Properties

Label 1200.2.v.j.257.2
Level $1200$
Weight $2$
Character 1200.257
Analytic conductor $9.582$
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] = [1200,2,Mod(257,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (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: \(9.58204824255\)
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 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1200.257
Dual form 1200.2.v.j.593.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.41421i) q^{3} +(-2.41421 - 2.41421i) q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(1.00000 + 1.41421i) q^{3} +(-2.41421 - 2.41421i) q^{7} +(-1.00000 + 2.82843i) q^{9} -0.828427i q^{11} +(-3.82843 + 3.82843i) q^{13} +(-1.82843 + 1.82843i) q^{17} +0.828427i q^{19} +(1.00000 - 5.82843i) q^{21} +(-4.41421 - 4.41421i) q^{23} +(-5.00000 + 1.41421i) q^{27} -3.65685 q^{29} -5.65685 q^{31} +(1.17157 - 0.828427i) q^{33} +(5.82843 + 5.82843i) q^{37} +(-9.24264 - 1.58579i) q^{39} +5.65685i q^{41} +(-0.414214 + 0.414214i) q^{43} +(-3.58579 + 3.58579i) q^{47} +4.65685i q^{49} +(-4.41421 - 0.757359i) q^{51} +(-3.00000 - 3.00000i) q^{53} +(-1.17157 + 0.828427i) q^{57} -4.00000 q^{59} +0.343146 q^{61} +(9.24264 - 4.41421i) q^{63} +(10.0711 + 10.0711i) q^{67} +(1.82843 - 10.6569i) q^{69} -10.4853i q^{71} +(4.65685 - 4.65685i) q^{73} +(-2.00000 + 2.00000i) q^{77} -0.828427i q^{79} +(-7.00000 - 5.65685i) q^{81} +(-3.24264 - 3.24264i) q^{83} +(-3.65685 - 5.17157i) q^{87} -15.6569 q^{89} +18.4853 q^{91} +(-5.65685 - 8.00000i) q^{93} +(-1.00000 - 1.00000i) q^{97} +(2.34315 + 0.828427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{17} + 4 q^{21} - 12 q^{23} - 20 q^{27} + 8 q^{29} + 16 q^{33} + 12 q^{37} - 20 q^{39} + 4 q^{43} - 20 q^{47} - 12 q^{51} - 12 q^{53} - 16 q^{57} - 16 q^{59} + 24 q^{61} + 20 q^{63} + 12 q^{67} - 4 q^{69} - 4 q^{73} - 8 q^{77} - 28 q^{81} + 4 q^{83} + 8 q^{87} - 40 q^{89} + 40 q^{91} - 4 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) 1.00000 + 1.41421i 0.577350 + 0.816497i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.41421 2.41421i −0.912487 0.912487i 0.0839804 0.996467i \(-0.473237\pi\)
−0.996467 + 0.0839804i \(0.973237\pi\)
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) 0.828427i 0.249780i −0.992171 0.124890i \(-0.960142\pi\)
0.992171 0.124890i \(-0.0398578\pi\)
\(12\) 0 0
\(13\) −3.82843 + 3.82843i −1.06181 + 1.06181i −0.0638555 + 0.997959i \(0.520340\pi\)
−0.997959 + 0.0638555i \(0.979660\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.82843 + 1.82843i −0.443459 + 0.443459i −0.893173 0.449714i \(-0.851526\pi\)
0.449714 + 0.893173i \(0.351526\pi\)
\(18\) 0 0
\(19\) 0.828427i 0.190054i 0.995475 + 0.0950271i \(0.0302938\pi\)
−0.995475 + 0.0950271i \(0.969706\pi\)
\(20\) 0 0
\(21\) 1.00000 5.82843i 0.218218 1.27187i
\(22\) 0 0
\(23\) −4.41421 4.41421i −0.920427 0.920427i 0.0766323 0.997059i \(-0.475583\pi\)
−0.997059 + 0.0766323i \(0.975583\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 1.17157 0.828427i 0.203945 0.144211i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.82843 + 5.82843i 0.958188 + 0.958188i 0.999160 0.0409727i \(-0.0130457\pi\)
−0.0409727 + 0.999160i \(0.513046\pi\)
\(38\) 0 0
\(39\) −9.24264 1.58579i −1.48001 0.253929i
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) −0.414214 + 0.414214i −0.0631670 + 0.0631670i −0.737985 0.674818i \(-0.764222\pi\)
0.674818 + 0.737985i \(0.264222\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.58579 + 3.58579i −0.523041 + 0.523041i −0.918488 0.395448i \(-0.870589\pi\)
0.395448 + 0.918488i \(0.370589\pi\)
\(48\) 0 0
\(49\) 4.65685i 0.665265i
\(50\) 0 0
\(51\) −4.41421 0.757359i −0.618114 0.106052i
\(52\) 0 0
\(53\) −3.00000 3.00000i −0.412082 0.412082i 0.470381 0.882463i \(-0.344116\pi\)
−0.882463 + 0.470381i \(0.844116\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.17157 + 0.828427i −0.155179 + 0.109728i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 0 0
\(63\) 9.24264 4.41421i 1.16446 0.556139i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0711 + 10.0711i 1.23038 + 1.23038i 0.963819 + 0.266558i \(0.0858863\pi\)
0.266558 + 0.963819i \(0.414114\pi\)
\(68\) 0 0
\(69\) 1.82843 10.6569i 0.220117 1.28293i
\(70\) 0 0
\(71\) 10.4853i 1.24437i −0.782869 0.622187i \(-0.786244\pi\)
0.782869 0.622187i \(-0.213756\pi\)
\(72\) 0 0
\(73\) 4.65685 4.65685i 0.545044 0.545044i −0.379960 0.925003i \(-0.624062\pi\)
0.925003 + 0.379960i \(0.124062\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 + 2.00000i −0.227921 + 0.227921i
\(78\) 0 0
\(79\) 0.828427i 0.0932053i −0.998914 0.0466027i \(-0.985161\pi\)
0.998914 0.0466027i \(-0.0148395\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) −3.24264 3.24264i −0.355926 0.355926i 0.506383 0.862309i \(-0.330982\pi\)
−0.862309 + 0.506383i \(0.830982\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.65685 5.17157i −0.392056 0.554451i
\(88\) 0 0
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 0 0
\(91\) 18.4853 1.93778
\(92\) 0 0
\(93\) −5.65685 8.00000i −0.586588 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 1.00000i −0.101535 0.101535i 0.654515 0.756049i \(-0.272873\pi\)
−0.756049 + 0.654515i \(0.772873\pi\)
\(98\) 0 0
\(99\) 2.34315 + 0.828427i 0.235495 + 0.0832601i
\(100\) 0 0
\(101\) 9.65685i 0.960893i −0.877024 0.480446i \(-0.840475\pi\)
0.877024 0.480446i \(-0.159525\pi\)
\(102\) 0 0
\(103\) −5.58579 + 5.58579i −0.550384 + 0.550384i −0.926552 0.376168i \(-0.877242\pi\)
0.376168 + 0.926552i \(0.377242\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.58579 9.58579i 0.926693 0.926693i −0.0707977 0.997491i \(-0.522554\pi\)
0.997491 + 0.0707977i \(0.0225545\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 0 0
\(111\) −2.41421 + 14.0711i −0.229147 + 1.33557i
\(112\) 0 0
\(113\) 9.48528 + 9.48528i 0.892300 + 0.892300i 0.994739 0.102439i \(-0.0326647\pi\)
−0.102439 + 0.994739i \(0.532665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −7.00000 14.6569i −0.647150 1.35503i
\(118\) 0 0
\(119\) 8.82843 0.809301
\(120\) 0 0
\(121\) 10.3137 0.937610
\(122\) 0 0
\(123\) −8.00000 + 5.65685i −0.721336 + 0.510061i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.58579 + 5.58579i 0.495658 + 0.495658i 0.910083 0.414425i \(-0.136017\pi\)
−0.414425 + 0.910083i \(0.636017\pi\)
\(128\) 0 0
\(129\) −1.00000 0.171573i −0.0880451 0.0151061i
\(130\) 0 0
\(131\) 8.82843i 0.771343i −0.922636 0.385672i \(-0.873970\pi\)
0.922636 0.385672i \(-0.126030\pi\)
\(132\) 0 0
\(133\) 2.00000 2.00000i 0.173422 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.82843 + 9.82843i −0.839699 + 0.839699i −0.988819 0.149120i \(-0.952356\pi\)
0.149120 + 0.988819i \(0.452356\pi\)
\(138\) 0 0
\(139\) 8.82843i 0.748817i 0.927264 + 0.374409i \(0.122154\pi\)
−0.927264 + 0.374409i \(0.877846\pi\)
\(140\) 0 0
\(141\) −8.65685 1.48528i −0.729039 0.125083i
\(142\) 0 0
\(143\) 3.17157 + 3.17157i 0.265220 + 0.265220i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.58579 + 4.65685i −0.543187 + 0.384091i
\(148\) 0 0
\(149\) 13.3137 1.09070 0.545351 0.838208i \(-0.316396\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(150\) 0 0
\(151\) 13.6569 1.11138 0.555690 0.831390i \(-0.312454\pi\)
0.555690 + 0.831390i \(0.312454\pi\)
\(152\) 0 0
\(153\) −3.34315 7.00000i −0.270277 0.565916i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.48528 5.48528i −0.437773 0.437773i 0.453489 0.891262i \(-0.350179\pi\)
−0.891262 + 0.453489i \(0.850179\pi\)
\(158\) 0 0
\(159\) 1.24264 7.24264i 0.0985478 0.574379i
\(160\) 0 0
\(161\) 21.3137i 1.67976i
\(162\) 0 0
\(163\) −0.414214 + 0.414214i −0.0324437 + 0.0324437i −0.723143 0.690699i \(-0.757303\pi\)
0.690699 + 0.723143i \(0.257303\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.24264 + 9.24264i −0.715217 + 0.715217i −0.967622 0.252405i \(-0.918779\pi\)
0.252405 + 0.967622i \(0.418779\pi\)
\(168\) 0 0
\(169\) 16.3137i 1.25490i
\(170\) 0 0
\(171\) −2.34315 0.828427i −0.179185 0.0633514i
\(172\) 0 0
\(173\) −0.656854 0.656854i −0.0499397 0.0499397i 0.681696 0.731636i \(-0.261243\pi\)
−0.731636 + 0.681696i \(0.761243\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 5.65685i −0.300658 0.425195i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −5.31371 −0.394965 −0.197482 0.980306i \(-0.563277\pi\)
−0.197482 + 0.980306i \(0.563277\pi\)
\(182\) 0 0
\(183\) 0.343146 + 0.485281i 0.0253661 + 0.0358730i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.51472 + 1.51472i 0.110767 + 0.110767i
\(188\) 0 0
\(189\) 15.4853 + 8.65685i 1.12639 + 0.629693i
\(190\) 0 0
\(191\) 4.14214i 0.299714i 0.988708 + 0.149857i \(0.0478814\pi\)
−0.988708 + 0.149857i \(0.952119\pi\)
\(192\) 0 0
\(193\) −14.6569 + 14.6569i −1.05502 + 1.05502i −0.0566281 + 0.998395i \(0.518035\pi\)
−0.998395 + 0.0566281i \(0.981965\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.6569 14.6569i 1.04426 1.04426i 0.0452834 0.998974i \(-0.485581\pi\)
0.998974 0.0452834i \(-0.0144191\pi\)
\(198\) 0 0
\(199\) 18.4853i 1.31039i 0.755461 + 0.655193i \(0.227413\pi\)
−0.755461 + 0.655193i \(0.772587\pi\)
\(200\) 0 0
\(201\) −4.17157 + 24.3137i −0.294240 + 1.71496i
\(202\) 0 0
\(203\) 8.82843 + 8.82843i 0.619634 + 0.619634i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.8995 8.07107i 1.17460 0.560978i
\(208\) 0 0
\(209\) 0.686292 0.0474718
\(210\) 0 0
\(211\) −20.9706 −1.44367 −0.721837 0.692064i \(-0.756702\pi\)
−0.721837 + 0.692064i \(0.756702\pi\)
\(212\) 0 0
\(213\) 14.8284 10.4853i 1.01603 0.718440i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 13.6569 + 13.6569i 0.927088 + 0.927088i
\(218\) 0 0
\(219\) 11.2426 + 1.92893i 0.759707 + 0.130345i
\(220\) 0 0
\(221\) 14.0000i 0.941742i
\(222\) 0 0
\(223\) −5.58579 + 5.58579i −0.374052 + 0.374052i −0.868951 0.494899i \(-0.835205\pi\)
0.494899 + 0.868951i \(0.335205\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.89949 4.89949i 0.325191 0.325191i −0.525563 0.850754i \(-0.676145\pi\)
0.850754 + 0.525563i \(0.176145\pi\)
\(228\) 0 0
\(229\) 14.3431i 0.947822i 0.880573 + 0.473911i \(0.157158\pi\)
−0.880573 + 0.473911i \(0.842842\pi\)
\(230\) 0 0
\(231\) −4.82843 0.828427i −0.317687 0.0545065i
\(232\) 0 0
\(233\) 11.8284 + 11.8284i 0.774906 + 0.774906i 0.978960 0.204054i \(-0.0654117\pi\)
−0.204054 + 0.978960i \(0.565412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.17157 0.828427i 0.0761018 0.0538121i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −0.343146 −0.0221040 −0.0110520 0.999939i \(-0.503518\pi\)
−0.0110520 + 0.999939i \(0.503518\pi\)
\(242\) 0 0
\(243\) 1.00000 15.5563i 0.0641500 0.997940i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.17157 3.17157i −0.201802 0.201802i
\(248\) 0 0
\(249\) 1.34315 7.82843i 0.0851184 0.496106i
\(250\) 0 0
\(251\) 26.4853i 1.67174i 0.548930 + 0.835868i \(0.315035\pi\)
−0.548930 + 0.835868i \(0.684965\pi\)
\(252\) 0 0
\(253\) −3.65685 + 3.65685i −0.229904 + 0.229904i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.48528 9.48528i 0.591676 0.591676i −0.346408 0.938084i \(-0.612599\pi\)
0.938084 + 0.346408i \(0.112599\pi\)
\(258\) 0 0
\(259\) 28.1421i 1.74867i
\(260\) 0 0
\(261\) 3.65685 10.3431i 0.226354 0.640225i
\(262\) 0 0
\(263\) 6.89949 + 6.89949i 0.425441 + 0.425441i 0.887072 0.461631i \(-0.152736\pi\)
−0.461631 + 0.887072i \(0.652736\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −15.6569 22.1421i −0.958184 1.35508i
\(268\) 0 0
\(269\) 16.6274 1.01379 0.506896 0.862007i \(-0.330793\pi\)
0.506896 + 0.862007i \(0.330793\pi\)
\(270\) 0 0
\(271\) −10.3431 −0.628301 −0.314151 0.949373i \(-0.601720\pi\)
−0.314151 + 0.949373i \(0.601720\pi\)
\(272\) 0 0
\(273\) 18.4853 + 26.1421i 1.11878 + 1.58219i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.17157 + 8.17157i 0.490982 + 0.490982i 0.908616 0.417633i \(-0.137140\pi\)
−0.417633 + 0.908616i \(0.637140\pi\)
\(278\) 0 0
\(279\) 5.65685 16.0000i 0.338667 0.957895i
\(280\) 0 0
\(281\) 5.65685i 0.337460i −0.985662 0.168730i \(-0.946033\pi\)
0.985662 0.168730i \(-0.0539665\pi\)
\(282\) 0 0
\(283\) 5.24264 5.24264i 0.311643 0.311643i −0.533903 0.845546i \(-0.679275\pi\)
0.845546 + 0.533903i \(0.179275\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.6569 13.6569i 0.806139 0.806139i
\(288\) 0 0
\(289\) 10.3137i 0.606689i
\(290\) 0 0
\(291\) 0.414214 2.41421i 0.0242816 0.141524i
\(292\) 0 0
\(293\) −16.6569 16.6569i −0.973104 0.973104i 0.0265438 0.999648i \(-0.491550\pi\)
−0.999648 + 0.0265438i \(0.991550\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.17157 + 4.14214i 0.0679816 + 0.240351i
\(298\) 0 0
\(299\) 33.7990 1.95465
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) 13.6569 9.65685i 0.784566 0.554772i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.89949 6.89949i −0.393775 0.393775i 0.482256 0.876031i \(-0.339818\pi\)
−0.876031 + 0.482256i \(0.839818\pi\)
\(308\) 0 0
\(309\) −13.4853 2.31371i −0.767151 0.131622i
\(310\) 0 0
\(311\) 5.51472i 0.312711i 0.987701 + 0.156356i \(0.0499746\pi\)
−0.987701 + 0.156356i \(0.950025\pi\)
\(312\) 0 0
\(313\) 2.31371 2.31371i 0.130779 0.130779i −0.638688 0.769466i \(-0.720522\pi\)
0.769466 + 0.638688i \(0.220522\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.65685 + 4.65685i −0.261555 + 0.261555i −0.825686 0.564131i \(-0.809211\pi\)
0.564131 + 0.825686i \(0.309211\pi\)
\(318\) 0 0
\(319\) 3.02944i 0.169616i
\(320\) 0 0
\(321\) 23.1421 + 3.97056i 1.29167 + 0.221615i
\(322\) 0 0
\(323\) −1.51472 1.51472i −0.0842812 0.0842812i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.65685 + 4.00000i −0.312825 + 0.221201i
\(328\) 0 0
\(329\) 17.3137 0.954536
\(330\) 0 0
\(331\) 9.65685 0.530789 0.265394 0.964140i \(-0.414498\pi\)
0.265394 + 0.964140i \(0.414498\pi\)
\(332\) 0 0
\(333\) −22.3137 + 10.6569i −1.22278 + 0.583992i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 1.00000i −0.0544735 0.0544735i 0.679345 0.733819i \(-0.262264\pi\)
−0.733819 + 0.679345i \(0.762264\pi\)
\(338\) 0 0
\(339\) −3.92893 + 22.8995i −0.213390 + 1.24373i
\(340\) 0 0
\(341\) 4.68629i 0.253777i
\(342\) 0 0
\(343\) −5.65685 + 5.65685i −0.305441 + 0.305441i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0711 + 12.0711i −0.648009 + 0.648009i −0.952511 0.304503i \(-0.901510\pi\)
0.304503 + 0.952511i \(0.401510\pi\)
\(348\) 0 0
\(349\) 9.65685i 0.516920i −0.966022 0.258460i \(-0.916785\pi\)
0.966022 0.258460i \(-0.0832149\pi\)
\(350\) 0 0
\(351\) 13.7279 24.5563i 0.732742 1.31072i
\(352\) 0 0
\(353\) −15.4853 15.4853i −0.824198 0.824198i 0.162509 0.986707i \(-0.448041\pi\)
−0.986707 + 0.162509i \(0.948041\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.82843 + 12.4853i 0.467250 + 0.660791i
\(358\) 0 0
\(359\) −35.3137 −1.86379 −0.931893 0.362733i \(-0.881844\pi\)
−0.931893 + 0.362733i \(0.881844\pi\)
\(360\) 0 0
\(361\) 18.3137 0.963879
\(362\) 0 0
\(363\) 10.3137 + 14.5858i 0.541329 + 0.765555i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.75736 4.75736i −0.248332 0.248332i 0.571954 0.820286i \(-0.306186\pi\)
−0.820286 + 0.571954i \(0.806186\pi\)
\(368\) 0 0
\(369\) −16.0000 5.65685i −0.832927 0.294484i
\(370\) 0 0
\(371\) 14.4853i 0.752038i
\(372\) 0 0
\(373\) −0.514719 + 0.514719i −0.0266511 + 0.0266511i −0.720307 0.693656i \(-0.755999\pi\)
0.693656 + 0.720307i \(0.255999\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0000 14.0000i 0.721037 0.721037i
\(378\) 0 0
\(379\) 29.7990i 1.53067i −0.643631 0.765336i \(-0.722573\pi\)
0.643631 0.765336i \(-0.277427\pi\)
\(380\) 0 0
\(381\) −2.31371 + 13.4853i −0.118535 + 0.690872i
\(382\) 0 0
\(383\) −12.4142 12.4142i −0.634337 0.634337i 0.314816 0.949153i \(-0.398057\pi\)
−0.949153 + 0.314816i \(0.898057\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.757359 1.58579i −0.0384987 0.0806101i
\(388\) 0 0
\(389\) 6.68629 0.339008 0.169504 0.985529i \(-0.445783\pi\)
0.169504 + 0.985529i \(0.445783\pi\)
\(390\) 0 0
\(391\) 16.1421 0.816343
\(392\) 0 0
\(393\) 12.4853 8.82843i 0.629799 0.445335i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.82843 7.82843i −0.392897 0.392897i 0.482821 0.875719i \(-0.339612\pi\)
−0.875719 + 0.482821i \(0.839612\pi\)
\(398\) 0 0
\(399\) 4.82843 + 0.828427i 0.241724 + 0.0414732i
\(400\) 0 0
\(401\) 16.0000i 0.799002i 0.916733 + 0.399501i \(0.130817\pi\)
−0.916733 + 0.399501i \(0.869183\pi\)
\(402\) 0 0
\(403\) 21.6569 21.6569i 1.07880 1.07880i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.82843 4.82843i 0.239336 0.239336i
\(408\) 0 0
\(409\) 21.6569i 1.07086i 0.844579 + 0.535431i \(0.179851\pi\)
−0.844579 + 0.535431i \(0.820149\pi\)
\(410\) 0 0
\(411\) −23.7279 4.07107i −1.17041 0.200811i
\(412\) 0 0
\(413\) 9.65685 + 9.65685i 0.475183 + 0.475183i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.4853 + 8.82843i −0.611407 + 0.432330i
\(418\) 0 0
\(419\) −29.9411 −1.46272 −0.731360 0.681992i \(-0.761114\pi\)
−0.731360 + 0.681992i \(0.761114\pi\)
\(420\) 0 0
\(421\) 30.9706 1.50941 0.754706 0.656063i \(-0.227779\pi\)
0.754706 + 0.656063i \(0.227779\pi\)
\(422\) 0 0
\(423\) −6.55635 13.7279i −0.318781 0.667474i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.828427 0.828427i −0.0400904 0.0400904i
\(428\) 0 0
\(429\) −1.31371 + 7.65685i −0.0634264 + 0.369676i
\(430\) 0 0
\(431\) 20.1421i 0.970213i 0.874455 + 0.485106i \(0.161219\pi\)
−0.874455 + 0.485106i \(0.838781\pi\)
\(432\) 0 0
\(433\) 15.0000 15.0000i 0.720854 0.720854i −0.247925 0.968779i \(-0.579749\pi\)
0.968779 + 0.247925i \(0.0797487\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.65685 3.65685i 0.174931 0.174931i
\(438\) 0 0
\(439\) 13.7990i 0.658590i 0.944227 + 0.329295i \(0.106811\pi\)
−0.944227 + 0.329295i \(0.893189\pi\)
\(440\) 0 0
\(441\) −13.1716 4.65685i −0.627218 0.221755i
\(442\) 0 0
\(443\) 0.0710678 + 0.0710678i 0.00337653 + 0.00337653i 0.708793 0.705416i \(-0.249240\pi\)
−0.705416 + 0.708793i \(0.749240\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.3137 + 18.8284i 0.629717 + 0.890554i
\(448\) 0 0
\(449\) 1.31371 0.0619977 0.0309989 0.999519i \(-0.490131\pi\)
0.0309989 + 0.999519i \(0.490131\pi\)
\(450\) 0 0
\(451\) 4.68629 0.220669
\(452\) 0 0
\(453\) 13.6569 + 19.3137i 0.641655 + 0.907437i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 1.00000i −0.0467780 0.0467780i 0.683331 0.730109i \(-0.260531\pi\)
−0.730109 + 0.683331i \(0.760531\pi\)
\(458\) 0 0
\(459\) 6.55635 11.7279i 0.306024 0.547413i
\(460\) 0 0
\(461\) 28.9706i 1.34929i −0.738141 0.674647i \(-0.764296\pi\)
0.738141 0.674647i \(-0.235704\pi\)
\(462\) 0 0
\(463\) −21.5858 + 21.5858i −1.00318 + 1.00318i −0.00318163 + 0.999995i \(0.501013\pi\)
−0.999995 + 0.00318163i \(0.998987\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.3848 + 23.3848i −1.08212 + 1.08212i −0.0858066 + 0.996312i \(0.527347\pi\)
−0.996312 + 0.0858066i \(0.972653\pi\)
\(468\) 0 0
\(469\) 48.6274i 2.24541i
\(470\) 0 0
\(471\) 2.27208 13.2426i 0.104692 0.610189i
\(472\) 0 0
\(473\) 0.343146 + 0.343146i 0.0157779 + 0.0157779i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.4853 5.48528i 0.525875 0.251154i
\(478\) 0 0
\(479\) 22.6274 1.03387 0.516937 0.856024i \(-0.327072\pi\)
0.516937 + 0.856024i \(0.327072\pi\)
\(480\) 0 0
\(481\) −44.6274 −2.03484
\(482\) 0 0
\(483\) −30.1421 + 21.3137i −1.37151 + 0.969807i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.5563 + 14.5563i 0.659611 + 0.659611i 0.955288 0.295677i \(-0.0955452\pi\)
−0.295677 + 0.955288i \(0.595545\pi\)
\(488\) 0 0
\(489\) −1.00000 0.171573i −0.0452216 0.00775879i
\(490\) 0 0
\(491\) 21.5147i 0.970946i −0.874252 0.485473i \(-0.838647\pi\)
0.874252 0.485473i \(-0.161353\pi\)
\(492\) 0 0
\(493\) 6.68629 6.68629i 0.301135 0.301135i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.3137 + 25.3137i −1.13548 + 1.13548i
\(498\) 0 0
\(499\) 34.7696i 1.55650i 0.627955 + 0.778249i \(0.283892\pi\)
−0.627955 + 0.778249i \(0.716108\pi\)
\(500\) 0 0
\(501\) −22.3137 3.82843i −0.996903 0.171042i
\(502\) 0 0
\(503\) 5.92893 + 5.92893i 0.264358 + 0.264358i 0.826822 0.562464i \(-0.190146\pi\)
−0.562464 + 0.826822i \(0.690146\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0711 16.3137i 1.02462 0.724517i
\(508\) 0 0
\(509\) −3.65685 −0.162087 −0.0810436 0.996711i \(-0.525825\pi\)
−0.0810436 + 0.996711i \(0.525825\pi\)
\(510\) 0 0
\(511\) −22.4853 −0.994690
\(512\) 0 0
\(513\) −1.17157 4.14214i −0.0517262 0.182880i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.97056 + 2.97056i 0.130645 + 0.130645i
\(518\) 0 0
\(519\) 0.272078 1.58579i 0.0119429 0.0696083i
\(520\) 0 0
\(521\) 24.0000i 1.05146i −0.850652 0.525730i \(-0.823792\pi\)
0.850652 0.525730i \(-0.176208\pi\)
\(522\) 0 0
\(523\) 26.8995 26.8995i 1.17623 1.17623i 0.195536 0.980696i \(-0.437355\pi\)
0.980696 0.195536i \(-0.0626448\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.3431 10.3431i 0.450555 0.450555i
\(528\) 0 0
\(529\) 15.9706i 0.694372i
\(530\) 0 0
\(531\) 4.00000 11.3137i 0.173585 0.490973i
\(532\) 0 0
\(533\) −21.6569 21.6569i −0.938062 0.938062i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 16.9706i −0.517838 0.732334i
\(538\) 0 0
\(539\) 3.85786 0.166170
\(540\) 0 0
\(541\) −29.3137 −1.26029 −0.630147 0.776476i \(-0.717006\pi\)
−0.630147 + 0.776476i \(0.717006\pi\)
\(542\) 0 0
\(543\) −5.31371 7.51472i −0.228033 0.322487i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.7279 + 15.7279i 0.672477 + 0.672477i 0.958287 0.285809i \(-0.0922623\pi\)
−0.285809 + 0.958287i \(0.592262\pi\)
\(548\) 0 0
\(549\) −0.343146 + 0.970563i −0.0146451 + 0.0414226i
\(550\) 0 0
\(551\) 3.02944i 0.129058i
\(552\) 0 0
\(553\) −2.00000 + 2.00000i −0.0850487 + 0.0850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.6274 15.6274i 0.662155 0.662155i −0.293733 0.955888i \(-0.594898\pi\)
0.955888 + 0.293733i \(0.0948976\pi\)
\(558\) 0 0
\(559\) 3.17157i 0.134143i
\(560\) 0 0
\(561\) −0.627417 + 3.65685i −0.0264896 + 0.154393i
\(562\) 0 0
\(563\) 1.44365 + 1.44365i 0.0608426 + 0.0608426i 0.736873 0.676031i \(-0.236301\pi\)
−0.676031 + 0.736873i \(0.736301\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.24264 + 30.5563i 0.136178 + 1.28325i
\(568\) 0 0
\(569\) −45.3137 −1.89965 −0.949825 0.312783i \(-0.898739\pi\)
−0.949825 + 0.312783i \(0.898739\pi\)
\(570\) 0 0
\(571\) 4.97056 0.208012 0.104006 0.994577i \(-0.466834\pi\)
0.104006 + 0.994577i \(0.466834\pi\)
\(572\) 0 0
\(573\) −5.85786 + 4.14214i −0.244716 + 0.173040i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −14.6569 14.6569i −0.610173 0.610173i 0.332818 0.942991i \(-0.392000\pi\)
−0.942991 + 0.332818i \(0.892000\pi\)
\(578\) 0 0
\(579\) −35.3848 6.07107i −1.47054 0.252305i
\(580\) 0 0
\(581\) 15.6569i 0.649556i
\(582\) 0 0
\(583\) −2.48528 + 2.48528i −0.102930 + 0.102930i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.5563 10.5563i 0.435707 0.435707i −0.454857 0.890564i \(-0.650310\pi\)
0.890564 + 0.454857i \(0.150310\pi\)
\(588\) 0 0
\(589\) 4.68629i 0.193095i
\(590\) 0 0
\(591\) 35.3848 + 6.07107i 1.45554 + 0.249730i
\(592\) 0 0
\(593\) −15.4853 15.4853i −0.635904 0.635904i 0.313638 0.949543i \(-0.398452\pi\)
−0.949543 + 0.313638i \(0.898452\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.1421 + 18.4853i −1.06993 + 0.756552i
\(598\) 0 0
\(599\) −41.9411 −1.71367 −0.856834 0.515592i \(-0.827572\pi\)
−0.856834 + 0.515592i \(0.827572\pi\)
\(600\) 0 0
\(601\) −14.9706 −0.610662 −0.305331 0.952246i \(-0.598767\pi\)
−0.305331 + 0.952246i \(0.598767\pi\)
\(602\) 0 0
\(603\) −38.5563 + 18.4142i −1.57014 + 0.749885i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −33.0416 33.0416i −1.34112 1.34112i −0.894948 0.446170i \(-0.852788\pi\)
−0.446170 0.894948i \(-0.647212\pi\)
\(608\) 0 0
\(609\) −3.65685 + 21.3137i −0.148183 + 0.863675i
\(610\) 0 0
\(611\) 27.4558i 1.11074i
\(612\) 0 0
\(613\) −9.48528 + 9.48528i −0.383107 + 0.383107i −0.872220 0.489113i \(-0.837320\pi\)
0.489113 + 0.872220i \(0.337320\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.1716 + 12.1716i −0.490009 + 0.490009i −0.908309 0.418300i \(-0.862626\pi\)
0.418300 + 0.908309i \(0.362626\pi\)
\(618\) 0 0
\(619\) 20.1421i 0.809581i 0.914410 + 0.404790i \(0.132656\pi\)
−0.914410 + 0.404790i \(0.867344\pi\)
\(620\) 0 0
\(621\) 28.3137 + 15.8284i 1.13619 + 0.635173i
\(622\) 0 0
\(623\) 37.7990 + 37.7990i 1.51438 + 1.51438i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.686292 + 0.970563i 0.0274078 + 0.0387605i
\(628\) 0 0
\(629\) −21.3137 −0.849833
\(630\) 0 0
\(631\) −31.5980 −1.25790 −0.628948 0.777447i \(-0.716514\pi\)
−0.628948 + 0.777447i \(0.716514\pi\)
\(632\) 0 0
\(633\) −20.9706 29.6569i −0.833505 1.17875i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −17.8284 17.8284i −0.706388 0.706388i
\(638\) 0 0
\(639\) 29.6569 + 10.4853i 1.17321 + 0.414791i
\(640\) 0 0
\(641\) 20.2843i 0.801181i −0.916257 0.400590i \(-0.868805\pi\)
0.916257 0.400590i \(-0.131195\pi\)
\(642\) 0 0
\(643\) −28.6985 + 28.6985i −1.13176 + 1.13176i −0.141873 + 0.989885i \(0.545312\pi\)
−0.989885 + 0.141873i \(0.954688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26.2132 + 26.2132i −1.03055 + 1.03055i −0.0310289 + 0.999518i \(0.509878\pi\)
−0.999518 + 0.0310289i \(0.990122\pi\)
\(648\) 0 0
\(649\) 3.31371i 0.130074i
\(650\) 0 0
\(651\) −5.65685 + 32.9706i −0.221710 + 1.29222i
\(652\) 0 0
\(653\) 26.6569 + 26.6569i 1.04316 + 1.04316i 0.999025 + 0.0441379i \(0.0140541\pi\)
0.0441379 + 0.999025i \(0.485946\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.51472 + 17.8284i 0.332191 + 0.695553i
\(658\) 0 0
\(659\) 10.6274 0.413985 0.206993 0.978342i \(-0.433632\pi\)
0.206993 + 0.978342i \(0.433632\pi\)
\(660\) 0 0
\(661\) −7.65685 −0.297817 −0.148909 0.988851i \(-0.547576\pi\)
−0.148909 + 0.988851i \(0.547576\pi\)
\(662\) 0 0
\(663\) 19.7990 14.0000i 0.768929 0.543715i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.1421 + 16.1421i 0.625026 + 0.625026i
\(668\) 0 0
\(669\) −13.4853 2.31371i −0.521371 0.0894531i
\(670\) 0 0
\(671\) 0.284271i 0.0109742i
\(672\) 0 0
\(673\) 3.68629 3.68629i 0.142096 0.142096i −0.632480 0.774576i \(-0.717963\pi\)
0.774576 + 0.632480i \(0.217963\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −35.2843 + 35.2843i −1.35608 + 1.35608i −0.477397 + 0.878688i \(0.658420\pi\)
−0.878688 + 0.477397i \(0.841580\pi\)
\(678\) 0 0
\(679\) 4.82843i 0.185298i
\(680\) 0 0
\(681\) 11.8284 + 2.02944i 0.453266 + 0.0777682i
\(682\) 0 0
\(683\) −22.5563 22.5563i −0.863095 0.863095i 0.128602 0.991696i \(-0.458951\pi\)
−0.991696 + 0.128602i \(0.958951\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −20.2843 + 14.3431i −0.773893 + 0.547225i
\(688\) 0 0
\(689\) 22.9706 0.875109
\(690\) 0 0
\(691\) 33.6569 1.28037 0.640184 0.768222i \(-0.278858\pi\)
0.640184 + 0.768222i \(0.278858\pi\)
\(692\) 0 0
\(693\) −3.65685 7.65685i −0.138912 0.290860i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10.3431 10.3431i −0.391775 0.391775i
\(698\) 0 0
\(699\) −4.89949 + 28.5563i −0.185316 + 1.08010i
\(700\) 0 0
\(701\) 4.00000i 0.151078i −0.997143 0.0755390i \(-0.975932\pi\)
0.997143 0.0755390i \(-0.0240677\pi\)
\(702\) 0 0
\(703\) −4.82843 + 4.82843i −0.182108 + 0.182108i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −23.3137 + 23.3137i −0.876802 + 0.876802i
\(708\) 0 0
\(709\) 32.2843i 1.21246i 0.795289 + 0.606231i \(0.207319\pi\)
−0.795289 + 0.606231i \(0.792681\pi\)
\(710\) 0 0
\(711\) 2.34315 + 0.828427i 0.0878748 + 0.0310684i
\(712\) 0 0
\(713\) 24.9706 + 24.9706i 0.935155 + 0.935155i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000 + 22.6274i 0.597531 + 0.845036i
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 26.9706 1.00444
\(722\) 0 0
\(723\) −0.343146 0.485281i −0.0127617 0.0180478i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −12.7574 12.7574i −0.473144 0.473144i 0.429786 0.902931i \(-0.358589\pi\)
−0.902931 + 0.429786i \(0.858589\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 1.51472i 0.0560239i
\(732\) 0 0
\(733\) 5.14214 5.14214i 0.189929 0.189929i −0.605736 0.795665i \(-0.707121\pi\)
0.795665 + 0.605736i \(0.207121\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.34315 8.34315i 0.307324 0.307324i
\(738\) 0 0
\(739\) 47.1716i 1.73523i −0.497233 0.867617i \(-0.665650\pi\)
0.497233 0.867617i \(-0.334350\pi\)
\(740\) 0 0
\(741\) 1.31371 7.65685i 0.0482603 0.281282i
\(742\) 0 0
\(743\) −37.3848 37.3848i −1.37151 1.37151i −0.858202 0.513313i \(-0.828418\pi\)
−0.513313 0.858202i \(-0.671582\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.4142 5.92893i 0.454212 0.216928i
\(748\) 0 0
\(749\) −46.2843 −1.69119
\(750\) 0 0
\(751\) 12.2843 0.448259 0.224130 0.974559i \(-0.428046\pi\)
0.224130 + 0.974559i \(0.428046\pi\)
\(752\) 0 0
\(753\) −37.4558 + 26.4853i −1.36497 + 0.965177i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −14.4558 14.4558i −0.525407 0.525407i 0.393793 0.919199i \(-0.371163\pi\)
−0.919199 + 0.393793i \(0.871163\pi\)
\(758\) 0 0
\(759\) −8.82843 1.51472i −0.320452 0.0549808i
\(760\) 0 0
\(761\) 12.6863i 0.459878i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738526\pi\)
\(762\) 0 0
\(763\) 9.65685 9.65685i 0.349602 0.349602i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.3137 15.3137i 0.552946 0.552946i
\(768\) 0 0
\(769\) 49.9411i 1.80092i −0.434936 0.900462i \(-0.643229\pi\)
0.434936 0.900462i \(-0.356771\pi\)
\(770\) 0 0
\(771\) 22.8995 + 3.92893i 0.824705 + 0.141497i
\(772\) 0 0
\(773\) 10.6569 + 10.6569i 0.383300 + 0.383300i 0.872290 0.488989i \(-0.162634\pi\)
−0.488989 + 0.872290i \(0.662634\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 39.7990 28.1421i 1.42778 1.00959i
\(778\) 0 0
\(779\) −4.68629 −0.167904
\(780\) 0 0
\(781\) −8.68629 −0.310820
\(782\) 0 0
\(783\) 18.2843 5.17157i 0.653427 0.184817i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −27.5858 27.5858i −0.983327 0.983327i 0.0165362 0.999863i \(-0.494736\pi\)
−0.999863 + 0.0165362i \(0.994736\pi\)
\(788\) 0 0
\(789\) −2.85786 + 16.6569i −0.101743 + 0.593000i
\(790\) 0 0
\(791\) 45.7990i 1.62842i
\(792\) 0 0
\(793\) −1.31371 + 1.31371i −0.0466512 + 0.0466512i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.65685 + 4.65685i −0.164954 + 0.164954i −0.784757 0.619803i \(-0.787212\pi\)
0.619803 + 0.784757i \(0.287212\pi\)
\(798\) 0 0
\(799\) 13.1127i 0.463894i
\(800\) 0 0
\(801\) 15.6569 44.2843i 0.553208 1.56471i
\(802\) 0 0
\(803\) −3.85786 3.85786i −0.136141 0.136141i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.6274 + 23.5147i 0.585313 + 0.827757i
\(808\) 0 0
\(809\) 32.3431 1.13712 0.568562 0.822640i \(-0.307500\pi\)
0.568562 + 0.822640i \(0.307500\pi\)
\(810\) 0 0
\(811\) −33.6569 −1.18185 −0.590926 0.806726i \(-0.701237\pi\)
−0.590926 + 0.806726i \(0.701237\pi\)
\(812\) 0 0
\(813\) −10.3431 14.6274i −0.362750 0.513006i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.343146 0.343146i −0.0120052 0.0120052i
\(818\) 0 0
\(819\) −18.4853 + 52.2843i −0.645928 + 1.82696i
\(820\) 0 0
\(821\) 37.9411i 1.32415i 0.749436 + 0.662077i \(0.230325\pi\)
−0.749436 + 0.662077i \(0.769675\pi\)
\(822\) 0 0
\(823\) 5.72792 5.72792i 0.199663 0.199663i −0.600193 0.799855i \(-0.704909\pi\)
0.799855 + 0.600193i \(0.204909\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.5269 27.5269i 0.957205 0.957205i −0.0419166 0.999121i \(-0.513346\pi\)
0.999121 + 0.0419166i \(0.0133464\pi\)
\(828\) 0 0
\(829\) 15.3137i 0.531867i 0.963991 + 0.265934i \(0.0856802\pi\)
−0.963991 + 0.265934i \(0.914320\pi\)
\(830\) 0 0
\(831\) −3.38478 + 19.7279i −0.117417 + 0.684354i
\(832\) 0 0
\(833\) −8.51472 8.51472i −0.295018 0.295018i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 28.2843 8.00000i 0.977647 0.276520i
\(838\) 0 0
\(839\) −12.6863 −0.437979 −0.218990 0.975727i \(-0.570276\pi\)
−0.218990 + 0.975727i \(0.570276\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) 8.00000 5.65685i 0.275535 0.194832i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −24.8995 24.8995i −0.855557 0.855557i
\(848\) 0 0
\(849\) 12.6569 + 2.17157i 0.434382 + 0.0745282i
\(850\) 0 0
\(851\) 51.4558i 1.76388i
\(852\) 0 0
\(853\) 24.4558 24.4558i 0.837352 0.837352i −0.151158 0.988510i \(-0.548300\pi\)
0.988510 + 0.151158i \(0.0483001\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 33.4853 33.4853i 1.14384 1.14384i 0.156093 0.987742i \(-0.450110\pi\)
0.987742 0.156093i \(-0.0498900\pi\)
\(858\) 0 0
\(859\) 52.4264i 1.78877i −0.447302 0.894383i \(-0.647615\pi\)
0.447302 0.894383i \(-0.352385\pi\)
\(860\) 0 0
\(861\) 32.9706 + 5.65685i 1.12363 + 0.192785i
\(862\) 0 0
\(863\) 3.58579 + 3.58579i 0.122062 + 0.122062i 0.765499 0.643437i \(-0.222492\pi\)
−0.643437 + 0.765499i \(0.722492\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −14.5858 + 10.3137i −0.495359 + 0.350272i
\(868\) 0 0
\(869\) −0.686292 −0.0232808
\(870\) 0 0
\(871\) −77.1127 −2.61286
\(872\) 0 0
\(873\) 3.82843 1.82843i 0.129573 0.0618829i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.4558 + 12.4558i 0.420604 + 0.420604i 0.885412 0.464808i \(-0.153877\pi\)
−0.464808 + 0.885412i \(0.653877\pi\)
\(878\) 0 0
\(879\) 6.89949 40.2132i 0.232714 1.35636i
\(880\) 0 0
\(881\) 8.97056i 0.302226i 0.988516 + 0.151113i \(0.0482857\pi\)
−0.988516 + 0.151113i \(0.951714\pi\)
\(882\) 0 0
\(883\) 31.5858 31.5858i 1.06295 1.06295i 0.0650653 0.997881i \(-0.479274\pi\)
0.997881 0.0650653i \(-0.0207256\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.72792 7.72792i 0.259478 0.259478i −0.565364 0.824842i \(-0.691264\pi\)
0.824842 + 0.565364i \(0.191264\pi\)
\(888\) 0 0
\(889\) 26.9706i 0.904564i
\(890\) 0 0
\(891\) −4.68629 + 5.79899i −0.156997 + 0.194273i
\(892\) 0 0
\(893\) −2.97056 2.97056i −0.0994061 0.0994061i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 33.7990 + 47.7990i 1.12852 + 1.59596i
\(898\) 0 0
\(899\) 20.6863 0.689926
\(900\) 0 0
\(901\) 10.9706 0.365482
\(902\) 0 0
\(903\) 2.00000 + 2.82843i 0.0665558 + 0.0941242i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.4142 + 28.4142i 0.943478 + 0.943478i 0.998486 0.0550075i \(-0.0175183\pi\)
−0.0550075 + 0.998486i \(0.517518\pi\)
\(908\) 0 0
\(909\) 27.3137 + 9.65685i 0.905939 + 0.320298i
\(910\) 0 0
\(911\) 40.8284i 1.35271i 0.736578 + 0.676353i \(0.236441\pi\)
−0.736578 + 0.676353i \(0.763559\pi\)
\(912\) 0 0
\(913\) −2.68629 + 2.68629i −0.0889033 + 0.0889033i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −21.3137 + 21.3137i −0.703841 + 0.703841i
\(918\) 0 0
\(919\) 8.82843i 0.291223i −0.989342 0.145611i \(-0.953485\pi\)
0.989342 0.145611i \(-0.0465149\pi\)
\(920\) 0 0
\(921\) 2.85786 16.6569i 0.0941698 0.548862i
\(922\) 0 0
\(923\) 40.1421 + 40.1421i 1.32129 + 1.32129i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −10.2132 21.3848i −0.335446 0.702368i
\(928\) 0 0
\(929\) −11.9411 −0.391776 −0.195888 0.980626i \(-0.562759\pi\)
−0.195888 + 0.980626i \(0.562759\pi\)
\(930\) 0 0
\(931\) −3.85786 −0.126436
\(932\) 0 0
\(933\) −7.79899 + 5.51472i −0.255327 + 0.180544i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.0000 + 31.0000i 1.01273 + 1.01273i 0.999918 + 0.0128079i \(0.00407699\pi\)
0.0128079 + 0.999918i \(0.495923\pi\)
\(938\) 0 0
\(939\) 5.58579 + 0.958369i 0.182285 + 0.0312752i
\(940\) 0 0
\(941\) 43.5980i 1.42125i 0.703569 + 0.710627i \(0.251589\pi\)
−0.703569 + 0.710627i \(0.748411\pi\)
\(942\) 0 0
\(943\) 24.9706 24.9706i 0.813153 0.813153i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.72792 + 1.72792i −0.0561499 + 0.0561499i −0.734624 0.678474i \(-0.762641\pi\)
0.678474 + 0.734624i \(0.262641\pi\)
\(948\) 0 0
\(949\) 35.6569i 1.15747i
\(950\) 0 0
\(951\) −11.2426 1.92893i −0.364568 0.0625499i
\(952\) 0 0
\(953\) −15.4853 15.4853i −0.501617 0.501617i 0.410323 0.911940i \(-0.365416\pi\)
−0.911940 + 0.410323i \(0.865416\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −4.28427 + 3.02944i −0.138491 + 0.0979278i
\(958\) 0 0
\(959\) 47.4558 1.53243
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 17.5269 + 36.6985i 0.564797 + 1.18259i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −18.4142 18.4142i −0.592161 0.592161i 0.346054 0.938215i \(-0.387522\pi\)
−0.938215 + 0.346054i \(0.887522\pi\)
\(968\) 0 0
\(969\) 0.627417 3.65685i 0.0201555 0.117475i
\(970\) 0 0
\(971\) 42.4853i 1.36342i 0.731624 + 0.681709i \(0.238763\pi\)
−0.731624 + 0.681709i \(0.761237\pi\)
\(972\) 0 0
\(973\) 21.3137 21.3137i 0.683286 0.683286i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.1421 23.1421i 0.740383 0.740383i −0.232269 0.972652i \(-0.574615\pi\)
0.972652 + 0.232269i \(0.0746150\pi\)
\(978\) 0 0
\(979\) 12.9706i 0.414541i
\(980\) 0 0
\(981\) −11.3137 4.00000i −0.361219 0.127710i
\(982\) 0 0
\(983\) −32.6985 32.6985i −1.04292 1.04292i −0.999037 0.0438830i \(-0.986027\pi\)
−0.0438830 0.999037i \(-0.513973\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 17.3137 + 24.4853i 0.551101 + 0.779375i
\(988\) 0 0
\(989\) 3.65685 0.116281
\(990\) 0 0
\(991\) 48.9706 1.55560 0.777801 0.628511i \(-0.216335\pi\)
0.777801 + 0.628511i \(0.216335\pi\)
\(992\) 0 0
\(993\) 9.65685 + 13.6569i 0.306451 + 0.433387i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 30.7990 + 30.7990i 0.975414 + 0.975414i 0.999705 0.0242911i \(-0.00773287\pi\)
−0.0242911 + 0.999705i \(0.507733\pi\)
\(998\) 0 0
\(999\) −37.3848 20.8995i −1.18280 0.661231i
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 1200.2.v.j.257.2 4
3.2 odd 2 1200.2.v.d.257.1 4
4.3 odd 2 600.2.r.b.257.1 4
5.2 odd 4 240.2.v.c.113.2 4
5.3 odd 4 1200.2.v.d.593.1 4
5.4 even 2 240.2.v.a.17.1 4
12.11 even 2 600.2.r.c.257.2 4
15.2 even 4 240.2.v.a.113.2 4
15.8 even 4 inner 1200.2.v.j.593.1 4
15.14 odd 2 240.2.v.c.17.2 4
20.3 even 4 600.2.r.c.593.2 4
20.7 even 4 120.2.r.b.113.1 yes 4
20.19 odd 2 120.2.r.c.17.2 yes 4
40.19 odd 2 960.2.v.a.257.1 4
40.27 even 4 960.2.v.f.833.2 4
40.29 even 2 960.2.v.i.257.2 4
40.37 odd 4 960.2.v.g.833.1 4
60.23 odd 4 600.2.r.b.593.2 4
60.47 odd 4 120.2.r.c.113.1 yes 4
60.59 even 2 120.2.r.b.17.1 4
120.29 odd 2 960.2.v.g.257.1 4
120.59 even 2 960.2.v.f.257.2 4
120.77 even 4 960.2.v.i.833.1 4
120.107 odd 4 960.2.v.a.833.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.r.b.17.1 4 60.59 even 2
120.2.r.b.113.1 yes 4 20.7 even 4
120.2.r.c.17.2 yes 4 20.19 odd 2
120.2.r.c.113.1 yes 4 60.47 odd 4
240.2.v.a.17.1 4 5.4 even 2
240.2.v.a.113.2 4 15.2 even 4
240.2.v.c.17.2 4 15.14 odd 2
240.2.v.c.113.2 4 5.2 odd 4
600.2.r.b.257.1 4 4.3 odd 2
600.2.r.b.593.2 4 60.23 odd 4
600.2.r.c.257.2 4 12.11 even 2
600.2.r.c.593.2 4 20.3 even 4
960.2.v.a.257.1 4 40.19 odd 2
960.2.v.a.833.2 4 120.107 odd 4
960.2.v.f.257.2 4 120.59 even 2
960.2.v.f.833.2 4 40.27 even 4
960.2.v.g.257.1 4 120.29 odd 2
960.2.v.g.833.1 4 40.37 odd 4
960.2.v.i.257.2 4 40.29 even 2
960.2.v.i.833.1 4 120.77 even 4
1200.2.v.d.257.1 4 3.2 odd 2
1200.2.v.d.593.1 4 5.3 odd 4
1200.2.v.j.257.2 4 1.1 even 1 trivial
1200.2.v.j.593.1 4 15.8 even 4 inner