Properties

Label 575.2.b.b.24.1
Level $575$
Weight $2$
Character 575.24
Analytic conductor $4.591$
Analytic rank $0$
Dimension $2$
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] = [575,2,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.2.b.b.24.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -1.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -3.00000i q^{18} +5.00000 q^{19} +1.00000i q^{22} -1.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +5.00000 q^{29} -2.00000 q^{31} -5.00000i q^{32} +3.00000 q^{36} -4.00000i q^{37} -5.00000i q^{38} -5.00000 q^{41} +9.00000i q^{43} -1.00000 q^{44} -1.00000 q^{46} -6.00000i q^{47} +6.00000 q^{49} -1.00000i q^{52} -2.00000i q^{53} +3.00000 q^{56} -5.00000i q^{58} -8.00000 q^{59} -8.00000 q^{61} +2.00000i q^{62} +3.00000i q^{63} -7.00000 q^{64} +8.00000i q^{67} -10.0000 q^{71} -9.00000i q^{72} +3.00000i q^{73} -4.00000 q^{74} +5.00000 q^{76} -1.00000i q^{77} +3.00000 q^{79} +9.00000 q^{81} +5.00000i q^{82} -3.00000i q^{83} +9.00000 q^{86} +3.00000i q^{88} -10.0000 q^{89} +1.00000 q^{91} -1.00000i q^{92} -6.00000 q^{94} -2.00000i q^{97} -6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} - 2 q^{16} + 10 q^{19} - 2 q^{26} + 10 q^{29} - 4 q^{31} + 6 q^{36} - 10 q^{41} - 2 q^{44} - 2 q^{46} + 12 q^{49} + 6 q^{56} - 16 q^{59} - 16 q^{61} - 14 q^{64} - 20 q^{71} - 8 q^{74} + 10 q^{76} + 6 q^{79} + 18 q^{81} + 18 q^{86} - 20 q^{89} + 2 q^{91} - 12 q^{94} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 5.00000i − 0.811107i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) − 5.00000i − 0.656532i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 2.00000i 0.254000i
\(63\) 3.00000i 0.377964i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) − 9.00000i − 1.06066i
\(73\) 3.00000i 0.351123i 0.984468 + 0.175562i \(0.0561742\pi\)
−0.984468 + 0.175562i \(0.943826\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 5.00000i 0.552158i
\(83\) − 3.00000i − 0.329293i −0.986353 0.164646i \(-0.947352\pi\)
0.986353 0.164646i \(-0.0526483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.00000 0.970495
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 17.0000i 1.67506i 0.546392 + 0.837530i \(0.316001\pi\)
−0.546392 + 0.837530i \(0.683999\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) − 3.00000i − 0.277350i
\(118\) 8.00000i 0.736460i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 8.00000i 0.724286i
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 5.00000i 0.433555i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0000i 0.839181i
\(143\) 1.00000i 0.0836242i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 3.00000 0.248282
\(147\) 0 0
\(148\) − 4.00000i − 0.328798i
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) − 15.0000i − 1.21666i
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) − 3.00000i − 0.238667i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) − 9.00000i − 0.707107i
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) 9.00000i 0.686244i
\(173\) − 15.0000i − 1.14043i −0.821496 0.570214i \(-0.806860\pi\)
0.821496 0.570214i \(-0.193140\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) − 1.00000i − 0.0741249i
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) − 17.0000i − 1.21120i −0.795769 0.605600i \(-0.792933\pi\)
0.795769 0.605600i \(-0.207067\pi\)
\(198\) 3.00000i 0.213201i
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 10.0000i − 0.703598i
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) − 3.00000i − 0.208514i
\(208\) 1.00000i 0.0693375i
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.00000i − 0.135769i
\(218\) − 4.00000i − 0.270914i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 15.0000i − 0.984798i
\(233\) 5.00000i 0.327561i 0.986497 + 0.163780i \(0.0523689\pi\)
−0.986497 + 0.163780i \(0.947631\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 10.0000i 0.642824i
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.00000i − 0.318142i
\(248\) 6.00000i 0.381000i
\(249\) 0 0
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 1.00000i 0.0628695i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 6.00000i 0.370681i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.00000 0.306570
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) − 33.0000i − 1.98278i −0.130950 0.991389i \(-0.541803\pi\)
0.130950 0.991389i \(-0.458197\pi\)
\(278\) 20.0000i 1.19952i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) − 32.0000i − 1.90220i −0.308879 0.951101i \(-0.599954\pi\)
0.308879 0.951101i \(-0.400046\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 1.00000 0.0591312
\(287\) − 5.00000i − 0.295141i
\(288\) − 15.0000i − 0.883883i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 3.00000i 0.175562i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) − 2.00000i − 0.115857i
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) − 6.00000i − 0.345261i
\(303\) 0 0
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 12.0000i 0.678280i 0.940736 + 0.339140i \(0.110136\pi\)
−0.940736 + 0.339140i \(0.889864\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) − 33.0000i − 1.85346i −0.375722 0.926732i \(-0.622605\pi\)
0.375722 0.926732i \(-0.377395\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 0 0
\(322\) − 1.00000i − 0.0557278i
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) 15.0000i 0.828236i
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) − 3.00000i − 0.164646i
\(333\) − 12.0000i − 0.657596i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) − 15.0000i − 0.811107i
\(343\) 13.0000i 0.701934i
\(344\) 27.0000 1.45574
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) 11.0000i 0.585471i 0.956193 + 0.292735i \(0.0945655\pi\)
−0.956193 + 0.292735i \(0.905434\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 6.00000i 0.317110i
\(359\) −33.0000 −1.74167 −0.870837 0.491572i \(-0.836422\pi\)
−0.870837 + 0.491572i \(0.836422\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 20.0000i 1.05118i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −15.0000 −0.780869
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) − 20.0000i − 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) − 5.00000i − 0.257513i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 15.0000i − 0.767467i
\(383\) 19.0000i 0.970855i 0.874277 + 0.485427i \(0.161336\pi\)
−0.874277 + 0.485427i \(0.838664\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 27.0000i 1.37249i
\(388\) − 2.00000i − 0.101535i
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 18.0000i − 0.909137i
\(393\) 0 0
\(394\) −17.0000 −0.856448
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) − 7.00000i − 0.350878i
\(399\) 0 0
\(400\) 0 0
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) 2.00000i 0.0996271i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 5.00000 0.248146
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −1.00000 −0.0494468 −0.0247234 0.999694i \(-0.507871\pi\)
−0.0247234 + 0.999694i \(0.507871\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 17.0000i 0.837530i
\(413\) − 8.00000i − 0.393654i
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 5.00000i 0.244558i
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 6.00000i 0.292075i
\(423\) − 18.0000i − 0.875190i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) − 32.0000i − 1.53782i −0.639356 0.768911i \(-0.720799\pi\)
0.639356 0.768911i \(-0.279201\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) − 5.00000i − 0.239182i
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 14.0000i 0.665160i 0.943075 + 0.332580i \(0.107919\pi\)
−0.943075 + 0.332580i \(0.892081\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) − 7.00000i − 0.330719i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) 0 0
\(460\) 0 0
\(461\) 7.00000 0.326023 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(462\) 0 0
\(463\) 28.0000i 1.30127i 0.759390 + 0.650635i \(0.225497\pi\)
−0.759390 + 0.650635i \(0.774503\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 5.00000 0.231621
\(467\) 33.0000i 1.52706i 0.645774 + 0.763529i \(0.276535\pi\)
−0.645774 + 0.763529i \(0.723465\pi\)
\(468\) − 3.00000i − 0.138675i
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 24.0000i 1.10469i
\(473\) − 9.00000i − 0.413820i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 10.0000i 0.457389i
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 30.0000i 1.36646i
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) 14.0000i 0.634401i 0.948359 + 0.317200i \(0.102743\pi\)
−0.948359 + 0.317200i \(0.897257\pi\)
\(488\) 24.0000i 1.08643i
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −5.00000 −0.224961
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) − 10.0000i − 0.448561i
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8.00000i 0.357057i
\(503\) 21.0000i 0.936344i 0.883637 + 0.468172i \(0.155087\pi\)
−0.883637 + 0.468172i \(0.844913\pi\)
\(504\) 9.00000 0.400892
\(505\) 0 0
\(506\) 1.00000 0.0444554
\(507\) 0 0
\(508\) 8.00000i 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 11.0000i 0.486136i
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000i 0.263880i
\(518\) − 4.00000i − 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) − 15.0000i − 0.656532i
\(523\) − 5.00000i − 0.218635i −0.994007 0.109317i \(-0.965134\pi\)
0.994007 0.109317i \(-0.0348665\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 5.00000i 0.216777i
\(533\) 5.00000i 0.216574i
\(534\) 0 0
\(535\) 0 0
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) 3.00000i 0.129339i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 31.0000 1.33279 0.666397 0.745597i \(-0.267836\pi\)
0.666397 + 0.745597i \(0.267836\pi\)
\(542\) 14.0000i 0.601351i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 18.0000i − 0.769624i −0.922995 0.384812i \(-0.874266\pi\)
0.922995 0.384812i \(-0.125734\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) 3.00000i 0.127573i
\(554\) −33.0000 −1.40204
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) − 16.0000i − 0.677942i −0.940797 0.338971i \(-0.889921\pi\)
0.940797 0.338971i \(-0.110079\pi\)
\(558\) 6.00000i 0.254000i
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) 0 0
\(562\) 30.0000i 1.26547i
\(563\) 41.0000i 1.72794i 0.503540 + 0.863972i \(0.332031\pi\)
−0.503540 + 0.863972i \(0.667969\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −32.0000 −1.34506
\(567\) 9.00000i 0.377964i
\(568\) 30.0000i 1.25877i
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 1.00000i 0.0418121i
\(573\) 0 0
\(574\) −5.00000 −0.208696
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) − 31.0000i − 1.29055i −0.763952 0.645273i \(-0.776743\pi\)
0.763952 0.645273i \(-0.223257\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 4.00000i 0.164399i
\(593\) 25.0000i 1.02663i 0.858201 + 0.513313i \(0.171582\pi\)
−0.858201 + 0.513313i \(0.828418\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) 1.00000i 0.0408930i
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 9.00000i 0.366813i
\(603\) 24.0000i 0.977356i
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) 0 0
\(607\) 2.00000i 0.0811775i 0.999176 + 0.0405887i \(0.0129233\pi\)
−0.999176 + 0.0405887i \(0.987077\pi\)
\(608\) − 25.0000i − 1.01388i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) − 28.0000i − 1.13091i −0.824779 0.565455i \(-0.808701\pi\)
0.824779 0.565455i \(-0.191299\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 28.0000i 1.12724i 0.826035 + 0.563619i \(0.190591\pi\)
−0.826035 + 0.563619i \(0.809409\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 28.0000i − 1.12270i
\(623\) − 10.0000i − 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) 12.0000 0.479616
\(627\) 0 0
\(628\) 14.0000i 0.558661i
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) − 9.00000i − 0.358001i
\(633\) 0 0
\(634\) −33.0000 −1.31060
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 5.00000i 0.197952i
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) − 31.0000i − 1.22252i −0.791430 0.611260i \(-0.790663\pi\)
0.791430 0.611260i \(-0.209337\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) − 38.0000i − 1.49393i −0.664861 0.746967i \(-0.731509\pi\)
0.664861 0.746967i \(-0.268491\pi\)
\(648\) − 27.0000i − 1.06066i
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 6.00000i 0.234978i
\(653\) 27.0000i 1.05659i 0.849060 + 0.528296i \(0.177169\pi\)
−0.849060 + 0.528296i \(0.822831\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 9.00000i 0.351123i
\(658\) − 6.00000i − 0.233904i
\(659\) 3.00000 0.116863 0.0584317 0.998291i \(-0.481390\pi\)
0.0584317 + 0.998291i \(0.481390\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) − 14.0000i − 0.544125i
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) − 5.00000i − 0.193601i
\(668\) 18.0000i 0.696441i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) − 19.0000i − 0.732396i −0.930537 0.366198i \(-0.880659\pi\)
0.930537 0.366198i \(-0.119341\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) − 12.0000i − 0.461197i −0.973049 0.230599i \(-0.925932\pi\)
0.973049 0.230599i \(-0.0740685\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) − 2.00000i − 0.0765840i
\(683\) − 2.00000i − 0.0765279i −0.999268 0.0382639i \(-0.987817\pi\)
0.999268 0.0382639i \(-0.0121828\pi\)
\(684\) 15.0000 0.573539
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) − 9.00000i − 0.343122i
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 15.0000i − 0.570214i
\(693\) − 3.00000i − 0.113961i
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 25.0000i − 0.946264i
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) − 20.0000i − 0.754314i
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 11.0000 0.413990
\(707\) 10.0000i 0.376089i
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 9.00000 0.337526
\(712\) 30.0000i 1.12430i
\(713\) 2.00000i 0.0749006i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) 33.0000i 1.23155i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −17.0000 −0.633113
\(722\) − 6.00000i − 0.223297i
\(723\) 0 0
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) − 3.00000i − 0.111187i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) − 8.00000i − 0.294684i
\(738\) 15.0000i 0.552158i
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 2.00000i − 0.0734223i
\(743\) 11.0000i 0.403551i 0.979432 + 0.201775i \(0.0646711\pi\)
−0.979432 + 0.201775i \(0.935329\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20.0000 −0.732252
\(747\) − 9.00000i − 0.329293i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 0 0
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000i 1.45382i 0.686730 + 0.726912i \(0.259045\pi\)
−0.686730 + 0.726912i \(0.740955\pi\)
\(758\) − 12.0000i − 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) −39.0000 −1.41375 −0.706874 0.707339i \(-0.749895\pi\)
−0.706874 + 0.707339i \(0.749895\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 19.0000 0.686498
\(767\) 8.00000i 0.288863i
\(768\) 0 0
\(769\) 52.0000 1.87517 0.937584 0.347759i \(-0.113057\pi\)
0.937584 + 0.347759i \(0.113057\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) − 20.0000i − 0.719350i −0.933078 0.359675i \(-0.882888\pi\)
0.933078 0.359675i \(-0.117112\pi\)
\(774\) 27.0000 0.970495
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) − 14.0000i − 0.501924i
\(779\) −25.0000 −0.895718
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) − 43.0000i − 1.53278i −0.642373 0.766392i \(-0.722050\pi\)
0.642373 0.766392i \(-0.277950\pi\)
\(788\) − 17.0000i − 0.605600i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 9.00000i 0.319801i
\(793\) 8.00000i 0.284088i
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) 8.00000i 0.283375i 0.989911 + 0.141687i \(0.0452527\pi\)
−0.989911 + 0.141687i \(0.954747\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) − 32.0000i − 1.12996i
\(803\) − 3.00000i − 0.105868i
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) − 30.0000i − 1.05540i
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 5.00000i 0.175466i
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 45.0000i 1.57435i
\(818\) 1.00000i 0.0349642i
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 1.00000 0.0349002 0.0174501 0.999848i \(-0.494445\pi\)
0.0174501 + 0.999848i \(0.494445\pi\)
\(822\) 0 0
\(823\) − 20.0000i − 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) 51.0000 1.77667
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 37.0000i 1.28662i 0.765607 + 0.643308i \(0.222439\pi\)
−0.765607 + 0.643308i \(0.777561\pi\)
\(828\) − 3.00000i − 0.104257i
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 7.00000i 0.242681i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) 0 0
\(838\) 9.00000i 0.310900i
\(839\) 9.00000 0.310715 0.155357 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) − 22.0000i − 0.758170i
\(843\) 0 0
\(844\) −6.00000 −0.206529
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) − 10.0000i − 0.343604i
\(848\) 2.00000i 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) − 37.0000i − 1.26686i −0.773802 0.633428i \(-0.781647\pi\)
0.773802 0.633428i \(-0.218353\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) − 26.0000i − 0.888143i −0.895991 0.444072i \(-0.853534\pi\)
0.895991 0.444072i \(-0.146466\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000i 0.272481i
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −32.0000 −1.08740
\(867\) 0 0
\(868\) − 2.00000i − 0.0678844i
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) − 12.0000i − 0.406371i
\(873\) − 6.00000i − 0.203069i
\(874\) −5.00000 −0.169128
\(875\) 0 0
\(876\) 0 0
\(877\) − 38.0000i − 1.28317i −0.767052 0.641584i \(-0.778277\pi\)
0.767052 0.641584i \(-0.221723\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 0 0
\(880\) 0 0
\(881\) −20.0000 −0.673817 −0.336909 0.941537i \(-0.609381\pi\)
−0.336909 + 0.941537i \(0.609381\pi\)
\(882\) − 18.0000i − 0.606092i
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 14.0000 0.470339
\(887\) 28.0000i 0.940148i 0.882627 + 0.470074i \(0.155773\pi\)
−0.882627 + 0.470074i \(0.844227\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) − 2.00000i − 0.0669650i
\(893\) − 30.0000i − 1.00391i
\(894\) 0 0
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) − 18.0000i − 0.600668i
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) 0 0
\(902\) − 5.00000i − 0.166482i
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 17.0000i 0.564476i 0.959344 + 0.282238i \(0.0910767\pi\)
−0.959344 + 0.282238i \(0.908923\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) −55.0000 −1.82223 −0.911116 0.412151i \(-0.864778\pi\)
−0.911116 + 0.412151i \(0.864778\pi\)
\(912\) 0 0
\(913\) 3.00000i 0.0992855i
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) − 6.00000i − 0.198137i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 7.00000i − 0.230533i
\(923\) 10.0000i 0.329154i
\(924\) 0 0
\(925\) 0 0
\(926\) 28.0000 0.920137
\(927\) 51.0000i 1.67506i
\(928\) − 25.0000i − 0.820665i
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 5.00000i 0.163780i
\(933\) 0 0
\(934\) 33.0000 1.07979
\(935\) 0 0
\(936\) −9.00000 −0.294174
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 5.00000i 0.162822i
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −9.00000 −0.292615
\(947\) − 22.0000i − 0.714904i −0.933932 0.357452i \(-0.883646\pi\)
0.933932 0.357452i \(-0.116354\pi\)
\(948\) 0 0
\(949\) 3.00000 0.0973841
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.0000i 0.907009i 0.891254 + 0.453504i \(0.149826\pi\)
−0.891254 + 0.453504i \(0.850174\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) − 9.00000i − 0.290777i
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000i 0.128965i
\(963\) 36.0000i 1.16008i
\(964\) −30.0000 −0.966235
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.00000i − 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) 30.0000i 0.964237i
\(969\) 0 0
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) − 20.0000i − 0.641171i
\(974\) 14.0000 0.448589
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) − 24.0000i − 0.765871i
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 5.00000i − 0.159071i
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 0 0
\(994\) −10.0000 −0.317181
\(995\) 0 0
\(996\) 0 0
\(997\) − 37.0000i − 1.17180i −0.810383 0.585901i \(-0.800741\pi\)
0.810383 0.585901i \(-0.199259\pi\)
\(998\) 10.0000i 0.316544i
\(999\) 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 575.2.b.b.24.1 2
5.2 odd 4 575.2.a.d.1.1 yes 1
5.3 odd 4 575.2.a.c.1.1 1
5.4 even 2 inner 575.2.b.b.24.2 2
15.2 even 4 5175.2.a.e.1.1 1
15.8 even 4 5175.2.a.u.1.1 1
20.3 even 4 9200.2.a.r.1.1 1
20.7 even 4 9200.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
575.2.a.c.1.1 1 5.3 odd 4
575.2.a.d.1.1 yes 1 5.2 odd 4
575.2.b.b.24.1 2 1.1 even 1 trivial
575.2.b.b.24.2 2 5.4 even 2 inner
5175.2.a.e.1.1 1 15.2 even 4
5175.2.a.u.1.1 1 15.8 even 4
9200.2.a.r.1.1 1 20.3 even 4
9200.2.a.u.1.1 1 20.7 even 4