669i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.13952 0.480256
\(222\) 0 0
\(223\) 13.3860 + 7.72842i 0.896394 + 0.517533i 0.876028 0.482259i \(-0.160184\pi\)
0.0203653 + 0.999793i \(0.493517\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.51731 −0.233452 −0.116726 0.993164i \(-0.537240\pi\)
−0.116726 + 0.993164i \(0.537240\pi\)
\(228\) 0 0
\(229\) −15.1834 −1.00335 −0.501674 0.865057i \(-0.667282\pi\)
−0.501674 + 0.865057i \(0.667282\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.6888 6.17117i −0.700246 0.404287i 0.107193 0.994238i \(-0.465814\pi\)
−0.807439 + 0.589951i \(0.799147\pi\)
\(234\) 0 0
\(235\) 6.57517 0.428917
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.8027i 1.28093i 0.767986 + 0.640466i \(0.221259\pi\)
−0.767986 + 0.640466i \(0.778741\pi\)
\(240\) 0 0
\(241\) 12.7113 7.33884i 0.818804 0.472737i −0.0312001 0.999513i \(-0.509933\pi\)
0.850004 + 0.526777i \(0.176600\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.36226 3.67325i −0.406470 0.234675i
\(246\) 0 0
\(247\) 6.53822 + 7.06115i 0.416017 + 0.449290i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.82239 + 2.20686i −0.241267 + 0.139296i −0.615759 0.787935i \(-0.711150\pi\)
0.374492 + 0.927230i \(0.377817\pi\)
\(252\) 0 0
\(253\) −3.65786 6.33560i −0.229968 0.398316i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.90044 10.2199i −0.368059 0.637497i 0.621203 0.783650i \(-0.286644\pi\)
−0.989262 + 0.146153i \(0.953311\pi\)
\(258\) 0 0
\(259\) 22.0030i 1.36720i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.83830 3.94810i −0.421668 0.243450i 0.274123 0.961695i \(-0.411613\pi\)
−0.695791 + 0.718245i \(0.744946\pi\)
\(264\) 0 0
\(265\) 1.31372i 0.0807010i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.3294 + 24.8193i −0.873681 + 1.51326i −0.0155207 + 0.999880i \(0.504941\pi\)
−0.858161 + 0.513381i \(0.828393\pi\)
\(270\) 0 0
\(271\) 3.13683 5.43315i 0.190549 0.330040i −0.754883 0.655859i \(-0.772307\pi\)
0.945432 + 0.325819i \(0.105640\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −19.9239 + 11.5031i −1.20146 + 0.693660i
\(276\) 0 0
\(277\) −7.56534 −0.454557 −0.227279 0.973830i \(-0.572983\pi\)
−0.227279 + 0.973830i \(0.572983\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.1342 + 21.0171i 0.723867 + 1.25377i 0.959439 + 0.281917i \(0.0909703\pi\)
−0.235572 + 0.971857i \(0.575696\pi\)
\(282\) 0 0
\(283\) 0.151128 0.261761i 0.00898362 0.0155601i −0.861499 0.507760i \(-0.830474\pi\)
0.870482 + 0.492200i \(0.163807\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.6175 28.7823i 0.980898 1.69896i
\(288\) 0 0
\(289\) −3.27107 5.66567i −0.192416 0.333274i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.09526 0.0639860 0.0319930 0.999488i \(-0.489815\pi\)
0.0319930 + 0.999488i \(0.489815\pi\)
\(294\) 0 0
\(295\) 0.321354 0.185534i 0.0187099 0.0108022i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.62631 2.81685i 0.0940519 0.162903i
\(300\) 0 0
\(301\) 6.57992 11.3968i 0.379260 0.656898i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.40118i 0.309270i
\(306\) 0 0
\(307\) −6.19400 3.57611i −0.353510 0.204099i 0.312720 0.949845i \(-0.398760\pi\)
−0.666230 + 0.745746i \(0.732093\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.76798i 0.156958i 0.996916 + 0.0784789i \(0.0250063\pi\)
−0.996916 + 0.0784789i \(0.974994\pi\)
\(312\) 0 0
\(313\) −5.28875 9.16039i −0.298938 0.517776i 0.676955 0.736024i \(-0.263299\pi\)
−0.975893 + 0.218248i \(0.929966\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.32100 16.1444i −0.523519 0.906762i −0.999625 0.0273741i \(-0.991285\pi\)
0.476106 0.879388i \(-0.342048\pi\)
\(318\) 0 0
\(319\) 9.06219 5.23206i 0.507385 0.292939i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.16881 + 13.4655i −0.231959 + 0.749242i
\(324\) 0 0
\(325\) −8.85829 5.11434i −0.491369 0.283692i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 41.1153 23.7379i 2.26676 1.30871i
\(330\) 0 0
\(331\) 1.01912i 0.0560161i 0.999608 + 0.0280081i \(0.00891641\pi\)
−0.999608 + 0.0280081i \(0.991084\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.21089 0.448609
\(336\) 0 0
\(337\) 14.2415 + 8.22233i 0.775784 + 0.447899i 0.834934 0.550350i \(-0.185506\pi\)
−0.0591502 + 0.998249i \(0.518839\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −48.9483 −2.65070
\(342\) 0 0
\(343\) −22.4300 −1.21111
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.18410 4.72509i −0.439346 0.253656i 0.263974 0.964530i \(-0.414967\pi\)
−0.703320 + 0.710873i \(0.748300\pi\)
\(348\) 0 0
\(349\) −22.6715 −1.21358 −0.606790 0.794863i \(-0.707543\pi\)
−0.606790 + 0.794863i \(0.707543\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.5827i 0.669708i −0.942270 0.334854i \(-0.891313\pi\)
0.942270 0.334854i \(-0.108687\pi\)
\(354\) 0 0
\(355\) 1.99570 1.15222i 0.105921 0.0611535i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.4835 + 10.0941i 0.922743 + 0.532746i 0.884509 0.466523i \(-0.154494\pi\)
0.0382336 + 0.999269i \(0.487827\pi\)
\(360\) 0 0
\(361\) −17.1354 + 8.20838i −0.901864 + 0.432020i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.84629 1.06596i 0.0966394 0.0557948i
\(366\) 0 0
\(367\) 4.61388 + 7.99148i 0.240843 + 0.417152i 0.960955 0.276706i \(-0.0892429\pi\)
−0.720112 + 0.693858i \(0.755910\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.74283 + 8.21483i 0.246236 + 0.426493i
\(372\) 0 0
\(373\) 2.81744i 0.145881i −0.997336 0.0729407i \(-0.976762\pi\)
0.997336 0.0729407i \(-0.0232384\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.02911 + 2.32621i 0.207510 + 0.119806i
\(378\) 0 0
\(379\) 12.9290i 0.664116i −0.943259 0.332058i \(-0.892257\pi\)
0.943259 0.332058i \(-0.107743\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.52287 + 13.0300i −0.384401 + 0.665801i −0.991686 0.128683i \(-0.958925\pi\)
0.607285 + 0.794484i \(0.292259\pi\)
\(384\) 0 0
\(385\) 6.57743 11.3924i 0.335217 0.580612i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.0322 6.94677i 0.610054 0.352215i −0.162932 0.986637i \(-0.552095\pi\)
0.772987 + 0.634422i \(0.218762\pi\)
\(390\) 0 0
\(391\) 4.76438 0.240945
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.82432 + 8.35597i 0.242738 + 0.420434i
\(396\) 0 0
\(397\) −10.0494 + 17.4060i −0.504364 + 0.873585i 0.495623 + 0.868538i \(0.334940\pi\)
−0.999987 + 0.00504676i \(0.998394\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.20805 14.2168i 0.409890 0.709951i −0.584987 0.811043i \(-0.698900\pi\)
0.994877 + 0.101092i \(0.0322336\pi\)
\(402\) 0 0
\(403\) −10.8814 18.8471i −0.542040 0.938841i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9812 1.23827
\(408\) 0 0
\(409\) −30.3916 + 17.5466i −1.50277 + 0.867623i −0.502773 + 0.864419i \(0.667687\pi\)
−0.999995 + 0.00320474i \(0.998980\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.33964 2.32033i 0.0659195 0.114176i
\(414\) 0 0
\(415\) 4.29347 7.43650i 0.210758 0.365044i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.77786i 0.477680i 0.971059 + 0.238840i \(0.0767671\pi\)
−0.971059 + 0.238840i \(0.923233\pi\)
\(420\) 0 0
\(421\) −18.1421 10.4744i −0.884192 0.510489i −0.0121540 0.999926i \(-0.503869\pi\)
−0.872038 + 0.489437i \(0.837202\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.9828i 0.726772i
\(426\) 0 0
\(427\) 19.4995 + 33.7742i 0.943649 + 1.63445i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.2100 21.1484i −0.588137 1.01868i −0.994476 0.104961i \(-0.966528\pi\)
0.406339 0.913722i \(-0.366805\pi\)
\(432\) 0 0
\(433\) −19.1998 + 11.0850i −0.922685 + 0.532712i −0.884491 0.466558i \(-0.845494\pi\)
−0.0381943 + 0.999270i \(0.512161\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.36312 + 4.71208i 0.208716 + 0.225410i
\(438\) 0 0
\(439\) −3.89573 2.24920i −0.185933 0.107349i 0.404144 0.914695i \(-0.367569\pi\)
−0.590077 + 0.807347i \(0.700903\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.6836 + 9.05495i −0.745152 + 0.430214i −0.823940 0.566678i \(-0.808228\pi\)
0.0787875 + 0.996891i \(0.474895\pi\)
\(444\) 0 0
\(445\) 8.22154i 0.389738i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.9207 −1.60082 −0.800409 0.599454i \(-0.795385\pi\)
−0.800409 + 0.599454i \(0.795385\pi\)
\(450\) 0 0
\(451\) 32.6781 + 18.8667i 1.53875 + 0.888398i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.84874 0.274193
\(456\) 0 0
\(457\) 31.7160 1.48361 0.741807 0.670614i \(-0.233969\pi\)
0.741807 + 0.670614i \(0.233969\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.65156 + 3.26293i 0.263219 + 0.151970i 0.625802 0.779982i \(-0.284772\pi\)
−0.362583 + 0.931952i \(0.618105\pi\)
\(462\) 0 0
\(463\) 23.5365 1.09383 0.546916 0.837187i \(-0.315802\pi\)
0.546916 + 0.837187i \(0.315802\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.7992i 1.28639i −0.765701 0.643196i \(-0.777608\pi\)
0.765701 0.643196i \(-0.222392\pi\)
\(468\) 0 0
\(469\) 51.3437 29.6433i 2.37083 1.36880i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.9394 + 7.47054i 0.594952 + 0.343496i
\(474\) 0 0
\(475\) 14.8183 13.7209i 0.679912 0.629559i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.7896 6.22939i 0.492991 0.284628i −0.232824 0.972519i \(-0.574797\pi\)
0.725814 + 0.687891i \(0.241463\pi\)
\(480\) 0 0
\(481\) 5.55342 + 9.61880i 0.253214 + 0.438579i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.09457 1.89585i −0.0497018 0.0860860i
\(486\) 0 0
\(487\) 19.2572i 0.872628i 0.899794 + 0.436314i \(0.143716\pi\)
−0.899794 + 0.436314i \(0.856284\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.6394 14.2256i −1.11196 0.641991i −0.172624 0.984988i \(-0.555225\pi\)
−0.939337 + 0.342997i \(0.888558\pi\)
\(492\) 0 0
\(493\) 6.81478i 0.306922i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.31958 14.4099i 0.373184 0.646374i
\(498\) 0 0
\(499\) −17.1417 + 29.6903i −0.767367 + 1.32912i 0.171618 + 0.985164i \(0.445100\pi\)
−0.938986 + 0.343956i \(0.888233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.7949 9.11919i 0.704260 0.406605i −0.104672 0.994507i \(-0.533379\pi\)
0.808932 + 0.587902i \(0.200046\pi\)
\(504\) 0 0
\(505\) 11.6614 0.518926
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.00706242 0.0122325i −0.000313036 0.000542195i 0.865869 0.500271i \(-0.166766\pi\)
−0.866182 + 0.499729i \(0.833433\pi\)
\(510\) 0 0
\(511\) 7.69673 13.3311i 0.340483 0.589734i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.997353 + 1.72747i −0.0439486 + 0.0761213i
\(516\) 0 0
\(517\) 26.9510 + 46.6804i 1.18530 + 2.05300i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.7469 1.04037 0.520186 0.854053i \(-0.325863\pi\)
0.520186 + 0.854053i \(0.325863\pi\)
\(522\) 0 0
\(523\) 26.6145 15.3659i 1.16377 0.671905i 0.211567 0.977363i \(-0.432143\pi\)
0.952205 + 0.305459i \(0.0988099\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.9389 27.6069i 0.694308 1.20258i
\(528\) 0 0
\(529\) −10.4147 + 18.0388i −0.452814 + 0.784297i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.7765i 0.726672i
\(534\) 0 0
\(535\) −1.05076 0.606655i −0.0454282 0.0262280i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 60.2252i 2.59408i
\(540\) 0 0
\(541\) −13.1670 22.8060i −0.566096 0.980506i −0.996947 0.0780835i \(-0.975120\pi\)
0.430851 0.902423i \(-0.358213\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.179939 0.311663i −0.00770774 0.0133502i
\(546\) 0 0
\(547\) −22.7124 + 13.1130i −0.971113 + 0.560672i −0.899575 0.436766i \(-0.856124\pi\)
−0.0715375 + 0.997438i \(0.522791\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.73998 + 6.24083i −0.287133 + 0.265868i
\(552\) 0 0
\(553\) 60.3341 + 34.8339i 2.56567 + 1.48129i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.8083 + 14.9004i −1.09353 + 0.631352i −0.934515 0.355924i \(-0.884166\pi\)
−0.159018 + 0.987276i \(0.550833\pi\)
\(558\) 0 0
\(559\) 6.64291i 0.280965i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.09466 0.214714 0.107357 0.994221i \(-0.465761\pi\)
0.107357 + 0.994221i \(0.465761\pi\)
\(564\) 0 0
\(565\) −3.74531 2.16236i −0.157566 0.0909710i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.90540 0.121801 0.0609003 0.998144i \(-0.480603\pi\)
0.0609003 + 0.998144i \(0.480603\pi\)
\(570\) 0 0
\(571\) −36.5319 −1.52881 −0.764406 0.644735i \(-0.776968\pi\)
−0.764406 + 0.644735i \(0.776968\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.91136 3.41293i −0.246521 0.142329i
\(576\) 0 0
\(577\) 40.6393 1.69184 0.845919 0.533312i \(-0.179053\pi\)
0.845919 + 0.533312i \(0.179053\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 62.0018i 2.57227i
\(582\) 0 0
\(583\) −9.32674 + 5.38480i −0.386274 + 0.223016i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.1443 + 9.89826i 0.707620 + 0.408545i 0.810179 0.586182i \(-0.199370\pi\)
−0.102559 + 0.994727i \(0.532703\pi\)
\(588\) 0 0
\(589\) 41.9004 9.51792i 1.72647 0.392179i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.51318 0.873633i 0.0621387 0.0358758i −0.468609 0.883406i \(-0.655245\pi\)
0.530747 + 0.847530i \(0.321911\pi\)
\(594\) 0 0
\(595\) 4.28357 + 7.41936i 0.175609 + 0.304164i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.607867 + 1.05286i 0.0248368 + 0.0430186i 0.878177 0.478336i \(-0.158760\pi\)
−0.853340 + 0.521355i \(0.825427\pi\)
\(600\) 0 0
\(601\) 39.9166i 1.62823i −0.580703 0.814116i \(-0.697222\pi\)
0.580703 0.814116i \(-0.302778\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.16417 + 4.13623i 0.291265 + 0.168162i
\(606\) 0 0
\(607\) 23.1840i 0.941008i −0.882398 0.470504i \(-0.844072\pi\)
0.882398 0.470504i \(-0.155928\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11.9826 + 20.7544i −0.484763 + 0.839635i
\(612\) 0 0
\(613\) 18.7145 32.4145i 0.755872 1.30921i −0.189068 0.981964i \(-0.560547\pi\)
0.944940 0.327244i \(-0.106120\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.1965 13.9699i 0.974115 0.562406i 0.0736271 0.997286i \(-0.476543\pi\)
0.900488 + 0.434880i \(0.143209\pi\)
\(618\) 0 0
\(619\) 20.3728 0.818850 0.409425 0.912344i \(-0.365729\pi\)
0.409425 + 0.912344i \(0.365729\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −29.6817 51.4103i −1.18917 2.05971i
\(624\) 0 0
\(625\) −9.81555 + 17.0010i −0.392622 + 0.680042i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.13455 + 14.0895i −0.324346 + 0.561784i
\(630\) 0 0
\(631\) −7.12585 12.3423i −0.283676 0.491341i 0.688611 0.725130i \(-0.258221\pi\)
−0.972287 + 0.233790i \(0.924887\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.85155 0.271895
\(636\) 0 0
\(637\) 23.1891 13.3883i 0.918787 0.530462i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.7247 35.8962i 0.818575 1.41781i −0.0881571 0.996107i \(-0.528098\pi\)
0.906732 0.421707i \(-0.138569\pi\)
\(642\) 0 0
\(643\) −24.3938 + 42.2512i −0.961996 + 1.66623i −0.244519 + 0.969644i \(0.578630\pi\)
−0.717477 + 0.696582i \(0.754703\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.1855i 0.950829i −0.879762 0.475414i \(-0.842298\pi\)
0.879762 0.475414i \(-0.157702\pi\)
\(648\) 0 0
\(649\) 2.63440 + 1.52097i 0.103409 + 0.0597032i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.3902i 0.758797i 0.925233 + 0.379399i \(0.123869\pi\)
−0.925233 + 0.379399i \(0.876131\pi\)
\(654\) 0 0
\(655\) 1.69485 + 2.93556i 0.0662231 + 0.114702i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −16.1706 28.0083i −0.629918 1.09105i −0.987568 0.157194i \(-0.949755\pi\)
0.357650 0.933856i \(-0.383578\pi\)
\(660\) 0 0
\(661\) 17.2738 9.97301i 0.671872 0.387905i −0.124914 0.992168i \(-0.539865\pi\)
0.796785 + 0.604262i \(0.206532\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.41512 + 11.0310i −0.132433 + 0.427765i
\(666\) 0 0
\(667\) 2.68873 + 1.55234i 0.104108 + 0.0601068i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −38.3457 + 22.1389i −1.48032 + 0.854662i
\(672\) 0 0
\(673\) 15.5000i 0.597480i 0.954335 + 0.298740i \(0.0965663\pi\)
−0.954335 + 0.298740i \(0.903434\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.6896 0.987332 0.493666 0.869652i \(-0.335657\pi\)
0.493666 + 0.869652i \(0.335657\pi\)
\(678\) 0 0
\(679\) −13.6889 7.90331i −0.525333 0.303301i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.9201 1.48924 0.744618 0.667491i \(-0.232632\pi\)
0.744618 + 0.667491i \(0.232632\pi\)
\(684\) 0 0
\(685\) −4.87923 −0.186426
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.14673 2.39412i −0.157978 0.0912086i
\(690\) 0 0
\(691\) 6.29825 0.239597 0.119798 0.992798i \(-0.461775\pi\)
0.119798 + 0.992798i \(0.461775\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.97146i 0.302375i
\(696\) 0 0
\(697\) −21.2817 + 12.2870i −0.806102 + 0.465403i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.2523 15.1568i −0.991537 0.572464i −0.0858038 0.996312i \(-0.527346\pi\)
−0.905733 + 0.423848i \(0.860679\pi\)
\(702\) 0 0
\(703\) −21.3843 + 4.85756i −0.806522 + 0.183206i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 72.9202 42.1005i 2.74245 1.58335i
\(708\) 0 0
\(709\) −0.526628 0.912146i −0.0197779 0.0342564i 0.855967 0.517030i \(-0.172963\pi\)
−0.875745 + 0.482774i \(0.839629\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.26142 12.5772i −0.271942 0.471018i
\(714\) 0 0
\(715\) 6.64039i 0.248336i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.0751 + 10.4356i 0.674086 + 0.389184i 0.797623 0.603156i \(-0.206090\pi\)
−0.123537 + 0.992340i \(0.539424\pi\)
\(720\) 0 0
\(721\) 14.4027i 0.536386i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.88171 8.45538i 0.181302 0.314025i
\(726\) 0 0
\(727\) 6.50923 11.2743i 0.241414 0.418141i −0.719703 0.694282i \(-0.755722\pi\)
0.961117 + 0.276141i \(0.0890555\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.42679 + 4.86521i −0.311676 + 0.179946i
\(732\) 0 0
\(733\) −52.1956 −1.92789 −0.963944 0.266104i \(-0.914263\pi\)
−0.963944 + 0.266104i \(0.914263\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.6556 + 58.2933i 1.23972 + 2.14726i
\(738\) 0 0
\(739\) 5.41460 9.37837i 0.199179 0.344989i −0.749083 0.662476i \(-0.769506\pi\)
0.948263 + 0.317487i \(0.102839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.7477 32.4720i 0.687786 1.19128i −0.284766 0.958597i \(-0.591916\pi\)
0.972552 0.232684i \(-0.0747507\pi\)
\(744\) 0 0
\(745\) 6.87385 + 11.9059i 0.251838 + 0.436197i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −8.76068 −0.320108
\(750\) 0 0
\(751\) 41.3564 23.8771i 1.50912 0.871289i 0.509174 0.860664i \(-0.329951\pi\)
0.999944 0.0106254i \(-0.00338223\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.45946 + 5.99196i −0.125903 + 0.218070i
\(756\) 0 0
\(757\) 5.28153 9.14788i 0.191960 0.332485i −0.753939 0.656944i \(-0.771849\pi\)
0.945900 + 0.324459i \(0.105182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.4566i 0.632801i 0.948626 + 0.316400i \(0.102474\pi\)
−0.948626 + 0.316400i \(0.897526\pi\)
\(762\) 0 0
\(763\) −2.25036 1.29925i −0.0814685 0.0470359i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.35247i 0.0488347i
\(768\) 0 0
\(769\) −2.34837 4.06750i −0.0846844 0.146678i 0.820572 0.571543i \(-0.193655\pi\)
−0.905257 + 0.424865i \(0.860322\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.20124 + 10.7409i 0.223043 + 0.386322i 0.955731 0.294243i \(-0.0950676\pi\)
−0.732687 + 0.680565i \(0.761734\pi\)
\(774\) 0 0
\(775\) −39.5520 + 22.8354i −1.42075 + 0.820270i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −31.6414 9.79592i −1.13367 0.350975i
\(780\) 0 0
\(781\) 16.3604 + 9.44567i 0.585421 + 0.337993i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.35634 2.51513i 0.155484 0.0897690i
\(786\) 0 0
\(787\) 0.833723i 0.0297190i −0.999890 0.0148595i \(-0.995270\pi\)
0.999890 0.0148595i \(-0.00473010\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.2265 −1.11029
\(792\) 0 0
\(793\) −17.0487 9.84310i −0.605419 0.349539i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −39.8381 −1.41114 −0.705569 0.708641i \(-0.749309\pi\)
−0.705569 + 0.708641i \(0.749309\pi\)
\(798\) 0 0
\(799\) −35.1038 −1.24188
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.1355 + 8.73851i 0.534122 + 0.308375i
\(804\) 0 0
\(805\) 3.90301 0.137563
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.2719i 1.02914i −0.857447 0.514572i \(-0.827951\pi\)
0.857447 0.514572i \(-0.172049\pi\)
\(810\) 0 0
\(811\) −16.6197 + 9.59539i −0.583597 + 0.336940i −0.762562 0.646916i \(-0.776059\pi\)
0.178965 + 0.983856i \(0.442725\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.90052 3.98402i −0.241715 0.139554i
\(816\) 0 0
\(817\) −12.5289 3.87884i −0.438330 0.135703i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.0207 10.9816i 0.663828 0.383261i −0.129906 0.991526i \(-0.541468\pi\)
0.793734 + 0.608265i \(0.208134\pi\)
\(822\) 0 0
\(823\) 3.18570 + 5.51779i 0.111046 + 0.192338i 0.916192 0.400739i \(-0.131246\pi\)
−0.805146 + 0.593077i \(0.797913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.935401 + 1.62016i 0.0325271 + 0.0563386i 0.881831 0.471566i \(-0.156311\pi\)
−0.849304 + 0.527905i \(0.822978\pi\)
\(828\) 0 0
\(829\) 2.41224i 0.0837806i 0.999122 + 0.0418903i \(0.0133380\pi\)
−0.999122 + 0.0418903i \(0.986662\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 33.9671 + 19.6109i 1.17689 + 0.679478i
\(834\) 0 0
\(835\) 13.4687i 0.466102i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.12136 + 15.7987i −0.314904 + 0.545430i −0.979417 0.201847i \(-0.935306\pi\)
0.664513 + 0.747277i \(0.268639\pi\)
\(840\) 0 0
\(841\) 12.2796 21.2689i 0.423434 0.733410i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.26261 2.46102i 0.146638 0.0846617i
\(846\) 0 0
\(847\) 59.7312 2.05239
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.70594 + 6.41887i 0.127038 + 0.220036i
\(852\) 0 0
\(853\) −6.34815 + 10.9953i −0.217357 + 0.376473i −0.953999 0.299810i \(-0.903077\pi\)
0.736642 + 0.676283i \(0.236410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.78818 + 10.0254i −0.197720 + 0.342462i −0.947789 0.318898i \(-0.896687\pi\)
0.750069 + 0.661360i \(0.230020\pi\)
\(858\) 0 0
\(859\) 11.8563 + 20.5357i 0.404532 + 0.700669i 0.994267 0.106927i \(-0.0341012\pi\)
−0.589735 + 0.807597i \(0.700768\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.59563 −0.258558 −0.129279 0.991608i \(-0.541266\pi\)
−0.129279 + 0.991608i \(0.541266\pi\)
\(864\) 0 0
\(865\) −5.91350 + 3.41416i −0.201065 + 0.116085i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −39.5488 + 68.5005i −1.34160 + 2.32372i
\(870\) 0 0
\(871\) −14.9635 + 25.9176i −0.507019 + 0.878183i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 25.5200i 0.862733i
\(876\) 0 0
\(877\) −9.21952 5.32289i −0.311321 0.179741i 0.336196 0.941792i \(-0.390859\pi\)
−0.647518 + 0.762051i \(0.724193\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47.9276i 1.61472i −0.590058 0.807361i \(-0.700895\pi\)
0.590058 0.807361i \(-0.299105\pi\)
\(882\) 0 0
\(883\) −24.9331 43.1854i −0.839067 1.45331i −0.890676 0.454639i \(-0.849768\pi\)
0.0516094 0.998667i \(-0.483565\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.4328 + 23.2663i 0.451030 + 0.781206i 0.998450 0.0556514i \(-0.0177235\pi\)
−0.547421 + 0.836858i \(0.684390\pi\)
\(888\) 0 0
\(889\) 42.8435 24.7357i 1.43693 0.829610i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32.1473 34.7184i −1.07577 1.16181i
\(894\) 0 0
\(895\) −13.1332 7.58246i −0.438995 0.253454i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.9899 10.3864i 0.599995 0.346407i
\(900\) 0 0
\(901\) 7.01373i 0.233661i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.47092 −0.148619
\(906\) 0 0
\(907\) −49.3174 28.4734i −1.63756 0.945444i −0.981671 0.190585i \(-0.938962\pi\)
−0.655887 0.754859i \(-0.727705\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −47.3317 −1.56817 −0.784084 0.620655i \(-0.786867\pi\)
−0.784084 + 0.620655i \(0.786867\pi\)
\(912\) 0 0
\(913\) 70.3940 2.32970
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.1962 + 12.2376i 0.699959 + 0.404121i
\(918\) 0 0
\(919\) −25.8656 −0.853229 −0.426614 0.904434i \(-0.640294\pi\)
−0.426614 + 0.904434i \(0.640294\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.39922i 0.276464i
\(924\) 0 0
\(925\) 20.1857 11.6542i 0.663703 0.383189i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 11.6190 + 6.70826i 0.381209 + 0.220091i 0.678344 0.734744i \(-0.262698\pi\)
−0.297135 + 0.954835i \(0.596031\pi\)
\(930\) 0 0
\(931\) 11.7107 + 51.5535i 0.383802 + 1.68960i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.42360 + 4.86337i −0.275481 + 0.159049i
\(936\) 0 0
\(937\) −10.5622 18.2943i −0.345053 0.597649i 0.640310 0.768116i \(-0.278806\pi\)
−0.985363 + 0.170467i \(0.945472\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4.45798 7.72145i −0.145326 0.251712i 0.784168 0.620548i \(-0.213090\pi\)
−0.929495 + 0.368836i \(0.879756\pi\)
\(942\) 0 0
\(943\) 11.1954i 0.364573i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.3309 + 22.1303i 1.24559 + 0.719139i 0.970226 0.242202i \(-0.0778698\pi\)
0.275360 + 0.961341i \(0.411203\pi\)
\(948\) 0 0
\(949\) 7.77040i 0.252238i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −21.4711 + 37.1890i −0.695517 + 1.20467i 0.274490 + 0.961590i \(0.411491\pi\)
−0.970006 + 0.243080i \(0.921842\pi\)
\(954\) 0 0
\(955\) 5.57683 9.65935i 0.180462 0.312569i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −30.5104 + 17.6152i −0.985233 + 0.568824i
\(960\) 0 0
\(961\) −66.1700 −2.13452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.21130 + 7.29419i 0.135567 + 0.234808i
\(966\) 0 0
\(967\) 6.84907 11.8629i 0.220251 0.381486i −0.734633 0.678465i \(-0.762646\pi\)
0.954884 + 0.296978i \(0.0959790\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.0241 26.0224i 0.482145 0.835100i −0.517645 0.855596i \(-0.673191\pi\)
0.999790 + 0.0204959i \(0.00652451\pi\)
\(972\) 0 0
\(973\) 28.7789 + 49.8465i 0.922609 + 1.59801i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.06375 −0.257982 −0.128991 0.991646i \(-0.541174\pi\)
−0.128991 + 0.991646i \(0.541174\pi\)
\(978\) 0 0
\(979\) 58.3689 33.6993i 1.86548 1.07703i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −30.7029 + 53.1790i −0.979271 + 1.69615i −0.314216 + 0.949352i \(0.601742\pi\)
−0.665055 + 0.746795i \(0.731592\pi\)
\(984\) 0 0
\(985\) −2.80528 + 4.85889i −0.0893838 + 0.154817i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.43298i 0.140961i
\(990\) 0 0
\(991\) 9.12690 + 5.26942i 0.289926 + 0.167389i 0.637908 0.770112i \(-0.279800\pi\)
−0.347983 + 0.937501i \(0.613133\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.44508i 0.267727i
\(996\) 0 0
\(997\) −16.1468 27.9670i −0.511373 0.885724i −0.999913 0.0131823i \(-0.995804\pi\)
0.488540 0.872541i \(-0.337530\pi\)
\(998\) 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 2736.2.dc.e.1889.4 20
3.2 odd 2 2736.2.dc.f.1889.7 20
4.3 odd 2 1368.2.cu.a.521.4 yes 20
12.11 even 2 1368.2.cu.b.521.7 yes 20
19.12 odd 6 2736.2.dc.f.449.7 20
57.50 even 6 inner 2736.2.dc.e.449.4 20
76.31 even 6 1368.2.cu.b.449.7 yes 20
228.107 odd 6 1368.2.cu.a.449.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.cu.a.449.4 20 228.107 odd 6
1368.2.cu.a.521.4 yes 20 4.3 odd 2
1368.2.cu.b.449.7 yes 20 76.31 even 6
1368.2.cu.b.521.7 yes 20 12.11 even 2
2736.2.dc.e.449.4 20 57.50 even 6 inner
2736.2.dc.e.1889.4 20 1.1 even 1 trivial
2736.2.dc.f.449.7 20 19.12 odd 6
2736.2.dc.f.1889.7 20 3.2 odd 2