Properties

Label 1344.2.s.b.911.2
Level $1344$
Weight $2$
Character 1344.911
Analytic conductor $10.732$
Analytic rank $0$
Dimension $4$
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] = [1344,2,Mod(239,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 911.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1344.911
Dual form 1344.2.s.b.239.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.41421i) q^{3} +(0.414214 - 0.414214i) q^{5} +1.00000 q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(1.00000 + 1.41421i) q^{3} +(0.414214 - 0.414214i) q^{5} +1.00000 q^{7} +(-1.00000 + 2.82843i) q^{9} +(3.82843 + 3.82843i) q^{11} +(-4.41421 + 4.41421i) q^{13} +(1.00000 + 0.171573i) q^{15} -1.17157i q^{17} +(-0.414214 - 0.414214i) q^{19} +(1.00000 + 1.41421i) q^{21} -7.65685i q^{23} +4.65685i q^{25} +(-5.00000 + 1.41421i) q^{27} +(-3.00000 - 3.00000i) q^{29} +6.48528i q^{31} +(-1.58579 + 9.24264i) q^{33} +(0.414214 - 0.414214i) q^{35} +(1.82843 + 1.82843i) q^{37} +(-10.6569 - 1.82843i) q^{39} -0.343146 q^{41} +(3.82843 - 3.82843i) q^{43} +(0.757359 + 1.58579i) q^{45} +5.65685 q^{47} +1.00000 q^{49} +(1.65685 - 1.17157i) q^{51} +(-5.82843 + 5.82843i) q^{53} +3.17157 q^{55} +(0.171573 - 1.00000i) q^{57} +(10.0711 + 10.0711i) q^{59} +(2.41421 - 2.41421i) q^{61} +(-1.00000 + 2.82843i) q^{63} +3.65685i q^{65} +(7.00000 + 7.00000i) q^{67} +(10.8284 - 7.65685i) q^{69} -6.00000i q^{71} -5.17157i q^{73} +(-6.58579 + 4.65685i) q^{75} +(3.82843 + 3.82843i) q^{77} +2.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} +(-8.89949 + 8.89949i) q^{83} +(-0.485281 - 0.485281i) q^{85} +(1.24264 - 7.24264i) q^{87} -15.6569 q^{89} +(-4.41421 + 4.41421i) q^{91} +(-9.17157 + 6.48528i) q^{93} -0.343146 q^{95} -2.00000 q^{97} +(-14.6569 + 7.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 12 q^{13} + 4 q^{15} + 4 q^{19} + 4 q^{21} - 20 q^{27} - 12 q^{29} - 12 q^{33} - 4 q^{35} - 4 q^{37} - 20 q^{39} - 24 q^{41} + 4 q^{43} + 20 q^{45} + 4 q^{49} - 16 q^{51} - 12 q^{53} + 24 q^{55} + 12 q^{57} + 12 q^{59} + 4 q^{61} - 4 q^{63} + 28 q^{67} + 32 q^{69} - 32 q^{75} + 4 q^{77} - 28 q^{81} + 4 q^{83} + 32 q^{85} - 12 q^{87} - 40 q^{89} - 12 q^{91} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 1.00000 + 1.41421i 0.577350 + 0.816497i
\(4\) 0 0
\(5\) 0.414214 0.414214i 0.185242 0.185242i −0.608394 0.793635i \(-0.708186\pi\)
0.793635 + 0.608394i \(0.208186\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) 3.82843 + 3.82843i 1.15431 + 1.15431i 0.985678 + 0.168636i \(0.0539362\pi\)
0.168636 + 0.985678i \(0.446064\pi\)
\(12\) 0 0
\(13\) −4.41421 + 4.41421i −1.22428 + 1.22428i −0.258188 + 0.966095i \(0.583125\pi\)
−0.966095 + 0.258188i \(0.916875\pi\)
\(14\) 0 0
\(15\) 1.00000 + 0.171573i 0.258199 + 0.0442999i
\(16\) 0 0
\(17\) 1.17157i 0.284148i −0.989856 0.142074i \(-0.954623\pi\)
0.989856 0.142074i \(-0.0453771\pi\)
\(18\) 0 0
\(19\) −0.414214 0.414214i −0.0950271 0.0950271i 0.657995 0.753022i \(-0.271405\pi\)
−0.753022 + 0.657995i \(0.771405\pi\)
\(20\) 0 0
\(21\) 1.00000 + 1.41421i 0.218218 + 0.308607i
\(22\) 0 0
\(23\) 7.65685i 1.59656i −0.602284 0.798282i \(-0.705742\pi\)
0.602284 0.798282i \(-0.294258\pi\)
\(24\) 0 0
\(25\) 4.65685i 0.931371i
\(26\) 0 0
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) −3.00000 3.00000i −0.557086 0.557086i 0.371391 0.928477i \(-0.378881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 6.48528i 1.16479i 0.812906 + 0.582395i \(0.197884\pi\)
−0.812906 + 0.582395i \(0.802116\pi\)
\(32\) 0 0
\(33\) −1.58579 + 9.24264i −0.276050 + 1.60894i
\(34\) 0 0
\(35\) 0.414214 0.414214i 0.0700149 0.0700149i
\(36\) 0 0
\(37\) 1.82843 + 1.82843i 0.300592 + 0.300592i 0.841245 0.540654i \(-0.181823\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(38\) 0 0
\(39\) −10.6569 1.82843i −1.70646 0.292783i
\(40\) 0 0
\(41\) −0.343146 −0.0535904 −0.0267952 0.999641i \(-0.508530\pi\)
−0.0267952 + 0.999641i \(0.508530\pi\)
\(42\) 0 0
\(43\) 3.82843 3.82843i 0.583830 0.583830i −0.352124 0.935953i \(-0.614540\pi\)
0.935953 + 0.352124i \(0.114540\pi\)
\(44\) 0 0
\(45\) 0.757359 + 1.58579i 0.112900 + 0.236395i
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.65685 1.17157i 0.232006 0.164053i
\(52\) 0 0
\(53\) −5.82843 + 5.82843i −0.800596 + 0.800596i −0.983189 0.182593i \(-0.941551\pi\)
0.182593 + 0.983189i \(0.441551\pi\)
\(54\) 0 0
\(55\) 3.17157 0.427655
\(56\) 0 0
\(57\) 0.171573 1.00000i 0.0227254 0.132453i
\(58\) 0 0
\(59\) 10.0711 + 10.0711i 1.31114 + 1.31114i 0.920575 + 0.390567i \(0.127721\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.41421 2.41421i 0.309108 0.309108i −0.535455 0.844564i \(-0.679860\pi\)
0.844564 + 0.535455i \(0.179860\pi\)
\(62\) 0 0
\(63\) −1.00000 + 2.82843i −0.125988 + 0.356348i
\(64\) 0 0
\(65\) 3.65685i 0.453577i
\(66\) 0 0
\(67\) 7.00000 + 7.00000i 0.855186 + 0.855186i 0.990766 0.135580i \(-0.0432899\pi\)
−0.135580 + 0.990766i \(0.543290\pi\)
\(68\) 0 0
\(69\) 10.8284 7.65685i 1.30359 0.921777i
\(70\) 0 0
\(71\) 6.00000i 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) 0 0
\(73\) 5.17157i 0.605287i −0.953104 0.302643i \(-0.902131\pi\)
0.953104 0.302643i \(-0.0978691\pi\)
\(74\) 0 0
\(75\) −6.58579 + 4.65685i −0.760461 + 0.537727i
\(76\) 0 0
\(77\) 3.82843 + 3.82843i 0.436290 + 0.436290i
\(78\) 0 0
\(79\) 2.00000i 0.225018i 0.993651 + 0.112509i \(0.0358886\pi\)
−0.993651 + 0.112509i \(0.964111\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) −8.89949 + 8.89949i −0.976846 + 0.976846i −0.999738 0.0228915i \(-0.992713\pi\)
0.0228915 + 0.999738i \(0.492713\pi\)
\(84\) 0 0
\(85\) −0.485281 0.485281i −0.0526362 0.0526362i
\(86\) 0 0
\(87\) 1.24264 7.24264i 0.133225 0.776493i
\(88\) 0 0
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 0 0
\(91\) −4.41421 + 4.41421i −0.462735 + 0.462735i
\(92\) 0 0
\(93\) −9.17157 + 6.48528i −0.951048 + 0.672492i
\(94\) 0 0
\(95\) −0.343146 −0.0352060
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −14.6569 + 7.00000i −1.47307 + 0.703526i
\(100\) 0 0
\(101\) 10.0711 10.0711i 1.00211 1.00211i 0.00211093 0.999998i \(-0.499328\pi\)
0.999998 0.00211093i \(-0.000671930\pi\)
\(102\) 0 0
\(103\) −11.3137 −1.11477 −0.557386 0.830253i \(-0.688196\pi\)
−0.557386 + 0.830253i \(0.688196\pi\)
\(104\) 0 0
\(105\) 1.00000 + 0.171573i 0.0975900 + 0.0167438i
\(106\) 0 0
\(107\) 5.00000 + 5.00000i 0.483368 + 0.483368i 0.906206 0.422837i \(-0.138966\pi\)
−0.422837 + 0.906206i \(0.638966\pi\)
\(108\) 0 0
\(109\) 5.82843 5.82843i 0.558262 0.558262i −0.370550 0.928812i \(-0.620831\pi\)
0.928812 + 0.370550i \(0.120831\pi\)
\(110\) 0 0
\(111\) −0.757359 + 4.41421i −0.0718854 + 0.418979i
\(112\) 0 0
\(113\) 1.65685i 0.155864i −0.996959 0.0779319i \(-0.975168\pi\)
0.996959 0.0779319i \(-0.0248317\pi\)
\(114\) 0 0
\(115\) −3.17157 3.17157i −0.295751 0.295751i
\(116\) 0 0
\(117\) −8.07107 16.8995i −0.746170 1.56236i
\(118\) 0 0
\(119\) 1.17157i 0.107398i
\(120\) 0 0
\(121\) 18.3137i 1.66488i
\(122\) 0 0
\(123\) −0.343146 0.485281i −0.0309404 0.0437563i
\(124\) 0 0
\(125\) 4.00000 + 4.00000i 0.357771 + 0.357771i
\(126\) 0 0
\(127\) 15.6569i 1.38932i −0.719338 0.694661i \(-0.755555\pi\)
0.719338 0.694661i \(-0.244445\pi\)
\(128\) 0 0
\(129\) 9.24264 + 1.58579i 0.813769 + 0.139621i
\(130\) 0 0
\(131\) 4.07107 4.07107i 0.355691 0.355691i −0.506531 0.862222i \(-0.669072\pi\)
0.862222 + 0.506531i \(0.169072\pi\)
\(132\) 0 0
\(133\) −0.414214 0.414214i −0.0359169 0.0359169i
\(134\) 0 0
\(135\) −1.48528 + 2.65685i −0.127833 + 0.228666i
\(136\) 0 0
\(137\) −7.65685 −0.654169 −0.327085 0.944995i \(-0.606066\pi\)
−0.327085 + 0.944995i \(0.606066\pi\)
\(138\) 0 0
\(139\) 4.41421 4.41421i 0.374409 0.374409i −0.494671 0.869080i \(-0.664712\pi\)
0.869080 + 0.494671i \(0.164712\pi\)
\(140\) 0 0
\(141\) 5.65685 + 8.00000i 0.476393 + 0.673722i
\(142\) 0 0
\(143\) −33.7990 −2.82641
\(144\) 0 0
\(145\) −2.48528 −0.206391
\(146\) 0 0
\(147\) 1.00000 + 1.41421i 0.0824786 + 0.116642i
\(148\) 0 0
\(149\) 9.48528 9.48528i 0.777065 0.777065i −0.202266 0.979331i \(-0.564831\pi\)
0.979331 + 0.202266i \(0.0648306\pi\)
\(150\) 0 0
\(151\) 17.6569 1.43689 0.718447 0.695581i \(-0.244853\pi\)
0.718447 + 0.695581i \(0.244853\pi\)
\(152\) 0 0
\(153\) 3.31371 + 1.17157i 0.267897 + 0.0947161i
\(154\) 0 0
\(155\) 2.68629 + 2.68629i 0.215768 + 0.215768i
\(156\) 0 0
\(157\) 2.89949 2.89949i 0.231405 0.231405i −0.581874 0.813279i \(-0.697680\pi\)
0.813279 + 0.581874i \(0.197680\pi\)
\(158\) 0 0
\(159\) −14.0711 2.41421i −1.11591 0.191460i
\(160\) 0 0
\(161\) 7.65685i 0.603445i
\(162\) 0 0
\(163\) 1.82843 + 1.82843i 0.143213 + 0.143213i 0.775078 0.631865i \(-0.217710\pi\)
−0.631865 + 0.775078i \(0.717710\pi\)
\(164\) 0 0
\(165\) 3.17157 + 4.48528i 0.246907 + 0.349179i
\(166\) 0 0
\(167\) 7.17157i 0.554953i 0.960732 + 0.277476i \(0.0894981\pi\)
−0.960732 + 0.277476i \(0.910502\pi\)
\(168\) 0 0
\(169\) 25.9706i 1.99774i
\(170\) 0 0
\(171\) 1.58579 0.757359i 0.121268 0.0579167i
\(172\) 0 0
\(173\) 4.89949 + 4.89949i 0.372502 + 0.372502i 0.868388 0.495886i \(-0.165157\pi\)
−0.495886 + 0.868388i \(0.665157\pi\)
\(174\) 0 0
\(175\) 4.65685i 0.352025i
\(176\) 0 0
\(177\) −4.17157 + 24.3137i −0.313555 + 1.82753i
\(178\) 0 0
\(179\) 15.0000 15.0000i 1.12115 1.12115i 0.129584 0.991568i \(-0.458636\pi\)
0.991568 0.129584i \(-0.0413643\pi\)
\(180\) 0 0
\(181\) −2.07107 2.07107i −0.153941 0.153941i 0.625934 0.779876i \(-0.284718\pi\)
−0.779876 + 0.625934i \(0.784718\pi\)
\(182\) 0 0
\(183\) 5.82843 + 1.00000i 0.430850 + 0.0739221i
\(184\) 0 0
\(185\) 1.51472 0.111364
\(186\) 0 0
\(187\) 4.48528 4.48528i 0.327996 0.327996i
\(188\) 0 0
\(189\) −5.00000 + 1.41421i −0.363696 + 0.102869i
\(190\) 0 0
\(191\) 6.34315 0.458974 0.229487 0.973312i \(-0.426295\pi\)
0.229487 + 0.973312i \(0.426295\pi\)
\(192\) 0 0
\(193\) 15.6569 1.12701 0.563503 0.826114i \(-0.309454\pi\)
0.563503 + 0.826114i \(0.309454\pi\)
\(194\) 0 0
\(195\) −5.17157 + 3.65685i −0.370344 + 0.261873i
\(196\) 0 0
\(197\) −5.82843 + 5.82843i −0.415258 + 0.415258i −0.883566 0.468307i \(-0.844864\pi\)
0.468307 + 0.883566i \(0.344864\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −2.89949 + 16.8995i −0.204515 + 1.19200i
\(202\) 0 0
\(203\) −3.00000 3.00000i −0.210559 0.210559i
\(204\) 0 0
\(205\) −0.142136 + 0.142136i −0.00992718 + 0.00992718i
\(206\) 0 0
\(207\) 21.6569 + 7.65685i 1.50526 + 0.532188i
\(208\) 0 0
\(209\) 3.17157i 0.219382i
\(210\) 0 0
\(211\) −5.00000 5.00000i −0.344214 0.344214i 0.513735 0.857949i \(-0.328262\pi\)
−0.857949 + 0.513735i \(0.828262\pi\)
\(212\) 0 0
\(213\) 8.48528 6.00000i 0.581402 0.411113i
\(214\) 0 0
\(215\) 3.17157i 0.216299i
\(216\) 0 0
\(217\) 6.48528i 0.440250i
\(218\) 0 0
\(219\) 7.31371 5.17157i 0.494215 0.349463i
\(220\) 0 0
\(221\) 5.17157 + 5.17157i 0.347878 + 0.347878i
\(222\) 0 0
\(223\) 7.17157i 0.480244i −0.970743 0.240122i \(-0.922813\pi\)
0.970743 0.240122i \(-0.0771875\pi\)
\(224\) 0 0
\(225\) −13.1716 4.65685i −0.878105 0.310457i
\(226\) 0 0
\(227\) 8.07107 8.07107i 0.535696 0.535696i −0.386566 0.922262i \(-0.626339\pi\)
0.922262 + 0.386566i \(0.126339\pi\)
\(228\) 0 0
\(229\) 10.8995 + 10.8995i 0.720259 + 0.720259i 0.968658 0.248399i \(-0.0799044\pi\)
−0.248399 + 0.968658i \(0.579904\pi\)
\(230\) 0 0
\(231\) −1.58579 + 9.24264i −0.104337 + 0.608121i
\(232\) 0 0
\(233\) −1.31371 −0.0860639 −0.0430320 0.999074i \(-0.513702\pi\)
−0.0430320 + 0.999074i \(0.513702\pi\)
\(234\) 0 0
\(235\) 2.34315 2.34315i 0.152850 0.152850i
\(236\) 0 0
\(237\) −2.82843 + 2.00000i −0.183726 + 0.129914i
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 0 0
\(241\) 0.343146 0.0221040 0.0110520 0.999939i \(-0.496482\pi\)
0.0110520 + 0.999939i \(0.496482\pi\)
\(242\) 0 0
\(243\) 1.00000 15.5563i 0.0641500 0.997940i
\(244\) 0 0
\(245\) 0.414214 0.414214i 0.0264631 0.0264631i
\(246\) 0 0
\(247\) 3.65685 0.232680
\(248\) 0 0
\(249\) −21.4853 3.68629i −1.36157 0.233609i
\(250\) 0 0
\(251\) −2.41421 2.41421i −0.152384 0.152384i 0.626798 0.779182i \(-0.284365\pi\)
−0.779182 + 0.626798i \(0.784365\pi\)
\(252\) 0 0
\(253\) 29.3137 29.3137i 1.84294 1.84294i
\(254\) 0 0
\(255\) 0.201010 1.17157i 0.0125877 0.0733667i
\(256\) 0 0
\(257\) 26.8284i 1.67351i −0.547576 0.836756i \(-0.684449\pi\)
0.547576 0.836756i \(-0.315551\pi\)
\(258\) 0 0
\(259\) 1.82843 + 1.82843i 0.113613 + 0.113613i
\(260\) 0 0
\(261\) 11.4853 5.48528i 0.710921 0.339530i
\(262\) 0 0
\(263\) 11.6569i 0.718792i −0.933185 0.359396i \(-0.882983\pi\)
0.933185 0.359396i \(-0.117017\pi\)
\(264\) 0 0
\(265\) 4.82843i 0.296608i
\(266\) 0 0
\(267\) −15.6569 22.1421i −0.958184 1.35508i
\(268\) 0 0
\(269\) 5.58579 + 5.58579i 0.340571 + 0.340571i 0.856582 0.516011i \(-0.172584\pi\)
−0.516011 + 0.856582i \(0.672584\pi\)
\(270\) 0 0
\(271\) 16.8284i 1.02225i 0.859505 + 0.511127i \(0.170772\pi\)
−0.859505 + 0.511127i \(0.829228\pi\)
\(272\) 0 0
\(273\) −10.6569 1.82843i −0.644982 0.110661i
\(274\) 0 0
\(275\) −17.8284 + 17.8284i −1.07509 + 1.07509i
\(276\) 0 0
\(277\) 6.31371 + 6.31371i 0.379354 + 0.379354i 0.870869 0.491515i \(-0.163557\pi\)
−0.491515 + 0.870869i \(0.663557\pi\)
\(278\) 0 0
\(279\) −18.3431 6.48528i −1.09818 0.388264i
\(280\) 0 0
\(281\) 6.68629 0.398871 0.199435 0.979911i \(-0.436089\pi\)
0.199435 + 0.979911i \(0.436089\pi\)
\(282\) 0 0
\(283\) 12.8995 12.8995i 0.766795 0.766795i −0.210746 0.977541i \(-0.567589\pi\)
0.977541 + 0.210746i \(0.0675892\pi\)
\(284\) 0 0
\(285\) −0.343146 0.485281i −0.0203262 0.0287456i
\(286\) 0 0
\(287\) −0.343146 −0.0202553
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) −2.00000 2.82843i −0.117242 0.165805i
\(292\) 0 0
\(293\) −9.24264 + 9.24264i −0.539961 + 0.539961i −0.923517 0.383557i \(-0.874699\pi\)
0.383557 + 0.923517i \(0.374699\pi\)
\(294\) 0 0
\(295\) 8.34315 0.485757
\(296\) 0 0
\(297\) −24.5563 13.7279i −1.42490 0.796575i
\(298\) 0 0
\(299\) 33.7990 + 33.7990i 1.95465 + 1.95465i
\(300\) 0 0
\(301\) 3.82843 3.82843i 0.220667 0.220667i
\(302\) 0 0
\(303\) 24.3137 + 4.17157i 1.39679 + 0.239651i
\(304\) 0 0
\(305\) 2.00000i 0.114520i
\(306\) 0 0
\(307\) 14.8995 + 14.8995i 0.850359 + 0.850359i 0.990177 0.139818i \(-0.0446518\pi\)
−0.139818 + 0.990177i \(0.544652\pi\)
\(308\) 0 0
\(309\) −11.3137 16.0000i −0.643614 0.910208i
\(310\) 0 0
\(311\) 1.51472i 0.0858918i 0.999077 + 0.0429459i \(0.0136743\pi\)
−0.999077 + 0.0429459i \(0.986326\pi\)
\(312\) 0 0
\(313\) 20.4853i 1.15790i −0.815364 0.578948i \(-0.803463\pi\)
0.815364 0.578948i \(-0.196537\pi\)
\(314\) 0 0
\(315\) 0.757359 + 1.58579i 0.0426724 + 0.0893489i
\(316\) 0 0
\(317\) −13.1421 13.1421i −0.738136 0.738136i 0.234081 0.972217i \(-0.424792\pi\)
−0.972217 + 0.234081i \(0.924792\pi\)
\(318\) 0 0
\(319\) 22.9706i 1.28610i
\(320\) 0 0
\(321\) −2.07107 + 12.0711i −0.115596 + 0.673741i
\(322\) 0 0
\(323\) −0.485281 + 0.485281i −0.0270018 + 0.0270018i
\(324\) 0 0
\(325\) −20.5563 20.5563i −1.14026 1.14026i
\(326\) 0 0
\(327\) 14.0711 + 2.41421i 0.778132 + 0.133506i
\(328\) 0 0
\(329\) 5.65685 0.311872
\(330\) 0 0
\(331\) −17.1421 + 17.1421i −0.942217 + 0.942217i −0.998419 0.0562024i \(-0.982101\pi\)
0.0562024 + 0.998419i \(0.482101\pi\)
\(332\) 0 0
\(333\) −7.00000 + 3.34315i −0.383598 + 0.183203i
\(334\) 0 0
\(335\) 5.79899 0.316833
\(336\) 0 0
\(337\) 25.3137 1.37893 0.689463 0.724321i \(-0.257847\pi\)
0.689463 + 0.724321i \(0.257847\pi\)
\(338\) 0 0
\(339\) 2.34315 1.65685i 0.127262 0.0899880i
\(340\) 0 0
\(341\) −24.8284 + 24.8284i −1.34453 + 1.34453i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.31371 7.65685i 0.0707277 0.412231i
\(346\) 0 0
\(347\) −13.1421 13.1421i −0.705507 0.705507i 0.260080 0.965587i \(-0.416251\pi\)
−0.965587 + 0.260080i \(0.916251\pi\)
\(348\) 0 0
\(349\) −2.07107 + 2.07107i −0.110862 + 0.110862i −0.760362 0.649500i \(-0.774978\pi\)
0.649500 + 0.760362i \(0.274978\pi\)
\(350\) 0 0
\(351\) 15.8284 28.3137i 0.844859 1.51127i
\(352\) 0 0
\(353\) 8.48528i 0.451626i 0.974171 + 0.225813i \(0.0725038\pi\)
−0.974171 + 0.225813i \(0.927496\pi\)
\(354\) 0 0
\(355\) −2.48528 2.48528i −0.131905 0.131905i
\(356\) 0 0
\(357\) 1.65685 1.17157i 0.0876900 0.0620062i
\(358\) 0 0
\(359\) 20.3431i 1.07367i −0.843687 0.536835i \(-0.819620\pi\)
0.843687 0.536835i \(-0.180380\pi\)
\(360\) 0 0
\(361\) 18.6569i 0.981940i
\(362\) 0 0
\(363\) −25.8995 + 18.3137i −1.35937 + 0.961220i
\(364\) 0 0
\(365\) −2.14214 2.14214i −0.112125 0.112125i
\(366\) 0 0
\(367\) 16.1421i 0.842613i −0.906918 0.421306i \(-0.861572\pi\)
0.906918 0.421306i \(-0.138428\pi\)
\(368\) 0 0
\(369\) 0.343146 0.970563i 0.0178635 0.0505255i
\(370\) 0 0
\(371\) −5.82843 + 5.82843i −0.302597 + 0.302597i
\(372\) 0 0
\(373\) −8.31371 8.31371i −0.430468 0.430468i 0.458320 0.888787i \(-0.348451\pi\)
−0.888787 + 0.458320i \(0.848451\pi\)
\(374\) 0 0
\(375\) −1.65685 + 9.65685i −0.0855596 + 0.498678i
\(376\) 0 0
\(377\) 26.4853 1.36406
\(378\) 0 0
\(379\) 27.1421 27.1421i 1.39420 1.39420i 0.578553 0.815645i \(-0.303618\pi\)
0.815645 0.578553i \(-0.196382\pi\)
\(380\) 0 0
\(381\) 22.1421 15.6569i 1.13438 0.802125i
\(382\) 0 0
\(383\) −19.3137 −0.986884 −0.493442 0.869779i \(-0.664262\pi\)
−0.493442 + 0.869779i \(0.664262\pi\)
\(384\) 0 0
\(385\) 3.17157 0.161638
\(386\) 0 0
\(387\) 7.00000 + 14.6569i 0.355830 + 0.745050i
\(388\) 0 0
\(389\) −19.9706 + 19.9706i −1.01255 + 1.01255i −0.0126275 + 0.999920i \(0.504020\pi\)
−0.999920 + 0.0126275i \(0.995980\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) 0 0
\(393\) 9.82843 + 1.68629i 0.495779 + 0.0850622i
\(394\) 0 0
\(395\) 0.828427 + 0.828427i 0.0416827 + 0.0416827i
\(396\) 0 0
\(397\) −23.7279 + 23.7279i −1.19087 + 1.19087i −0.214047 + 0.976823i \(0.568665\pi\)
−0.976823 + 0.214047i \(0.931335\pi\)
\(398\) 0 0
\(399\) 0.171573 1.00000i 0.00858939 0.0500626i
\(400\) 0 0
\(401\) 19.3137i 0.964481i 0.876039 + 0.482240i \(0.160177\pi\)
−0.876039 + 0.482240i \(0.839823\pi\)
\(402\) 0 0
\(403\) −28.6274 28.6274i −1.42603 1.42603i
\(404\) 0 0
\(405\) −5.24264 + 0.556349i −0.260509 + 0.0276452i
\(406\) 0 0
\(407\) 14.0000i 0.693954i
\(408\) 0 0
\(409\) 25.1716i 1.24465i 0.782757 + 0.622327i \(0.213813\pi\)
−0.782757 + 0.622327i \(0.786187\pi\)
\(410\) 0 0
\(411\) −7.65685 10.8284i −0.377685 0.534127i
\(412\) 0 0
\(413\) 10.0711 + 10.0711i 0.495565 + 0.495565i
\(414\) 0 0
\(415\) 7.37258i 0.361906i
\(416\) 0 0
\(417\) 10.6569 + 1.82843i 0.521868 + 0.0895385i
\(418\) 0 0
\(419\) 13.2426 13.2426i 0.646945 0.646945i −0.305308 0.952254i \(-0.598759\pi\)
0.952254 + 0.305308i \(0.0987595\pi\)
\(420\) 0 0
\(421\) −1.68629 1.68629i −0.0821848 0.0821848i 0.664819 0.747004i \(-0.268509\pi\)
−0.747004 + 0.664819i \(0.768509\pi\)
\(422\) 0 0
\(423\) −5.65685 + 16.0000i −0.275046 + 0.777947i
\(424\) 0 0
\(425\) 5.45584 0.264647
\(426\) 0 0
\(427\) 2.41421 2.41421i 0.116832 0.116832i
\(428\) 0 0
\(429\) −33.7990 47.7990i −1.63183 2.30776i
\(430\) 0 0
\(431\) 3.31371 0.159616 0.0798079 0.996810i \(-0.474569\pi\)
0.0798079 + 0.996810i \(0.474569\pi\)
\(432\) 0 0
\(433\) −14.6863 −0.705778 −0.352889 0.935665i \(-0.614801\pi\)
−0.352889 + 0.935665i \(0.614801\pi\)
\(434\) 0 0
\(435\) −2.48528 3.51472i −0.119160 0.168518i
\(436\) 0 0
\(437\) −3.17157 + 3.17157i −0.151717 + 0.151717i
\(438\) 0 0
\(439\) 0.970563 0.0463224 0.0231612 0.999732i \(-0.492627\pi\)
0.0231612 + 0.999732i \(0.492627\pi\)
\(440\) 0 0
\(441\) −1.00000 + 2.82843i −0.0476190 + 0.134687i
\(442\) 0 0
\(443\) −16.6569 16.6569i −0.791391 0.791391i 0.190329 0.981720i \(-0.439044\pi\)
−0.981720 + 0.190329i \(0.939044\pi\)
\(444\) 0 0
\(445\) −6.48528 + 6.48528i −0.307432 + 0.307432i
\(446\) 0 0
\(447\) 22.8995 + 3.92893i 1.08311 + 0.185832i
\(448\) 0 0
\(449\) 12.9706i 0.612119i −0.952012 0.306059i \(-0.900989\pi\)
0.952012 0.306059i \(-0.0990106\pi\)
\(450\) 0 0
\(451\) −1.31371 1.31371i −0.0618601 0.0618601i
\(452\) 0 0
\(453\) 17.6569 + 24.9706i 0.829591 + 1.17322i
\(454\) 0 0
\(455\) 3.65685i 0.171436i
\(456\) 0 0
\(457\) 29.9411i 1.40059i −0.713855 0.700293i \(-0.753053\pi\)
0.713855 0.700293i \(-0.246947\pi\)
\(458\) 0 0
\(459\) 1.65685 + 5.85786i 0.0773353 + 0.273422i
\(460\) 0 0
\(461\) −22.4142 22.4142i −1.04393 1.04393i −0.998989 0.0449445i \(-0.985689\pi\)
−0.0449445 0.998989i \(-0.514311\pi\)
\(462\) 0 0
\(463\) 11.6569i 0.541740i −0.962616 0.270870i \(-0.912689\pi\)
0.962616 0.270870i \(-0.0873113\pi\)
\(464\) 0 0
\(465\) −1.11270 + 6.48528i −0.0516002 + 0.300748i
\(466\) 0 0
\(467\) 14.4142 14.4142i 0.667010 0.667010i −0.290013 0.957023i \(-0.593659\pi\)
0.957023 + 0.290013i \(0.0936595\pi\)
\(468\) 0 0
\(469\) 7.00000 + 7.00000i 0.323230 + 0.323230i
\(470\) 0 0
\(471\) 7.00000 + 1.20101i 0.322543 + 0.0553396i
\(472\) 0 0
\(473\) 29.3137 1.34785
\(474\) 0 0
\(475\) 1.92893 1.92893i 0.0885055 0.0885055i
\(476\) 0 0
\(477\) −10.6569 22.3137i −0.487944 1.02167i
\(478\) 0 0
\(479\) −27.3137 −1.24800 −0.623998 0.781426i \(-0.714493\pi\)
−0.623998 + 0.781426i \(0.714493\pi\)
\(480\) 0 0
\(481\) −16.1421 −0.736018
\(482\) 0 0
\(483\) 10.8284 7.65685i 0.492710 0.348399i
\(484\) 0 0
\(485\) −0.828427 + 0.828427i −0.0376169 + 0.0376169i
\(486\) 0 0
\(487\) 16.2843 0.737911 0.368955 0.929447i \(-0.379716\pi\)
0.368955 + 0.929447i \(0.379716\pi\)
\(488\) 0 0
\(489\) −0.757359 + 4.41421i −0.0342490 + 0.199618i
\(490\) 0 0
\(491\) −10.3137 10.3137i −0.465451 0.465451i 0.434986 0.900437i \(-0.356753\pi\)
−0.900437 + 0.434986i \(0.856753\pi\)
\(492\) 0 0
\(493\) −3.51472 + 3.51472i −0.158295 + 0.158295i
\(494\) 0 0
\(495\) −3.17157 + 8.97056i −0.142552 + 0.403197i
\(496\) 0 0
\(497\) 6.00000i 0.269137i
\(498\) 0 0
\(499\) −5.68629 5.68629i −0.254553 0.254553i 0.568281 0.822834i \(-0.307609\pi\)
−0.822834 + 0.568281i \(0.807609\pi\)
\(500\) 0 0
\(501\) −10.1421 + 7.17157i −0.453117 + 0.320402i
\(502\) 0 0
\(503\) 16.8284i 0.750342i −0.926956 0.375171i \(-0.877584\pi\)
0.926956 0.375171i \(-0.122416\pi\)
\(504\) 0 0
\(505\) 8.34315i 0.371265i
\(506\) 0 0
\(507\) 36.7279 25.9706i 1.63114 1.15339i
\(508\) 0 0
\(509\) 7.24264 + 7.24264i 0.321024 + 0.321024i 0.849160 0.528136i \(-0.177109\pi\)
−0.528136 + 0.849160i \(0.677109\pi\)
\(510\) 0 0
\(511\) 5.17157i 0.228777i
\(512\) 0 0
\(513\) 2.65685 + 1.48528i 0.117303 + 0.0655768i
\(514\) 0 0
\(515\) −4.68629 + 4.68629i −0.206503 + 0.206503i
\(516\) 0 0
\(517\) 21.6569 + 21.6569i 0.952467 + 0.952467i
\(518\) 0 0
\(519\) −2.02944 + 11.8284i −0.0890824 + 0.519210i
\(520\) 0 0
\(521\) 13.3137 0.583284 0.291642 0.956528i \(-0.405798\pi\)
0.291642 + 0.956528i \(0.405798\pi\)
\(522\) 0 0
\(523\) −21.0416 + 21.0416i −0.920086 + 0.920086i −0.997035 0.0769488i \(-0.975482\pi\)
0.0769488 + 0.997035i \(0.475482\pi\)
\(524\) 0 0
\(525\) −6.58579 + 4.65685i −0.287427 + 0.203242i
\(526\) 0 0
\(527\) 7.59798 0.330973
\(528\) 0 0
\(529\) −35.6274 −1.54902
\(530\) 0 0
\(531\) −38.5563 + 18.4142i −1.67320 + 0.799109i
\(532\) 0 0
\(533\) 1.51472 1.51472i 0.0656097 0.0656097i
\(534\) 0 0
\(535\) 4.14214 0.179080
\(536\) 0 0
\(537\) 36.2132 + 6.21320i 1.56272 + 0.268120i
\(538\) 0 0
\(539\) 3.82843 + 3.82843i 0.164902 + 0.164902i
\(540\) 0 0
\(541\) −21.0000 + 21.0000i −0.902861 + 0.902861i −0.995683 0.0928222i \(-0.970411\pi\)
0.0928222 + 0.995683i \(0.470411\pi\)
\(542\) 0 0
\(543\) 0.857864 5.00000i 0.0368145 0.214571i
\(544\) 0 0
\(545\) 4.82843i 0.206827i
\(546\) 0 0
\(547\) 17.1421 + 17.1421i 0.732945 + 0.732945i 0.971202 0.238257i \(-0.0765761\pi\)
−0.238257 + 0.971202i \(0.576576\pi\)
\(548\) 0 0
\(549\) 4.41421 + 9.24264i 0.188394 + 0.394466i
\(550\) 0 0
\(551\) 2.48528i 0.105877i
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) 1.51472 + 2.14214i 0.0642962 + 0.0909286i
\(556\) 0 0
\(557\) 15.1421 + 15.1421i 0.641593 + 0.641593i 0.950947 0.309354i \(-0.100113\pi\)
−0.309354 + 0.950947i \(0.600113\pi\)
\(558\) 0 0
\(559\) 33.7990i 1.42954i
\(560\) 0 0
\(561\) 10.8284 + 1.85786i 0.457177 + 0.0784391i
\(562\) 0 0
\(563\) 19.5858 19.5858i 0.825442 0.825442i −0.161440 0.986883i \(-0.551614\pi\)
0.986883 + 0.161440i \(0.0516138\pi\)
\(564\) 0 0
\(565\) −0.686292 0.686292i −0.0288725 0.0288725i
\(566\) 0 0
\(567\) −7.00000 5.65685i −0.293972 0.237566i
\(568\) 0 0
\(569\) 32.6274 1.36781 0.683906 0.729570i \(-0.260280\pi\)
0.683906 + 0.729570i \(0.260280\pi\)
\(570\) 0 0
\(571\) 6.17157 6.17157i 0.258272 0.258272i −0.566079 0.824351i \(-0.691540\pi\)
0.824351 + 0.566079i \(0.191540\pi\)
\(572\) 0 0
\(573\) 6.34315 + 8.97056i 0.264989 + 0.374751i
\(574\) 0 0
\(575\) 35.6569 1.48699
\(576\) 0 0
\(577\) −0.343146 −0.0142853 −0.00714267 0.999974i \(-0.502274\pi\)
−0.00714267 + 0.999974i \(0.502274\pi\)
\(578\) 0 0
\(579\) 15.6569 + 22.1421i 0.650677 + 0.920196i
\(580\) 0 0
\(581\) −8.89949 + 8.89949i −0.369213 + 0.369213i
\(582\) 0 0
\(583\) −44.6274 −1.84828
\(584\) 0 0
\(585\) −10.3431 3.65685i −0.427636 0.151192i
\(586\) 0 0
\(587\) −16.0711 16.0711i −0.663324 0.663324i 0.292838 0.956162i \(-0.405400\pi\)
−0.956162 + 0.292838i \(0.905400\pi\)
\(588\) 0 0
\(589\) 2.68629 2.68629i 0.110687 0.110687i
\(590\) 0 0
\(591\) −14.0711 2.41421i −0.578806 0.0993075i
\(592\) 0 0
\(593\) 1.85786i 0.0762933i −0.999272 0.0381467i \(-0.987855\pi\)
0.999272 0.0381467i \(-0.0121454\pi\)
\(594\) 0 0
\(595\) −0.485281 0.485281i −0.0198946 0.0198946i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 42.9706i 1.75573i 0.478909 + 0.877865i \(0.341033\pi\)
−0.478909 + 0.877865i \(0.658967\pi\)
\(600\) 0 0
\(601\) 16.4853i 0.672449i 0.941782 + 0.336224i \(0.109150\pi\)
−0.941782 + 0.336224i \(0.890850\pi\)
\(602\) 0 0
\(603\) −26.7990 + 12.7990i −1.09134 + 0.521215i
\(604\) 0 0
\(605\) 7.58579 + 7.58579i 0.308406 + 0.308406i
\(606\) 0 0
\(607\) 21.1127i 0.856938i 0.903556 + 0.428469i \(0.140947\pi\)
−0.903556 + 0.428469i \(0.859053\pi\)
\(608\) 0 0
\(609\) 1.24264 7.24264i 0.0503543 0.293487i
\(610\) 0 0
\(611\) −24.9706 + 24.9706i −1.01020 + 1.01020i
\(612\) 0 0
\(613\) −14.1716 14.1716i −0.572384 0.572384i 0.360410 0.932794i \(-0.382637\pi\)
−0.932794 + 0.360410i \(0.882637\pi\)
\(614\) 0 0
\(615\) −0.343146 0.0588745i −0.0138370 0.00237405i
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −26.4142 + 26.4142i −1.06168 + 1.06168i −0.0637083 + 0.997969i \(0.520293\pi\)
−0.997969 + 0.0637083i \(0.979707\pi\)
\(620\) 0 0
\(621\) 10.8284 + 38.2843i 0.434530 + 1.53629i
\(622\) 0 0
\(623\) −15.6569 −0.627279
\(624\) 0 0
\(625\) −19.9706 −0.798823
\(626\) 0 0
\(627\) 4.48528 3.17157i 0.179125 0.126660i
\(628\) 0 0
\(629\) 2.14214 2.14214i 0.0854125 0.0854125i
\(630\) 0 0
\(631\) 13.6569 0.543671 0.271835 0.962344i \(-0.412369\pi\)
0.271835 + 0.962344i \(0.412369\pi\)
\(632\) 0 0
\(633\) 2.07107 12.0711i 0.0823176 0.479782i
\(634\) 0 0
\(635\) −6.48528 6.48528i −0.257361 0.257361i
\(636\) 0 0
\(637\) −4.41421 + 4.41421i −0.174898 + 0.174898i
\(638\) 0 0
\(639\) 16.9706 + 6.00000i 0.671345 + 0.237356i
\(640\) 0 0
\(641\) 45.2548i 1.78746i −0.448607 0.893729i \(-0.648080\pi\)
0.448607 0.893729i \(-0.351920\pi\)
\(642\) 0 0
\(643\) 21.0416 + 21.0416i 0.829801 + 0.829801i 0.987489 0.157688i \(-0.0504040\pi\)
−0.157688 + 0.987489i \(0.550404\pi\)
\(644\) 0 0
\(645\) 4.48528 3.17157i 0.176608 0.124881i
\(646\) 0 0
\(647\) 16.1421i 0.634613i 0.948323 + 0.317306i \(0.102778\pi\)
−0.948323 + 0.317306i \(0.897222\pi\)
\(648\) 0 0
\(649\) 77.1127i 3.02694i
\(650\) 0 0
\(651\) −9.17157 + 6.48528i −0.359462 + 0.254178i
\(652\) 0 0
\(653\) 11.3431 + 11.3431i 0.443892 + 0.443892i 0.893318 0.449426i \(-0.148371\pi\)
−0.449426 + 0.893318i \(0.648371\pi\)
\(654\) 0 0
\(655\) 3.37258i 0.131778i
\(656\) 0 0
\(657\) 14.6274 + 5.17157i 0.570670 + 0.201762i
\(658\) 0 0
\(659\) −25.4853 + 25.4853i −0.992766 + 0.992766i −0.999974 0.00720841i \(-0.997705\pi\)
0.00720841 + 0.999974i \(0.497705\pi\)
\(660\) 0 0
\(661\) 11.3848 + 11.3848i 0.442816 + 0.442816i 0.892957 0.450141i \(-0.148626\pi\)
−0.450141 + 0.892957i \(0.648626\pi\)
\(662\) 0 0
\(663\) −2.14214 + 12.4853i −0.0831937 + 0.484888i
\(664\) 0 0
\(665\) −0.343146 −0.0133066
\(666\) 0 0
\(667\) −22.9706 + 22.9706i −0.889424 + 0.889424i
\(668\) 0 0
\(669\) 10.1421 7.17157i 0.392118 0.277269i
\(670\) 0 0
\(671\) 18.4853 0.713616
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) −6.58579 23.2843i −0.253487 0.896212i
\(676\) 0 0
\(677\) 20.8995 20.8995i 0.803233 0.803233i −0.180367 0.983599i \(-0.557728\pi\)
0.983599 + 0.180367i \(0.0577284\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 19.4853 + 3.34315i 0.746678 + 0.128110i
\(682\) 0 0
\(683\) 0.313708 + 0.313708i 0.0120037 + 0.0120037i 0.713083 0.701079i \(-0.247298\pi\)
−0.701079 + 0.713083i \(0.747298\pi\)
\(684\) 0 0
\(685\) −3.17157 + 3.17157i −0.121180 + 0.121180i
\(686\) 0 0
\(687\) −4.51472 + 26.3137i −0.172247 + 1.00393i
\(688\) 0 0
\(689\) 51.4558i 1.96031i
\(690\) 0 0
\(691\) 14.8995 + 14.8995i 0.566803 + 0.566803i 0.931232 0.364428i \(-0.118735\pi\)
−0.364428 + 0.931232i \(0.618735\pi\)
\(692\) 0 0
\(693\) −14.6569 + 7.00000i −0.556768 + 0.265908i
\(694\) 0 0
\(695\) 3.65685i 0.138712i
\(696\) 0 0
\(697\) 0.402020i 0.0152276i
\(698\) 0 0
\(699\) −1.31371 1.85786i −0.0496890 0.0702709i
\(700\) 0 0
\(701\) −0.656854 0.656854i −0.0248090 0.0248090i 0.694593 0.719402i \(-0.255584\pi\)
−0.719402 + 0.694593i \(0.755584\pi\)
\(702\) 0 0
\(703\) 1.51472i 0.0571287i
\(704\) 0 0
\(705\) 5.65685 + 0.970563i 0.213049 + 0.0365535i
\(706\) 0 0
\(707\) 10.0711 10.0711i 0.378761 0.378761i
\(708\) 0 0
\(709\) −2.17157 2.17157i −0.0815551 0.0815551i 0.665152 0.746708i \(-0.268367\pi\)
−0.746708 + 0.665152i \(0.768367\pi\)
\(710\) 0 0
\(711\) −5.65685 2.00000i −0.212149 0.0750059i
\(712\) 0 0
\(713\) 49.6569 1.85966
\(714\) 0 0
\(715\) −14.0000 + 14.0000i −0.523570 + 0.523570i
\(716\) 0 0
\(717\) −11.3137 16.0000i −0.422518 0.597531i
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) −11.3137 −0.421345
\(722\) 0 0
\(723\) 0.343146 + 0.485281i 0.0127617 + 0.0180478i
\(724\) 0 0
\(725\) 13.9706 13.9706i 0.518854 0.518854i
\(726\) 0 0
\(727\) 35.3137 1.30971 0.654856 0.755753i \(-0.272729\pi\)
0.654856 + 0.755753i \(0.272729\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) −4.48528 4.48528i −0.165894 0.165894i
\(732\) 0 0
\(733\) −2.27208 + 2.27208i −0.0839211 + 0.0839211i −0.747821 0.663900i \(-0.768900\pi\)
0.663900 + 0.747821i \(0.268900\pi\)
\(734\) 0 0
\(735\) 1.00000 + 0.171573i 0.0368856 + 0.00632856i
\(736\) 0 0
\(737\) 53.5980i 1.97431i
\(738\) 0 0
\(739\) 21.8284 + 21.8284i 0.802972 + 0.802972i 0.983559 0.180587i \(-0.0577998\pi\)
−0.180587 + 0.983559i \(0.557800\pi\)
\(740\) 0 0
\(741\) 3.65685 + 5.17157i 0.134338 + 0.189982i
\(742\) 0 0
\(743\) 4.34315i 0.159335i −0.996822 0.0796673i \(-0.974614\pi\)
0.996822 0.0796673i \(-0.0253858\pi\)
\(744\) 0 0
\(745\) 7.85786i 0.287890i
\(746\) 0 0
\(747\) −16.2721 34.0711i −0.595364 1.24660i
\(748\) 0 0
\(749\) 5.00000 + 5.00000i 0.182696 + 0.182696i
\(750\) 0 0
\(751\) 18.6863i 0.681872i 0.940086 + 0.340936i \(0.110744\pi\)
−0.940086 + 0.340936i \(0.889256\pi\)
\(752\) 0 0
\(753\) 1.00000 5.82843i 0.0364420 0.212400i
\(754\) 0 0
\(755\) 7.31371 7.31371i 0.266173 0.266173i
\(756\) 0 0
\(757\) 14.5147 + 14.5147i 0.527546 + 0.527546i 0.919840 0.392294i \(-0.128318\pi\)
−0.392294 + 0.919840i \(0.628318\pi\)
\(758\) 0 0
\(759\) 70.7696 + 12.1421i 2.56877 + 0.440732i
\(760\) 0 0
\(761\) −36.6274 −1.32774 −0.663871 0.747847i \(-0.731088\pi\)
−0.663871 + 0.747847i \(0.731088\pi\)
\(762\) 0 0
\(763\) 5.82843 5.82843i 0.211003 0.211003i
\(764\) 0 0
\(765\) 1.85786 0.887302i 0.0671712 0.0320805i
\(766\) 0 0
\(767\) −88.9117 −3.21041
\(768\) 0 0
\(769\) 24.3431 0.877836 0.438918 0.898527i \(-0.355362\pi\)
0.438918 + 0.898527i \(0.355362\pi\)
\(770\) 0 0
\(771\) 37.9411 26.8284i 1.36642 0.966202i
\(772\) 0 0
\(773\) −36.5563 + 36.5563i −1.31484 + 1.31484i −0.397039 + 0.917802i \(0.629962\pi\)
−0.917802 + 0.397039i \(0.870038\pi\)
\(774\) 0 0
\(775\) −30.2010 −1.08485
\(776\) 0 0
\(777\) −0.757359 + 4.41421i −0.0271701 + 0.158359i
\(778\) 0 0
\(779\) 0.142136 + 0.142136i 0.00509254 + 0.00509254i
\(780\) 0 0
\(781\) 22.9706 22.9706i 0.821951 0.821951i
\(782\) 0 0
\(783\) 19.2426 + 10.7574i 0.687676 + 0.384437i
\(784\) 0 0
\(785\) 2.40202i 0.0857318i
\(786\) 0 0
\(787\) −23.0416 23.0416i −0.821345 0.821345i 0.164956 0.986301i \(-0.447252\pi\)
−0.986301 + 0.164956i \(0.947252\pi\)
\(788\) 0 0
\(789\) 16.4853 11.6569i 0.586892 0.414995i
\(790\) 0 0
\(791\) 1.65685i 0.0589110i
\(792\) 0 0
\(793\) 21.3137i 0.756872i
\(794\) 0 0
\(795\) −6.82843 + 4.82843i −0.242179 + 0.171247i
\(796\) 0 0
\(797\) −34.4142 34.4142i −1.21901 1.21901i −0.967978 0.251036i \(-0.919229\pi\)
−0.251036 0.967978i \(-0.580771\pi\)
\(798\) 0 0
\(799\) 6.62742i 0.234461i
\(800\) 0 0
\(801\) 15.6569 44.2843i 0.553208 1.56471i
\(802\) 0 0
\(803\) 19.7990 19.7990i 0.698691 0.698691i
\(804\) 0 0
\(805\) −3.17157 3.17157i −0.111783 0.111783i
\(806\) 0 0
\(807\) −2.31371 + 13.4853i −0.0814464 + 0.474704i
\(808\) 0 0
\(809\) 22.6863 0.797608 0.398804 0.917036i \(-0.369425\pi\)
0.398804 + 0.917036i \(0.369425\pi\)
\(810\) 0 0
\(811\) −16.0711 + 16.0711i −0.564332 + 0.564332i −0.930535 0.366203i \(-0.880657\pi\)
0.366203 + 0.930535i \(0.380657\pi\)
\(812\) 0 0
\(813\) −23.7990 + 16.8284i −0.834667 + 0.590199i
\(814\) 0 0
\(815\) 1.51472 0.0530583
\(816\) 0 0
\(817\) −3.17157 −0.110959
\(818\) 0 0
\(819\) −8.07107 16.8995i −0.282026 0.590516i
\(820\) 0 0
\(821\) −15.4853 + 15.4853i −0.540440 + 0.540440i −0.923658 0.383218i \(-0.874816\pi\)
0.383218 + 0.923658i \(0.374816\pi\)
\(822\) 0 0
\(823\) −24.9706 −0.870419 −0.435210 0.900329i \(-0.643326\pi\)
−0.435210 + 0.900329i \(0.643326\pi\)
\(824\) 0 0
\(825\) −43.0416 7.38478i −1.49852 0.257105i
\(826\) 0 0
\(827\) 9.00000 + 9.00000i 0.312961 + 0.312961i 0.846055 0.533095i \(-0.178971\pi\)
−0.533095 + 0.846055i \(0.678971\pi\)
\(828\) 0 0
\(829\) −24.8995 + 24.8995i −0.864795 + 0.864795i −0.991891 0.127095i \(-0.959435\pi\)
0.127095 + 0.991891i \(0.459435\pi\)
\(830\) 0 0
\(831\) −2.61522 + 15.2426i −0.0907211 + 0.528761i
\(832\) 0 0
\(833\) 1.17157i 0.0405926i
\(834\) 0 0
\(835\) 2.97056 + 2.97056i 0.102801 + 0.102801i
\(836\) 0 0
\(837\) −9.17157 32.4264i −0.317016 1.12082i
\(838\) 0 0
\(839\) 51.4558i 1.77645i 0.459406 + 0.888227i \(0.348062\pi\)
−0.459406 + 0.888227i \(0.651938\pi\)
\(840\) 0 0
\(841\) 11.0000i 0.379310i
\(842\) 0 0
\(843\) 6.68629 + 9.45584i 0.230288 + 0.325677i
\(844\) 0 0
\(845\) −10.7574 10.7574i −0.370064 0.370064i
\(846\) 0 0
\(847\) 18.3137i 0.629266i
\(848\) 0 0
\(849\) 31.1421 + 5.34315i 1.06880 + 0.183376i
\(850\) 0 0
\(851\) 14.0000 14.0000i 0.479914 0.479914i
\(852\) 0 0
\(853\) −28.4142 28.4142i −0.972884 0.972884i 0.0267578 0.999642i \(-0.491482\pi\)
−0.999642 + 0.0267578i \(0.991482\pi\)
\(854\) 0 0
\(855\) 0.343146 0.970563i 0.0117353 0.0331925i
\(856\) 0 0
\(857\) −31.9411 −1.09109 −0.545544 0.838082i \(-0.683677\pi\)
−0.545544 + 0.838082i \(0.683677\pi\)
\(858\) 0 0
\(859\) −22.8995 + 22.8995i −0.781321 + 0.781321i −0.980054 0.198733i \(-0.936317\pi\)
0.198733 + 0.980054i \(0.436317\pi\)
\(860\) 0 0
\(861\) −0.343146 0.485281i −0.0116944 0.0165383i
\(862\) 0 0
\(863\) 39.3137 1.33825 0.669127 0.743148i \(-0.266668\pi\)
0.669127 + 0.743148i \(0.266668\pi\)
\(864\) 0 0
\(865\) 4.05887 0.138006
\(866\) 0 0
\(867\) 15.6274 + 22.1005i 0.530735 + 0.750573i
\(868\) 0 0
\(869\) −7.65685 + 7.65685i −0.259741 + 0.259741i
\(870\) 0 0
\(871\) −61.7990 −2.09398
\(872\) 0 0
\(873\) 2.00000 5.65685i 0.0676897 0.191456i
\(874\) 0 0
\(875\) 4.00000 + 4.00000i 0.135225 + 0.135225i
\(876\) 0 0
\(877\) 6.51472 6.51472i 0.219986 0.219986i −0.588506 0.808493i \(-0.700284\pi\)
0.808493 + 0.588506i \(0.200284\pi\)
\(878\) 0 0
\(879\) −22.3137 3.82843i −0.752623 0.129130i
\(880\) 0 0
\(881\) 48.4853i 1.63351i 0.576984 + 0.816755i \(0.304229\pi\)
−0.576984 + 0.816755i \(0.695771\pi\)
\(882\) 0 0
\(883\) −38.9411 38.9411i −1.31047 1.31047i −0.921065 0.389408i \(-0.872680\pi\)
−0.389408 0.921065i \(-0.627320\pi\)
\(884\) 0 0
\(885\) 8.34315 + 11.7990i 0.280452 + 0.396619i
\(886\) 0 0
\(887\) 39.1716i 1.31525i 0.753344 + 0.657626i \(0.228439\pi\)
−0.753344 + 0.657626i \(0.771561\pi\)
\(888\) 0 0
\(889\) 15.6569i 0.525114i
\(890\) 0 0
\(891\) −5.14214 48.4558i −0.172268 1.62333i
\(892\) 0 0
\(893\) −2.34315 2.34315i −0.0784104 0.0784104i
\(894\) 0 0
\(895\) 12.4264i 0.415369i
\(896\) 0 0
\(897\) −14.0000 + 81.5980i −0.467446 + 2.72448i
\(898\) 0 0
\(899\) 19.4558 19.4558i 0.648889 0.648889i
\(900\) 0 0
\(901\) 6.82843 + 6.82843i 0.227488 + 0.227488i
\(902\) 0 0
\(903\) 9.24264 + 1.58579i 0.307576 + 0.0527717i
\(904\) 0 0
\(905\) −1.71573 −0.0570328
\(906\) 0 0
\(907\) 20.3137 20.3137i 0.674506 0.674506i −0.284246 0.958751i \(-0.591743\pi\)
0.958751 + 0.284246i \(0.0917432\pi\)
\(908\) 0 0
\(909\) 18.4142 + 38.5563i 0.610761 + 1.27883i
\(910\) 0 0
\(911\) 44.2843 1.46720 0.733602 0.679580i \(-0.237838\pi\)
0.733602 + 0.679580i \(0.237838\pi\)
\(912\) 0 0
\(913\) −68.1421 −2.25518
\(914\) 0 0
\(915\) 2.82843 2.00000i 0.0935049 0.0661180i
\(916\) 0 0
\(917\) 4.07107 4.07107i 0.134439 0.134439i
\(918\) 0 0
\(919\) 2.34315 0.0772932 0.0386466 0.999253i \(-0.487695\pi\)
0.0386466 + 0.999253i \(0.487695\pi\)
\(920\) 0 0
\(921\) −6.17157 + 35.9706i −0.203360 + 1.18527i
\(922\) 0 0
\(923\) 26.4853 + 26.4853i 0.871774 + 0.871774i
\(924\) 0 0
\(925\) −8.51472 + 8.51472i −0.279962 + 0.279962i
\(926\) 0 0
\(927\) 11.3137 32.0000i 0.371591 1.05102i
\(928\) 0 0
\(929\) 19.1127i 0.627067i −0.949577 0.313534i \(-0.898487\pi\)
0.949577 0.313534i \(-0.101513\pi\)
\(930\) 0 0
\(931\) −0.414214 0.414214i −0.0135753 0.0135753i
\(932\) 0 0
\(933\) −2.14214 + 1.51472i −0.0701304 + 0.0495897i
\(934\) 0 0
\(935\) 3.71573i 0.121517i
\(936\) 0 0
\(937\) 1.85786i 0.0606938i −0.999539 0.0303469i \(-0.990339\pi\)
0.999539 0.0303469i \(-0.00966120\pi\)
\(938\) 0 0
\(939\) 28.9706 20.4853i 0.945419 0.668512i
\(940\) 0 0
\(941\) 42.3553 + 42.3553i 1.38074 + 1.38074i 0.843297 + 0.537447i \(0.180611\pi\)
0.537447 + 0.843297i \(0.319389\pi\)
\(942\) 0 0
\(943\) 2.62742i 0.0855605i
\(944\) 0 0
\(945\) −1.48528 + 2.65685i −0.0483162 + 0.0864275i
\(946\) 0 0
\(947\) −19.1421 + 19.1421i −0.622036 + 0.622036i −0.946052 0.324016i \(-0.894967\pi\)
0.324016 + 0.946052i \(0.394967\pi\)
\(948\) 0 0
\(949\) 22.8284 + 22.8284i 0.741042 + 0.741042i
\(950\) 0 0
\(951\) 5.44365 31.7279i 0.176522 1.02885i
\(952\) 0 0
\(953\) −20.3431 −0.658979 −0.329490 0.944159i \(-0.606877\pi\)
−0.329490 + 0.944159i \(0.606877\pi\)
\(954\) 0 0
\(955\) 2.62742 2.62742i 0.0850212 0.0850212i
\(956\) 0 0
\(957\) 32.4853 22.9706i 1.05010 0.742533i
\(958\) 0 0
\(959\) −7.65685 −0.247253
\(960\) 0 0
\(961\) −11.0589 −0.356738
\(962\) 0 0
\(963\) −19.1421 + 9.14214i −0.616847 + 0.294601i
\(964\) 0 0
\(965\) 6.48528 6.48528i 0.208769 0.208769i
\(966\) 0 0
\(967\) 23.3137 0.749718 0.374859 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(968\) 0 0
\(969\) −1.17157 0.201010i −0.0376363 0.00645738i
\(970\) 0 0
\(971\) −2.89949 2.89949i −0.0930492 0.0930492i 0.659050 0.752099i \(-0.270959\pi\)
−0.752099 + 0.659050i \(0.770959\pi\)
\(972\) 0 0
\(973\) 4.41421 4.41421i 0.141513 0.141513i
\(974\) 0 0
\(975\) 8.51472 49.6274i 0.272689 1.58935i
\(976\) 0 0
\(977\) 8.68629i 0.277899i −0.990299 0.138950i \(-0.955627\pi\)
0.990299 0.138950i \(-0.0443726\pi\)
\(978\) 0 0
\(979\) −59.9411 59.9411i −1.91573 1.91573i
\(980\) 0 0
\(981\) 10.6569 + 22.3137i 0.340247 + 0.712422i
\(982\) 0 0
\(983\) 37.5147i 1.19653i −0.801297 0.598267i \(-0.795856\pi\)
0.801297 0.598267i \(-0.204144\pi\)
\(984\) 0 0
\(985\) 4.82843i 0.153846i
\(986\) 0 0
\(987\) 5.65685 + 8.00000i 0.180060 + 0.254643i
\(988\) 0 0
\(989\) −29.3137 29.3137i −0.932122 0.932122i
\(990\) 0 0
\(991\) 22.6863i 0.720654i 0.932826 + 0.360327i \(0.117335\pi\)
−0.932826 + 0.360327i \(0.882665\pi\)
\(992\) 0 0
\(993\) −41.3848 7.10051i −1.31331 0.225328i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −19.9289 19.9289i −0.631156 0.631156i 0.317202 0.948358i \(-0.397257\pi\)
−0.948358 + 0.317202i \(0.897257\pi\)
\(998\) 0 0
\(999\) −11.7279 6.55635i −0.371055 0.207434i
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 1344.2.s.b.911.2 4
3.2 odd 2 1344.2.s.a.911.2 4
4.3 odd 2 336.2.s.a.155.1 4
12.11 even 2 336.2.s.b.155.1 yes 4
16.3 odd 4 1344.2.s.a.239.2 4
16.13 even 4 336.2.s.b.323.1 yes 4
48.29 odd 4 336.2.s.a.323.2 yes 4
48.35 even 4 inner 1344.2.s.b.239.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.s.a.155.1 4 4.3 odd 2
336.2.s.a.323.2 yes 4 48.29 odd 4
336.2.s.b.155.1 yes 4 12.11 even 2
336.2.s.b.323.1 yes 4 16.13 even 4
1344.2.s.a.239.2 4 16.3 odd 4
1344.2.s.a.911.2 4 3.2 odd 2
1344.2.s.b.239.1 4 48.35 even 4 inner
1344.2.s.b.911.2 4 1.1 even 1 trivial