Properties

Label 280.2.g.a.169.1
Level $280$
Weight $2$
Character 280.169
Analytic conductor $2.236$
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] = [280,2,Mod(169,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("280.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.g (of order \(2\), degree \(1\), 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: \(2.23581125660\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 280.169
Dual form 280.2.g.a.169.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +1.00000i q^{13} +(1.00000 - 2.00000i) q^{15} -3.00000i q^{17} +4.00000 q^{19} -1.00000 q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -5.00000i q^{27} +1.00000 q^{29} -6.00000 q^{31} +1.00000i q^{33} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +1.00000 q^{39} -10.0000 q^{41} +(4.00000 + 2.00000i) q^{45} +9.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} +14.0000i q^{53} +(-2.00000 - 1.00000i) q^{55} -4.00000i q^{57} -6.00000 q^{59} -4.00000 q^{61} -2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} +10.0000i q^{67} -2.00000 q^{69} -16.0000 q^{71} -10.0000i q^{73} +(4.00000 - 3.00000i) q^{75} +1.00000i q^{77} +11.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(3.00000 - 6.00000i) q^{85} -1.00000i q^{87} -12.0000 q^{89} +1.00000 q^{91} +6.00000i q^{93} +(8.00000 + 4.00000i) q^{95} -19.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 4 q^{9} - 2 q^{11} + 2 q^{15} + 8 q^{19} - 2 q^{21} + 6 q^{25} + 2 q^{29} - 12 q^{31} + 2 q^{35} + 2 q^{39} - 20 q^{41} + 8 q^{45} - 2 q^{49} - 6 q^{51} - 4 q^{55} - 12 q^{59} - 8 q^{61} - 2 q^{65} - 4 q^{69} - 32 q^{71} + 8 q^{75} + 22 q^{79} + 2 q^{81} + 6 q^{85} - 24 q^{89} + 2 q^{91} + 16 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 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 \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 1.00000 2.00000i 0.258199 0.516398i
\(16\) 0 0
\(17\) 3.00000i 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 4.00000 + 2.00000i 0.596285 + 0.298142i
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −2.00000 1.00000i −0.269680 0.134840i
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −1.00000 + 2.00000i −0.124035 + 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 4.00000 3.00000i 0.461880 0.346410i
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 3.00000 6.00000i 0.325396 0.650791i
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) 8.00000 + 4.00000i 0.820783 + 0.410391i
\(96\) 0 0
\(97\) 19.0000i 1.92916i −0.263795 0.964579i \(-0.584974\pi\)
0.263795 0.964579i \(-0.415026\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1.00000i 0.0985329i −0.998786 0.0492665i \(-0.984312\pi\)
0.998786 0.0492665i \(-0.0156884\pi\)
\(104\) 0 0
\(105\) −2.00000 1.00000i −0.195180 0.0975900i
\(106\) 0 0
\(107\) 16.0000i 1.54678i 0.633932 + 0.773389i \(0.281440\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 14.0000i 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 2.00000 4.00000i 0.186501 0.373002i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 5.00000 10.0000i 0.430331 0.860663i
\(136\) 0 0
\(137\) 8.00000i 0.683486i −0.939793 0.341743i \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 1.00000i 0.0836242i
\(144\) 0 0
\(145\) 2.00000 + 1.00000i 0.166091 + 0.0830455i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −12.0000 6.00000i −0.963863 0.481932i
\(156\) 0 0
\(157\) 14.0000i 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 10.0000i 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) −1.00000 + 2.00000i −0.0778499 + 0.155700i
\(166\) 0 0
\(167\) 15.0000i 1.16073i −0.814355 0.580367i \(-0.802909\pi\)
0.814355 0.580367i \(-0.197091\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 0 0
\(173\) 15.0000i 1.14043i 0.821496 + 0.570214i \(0.193140\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(174\) 0 0
\(175\) 4.00000 3.00000i 0.302372 0.226779i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 0 0
\(185\) −2.00000 + 4.00000i −0.147043 + 0.294086i
\(186\) 0 0
\(187\) 3.00000i 0.219382i
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 0 0
\(195\) 2.00000 + 1.00000i 0.143223 + 0.0716115i
\(196\) 0 0
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 0 0
\(205\) −20.0000 10.0000i −1.39686 0.698430i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 15.0000i 1.00447i 0.864730 + 0.502237i \(0.167490\pi\)
−0.864730 + 0.502237i \(0.832510\pi\)
\(224\) 0 0
\(225\) 6.00000 + 8.00000i 0.400000 + 0.533333i
\(226\) 0 0
\(227\) 7.00000i 0.464606i −0.972643 0.232303i \(-0.925374\pi\)
0.972643 0.232303i \(-0.0746261\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −9.00000 + 18.0000i −0.587095 + 1.17419i
\(236\) 0 0
\(237\) 11.0000i 0.714527i
\(238\) 0 0
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) −2.00000 1.00000i −0.127775 0.0638877i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) −6.00000 3.00000i −0.375735 0.187867i
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 24.0000i 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) −14.0000 + 28.0000i −0.860013 + 1.72003i
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 1.00000i 0.0605228i
\(274\) 0 0
\(275\) −3.00000 4.00000i −0.180907 0.241209i
\(276\) 0 0
\(277\) 18.0000i 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 0 0
\(283\) 21.0000i 1.24832i 0.781296 + 0.624160i \(0.214559\pi\)
−0.781296 + 0.624160i \(0.785441\pi\)
\(284\) 0 0
\(285\) 4.00000 8.00000i 0.236940 0.473879i
\(286\) 0 0
\(287\) 10.0000i 0.590281i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −19.0000 −1.11380
\(292\) 0 0
\(293\) 31.0000i 1.81104i 0.424304 + 0.905520i \(0.360519\pi\)
−0.424304 + 0.905520i \(0.639481\pi\)
\(294\) 0 0
\(295\) −12.0000 6.00000i −0.698667 0.349334i
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 4.00000i −0.458079 0.229039i
\(306\) 0 0
\(307\) 25.0000i 1.42683i −0.700744 0.713413i \(-0.747149\pi\)
0.700744 0.713413i \(-0.252851\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 7.00000i 0.395663i −0.980236 0.197832i \(-0.936610\pi\)
0.980236 0.197832i \(-0.0633900\pi\)
\(314\) 0 0
\(315\) 2.00000 4.00000i 0.112687 0.225374i
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) −4.00000 + 3.00000i −0.221880 + 0.166410i
\(326\) 0 0
\(327\) 15.0000i 0.829502i
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) −10.0000 + 20.0000i −0.546358 + 1.09272i
\(336\) 0 0
\(337\) 34.0000i 1.85210i 0.377403 + 0.926049i \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) 0 0
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −4.00000 2.00000i −0.215353 0.107676i
\(346\) 0 0
\(347\) 22.0000i 1.18102i −0.807030 0.590511i \(-0.798926\pi\)
0.807030 0.590511i \(-0.201074\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 9.00000i 0.479022i −0.970894 0.239511i \(-0.923013\pi\)
0.970894 0.239511i \(-0.0769871\pi\)
\(354\) 0 0
\(355\) −32.0000 16.0000i −1.69838 0.849192i
\(356\) 0 0
\(357\) 3.00000i 0.158777i
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 10.0000 20.0000i 0.523424 1.04685i
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 0 0
\(369\) −20.0000 −1.04116
\(370\) 0 0
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) 20.0000i 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 11.0000 2.00000i 0.568038 0.103280i
\(376\) 0 0
\(377\) 1.00000i 0.0515026i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) −1.00000 + 2.00000i −0.0509647 + 0.101929i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 22.0000 + 11.0000i 1.10694 + 0.553470i
\(396\) 0 0
\(397\) 15.0000i 0.752828i −0.926451 0.376414i \(-0.877157\pi\)
0.926451 0.376414i \(-0.122843\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) 0 0
\(405\) 2.00000 + 1.00000i 0.0993808 + 0.0496904i
\(406\) 0 0
\(407\) 2.00000i 0.0991363i
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 6.00000i 0.295241i
\(414\) 0 0
\(415\) 4.00000 8.00000i 0.196352 0.392705i
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 0 0
\(423\) 18.0000i 0.875190i
\(424\) 0 0
\(425\) 12.0000 9.00000i 0.582086 0.436564i
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 0 0
\(435\) 1.00000 2.00000i 0.0479463 0.0958927i
\(436\) 0 0
\(437\) 8.00000i 0.382692i
\(438\) 0 0
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −24.0000 12.0000i −1.13771 0.568855i
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 13.0000i 0.610793i
\(454\) 0 0
\(455\) 2.00000 + 1.00000i 0.0937614 + 0.0468807i
\(456\) 0 0
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) −6.00000 + 12.0000i −0.278243 + 0.556487i
\(466\) 0 0
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 12.0000 + 16.0000i 0.550598 + 0.734130i
\(476\) 0 0
\(477\) 28.0000i 1.28203i
\(478\) 0 0
\(479\) −2.00000 −0.0913823 −0.0456912 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) 19.0000 38.0000i 0.862746 1.72549i
\(486\) 0 0
\(487\) 38.0000i 1.72194i −0.508652 0.860972i \(-0.669856\pi\)
0.508652 0.860972i \(-0.330144\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 5.00000 0.225647 0.112823 0.993615i \(-0.464011\pi\)
0.112823 + 0.993615i \(0.464011\pi\)
\(492\) 0 0
\(493\) 3.00000i 0.135113i
\(494\) 0 0
\(495\) −4.00000 2.00000i −0.179787 0.0898933i
\(496\) 0 0
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) −23.0000 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(500\) 0 0
\(501\) −15.0000 −0.670151
\(502\) 0 0
\(503\) 29.0000i 1.29305i 0.762894 + 0.646523i \(0.223778\pi\)
−0.762894 + 0.646523i \(0.776222\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000i 0.532939i
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 20.0000i 0.883022i
\(514\) 0 0
\(515\) 1.00000 2.00000i 0.0440653 0.0881305i
\(516\) 0 0
\(517\) 9.00000i 0.395820i
\(518\) 0 0
\(519\) 15.0000 0.658427
\(520\) 0 0
\(521\) −8.00000 −0.350486 −0.175243 0.984525i \(-0.556071\pi\)
−0.175243 + 0.984525i \(0.556071\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) −3.00000 4.00000i −0.130931 0.174574i
\(526\) 0 0
\(527\) 18.0000i 0.784092i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) −16.0000 + 32.0000i −0.691740 + 1.38348i
\(536\) 0 0
\(537\) 20.0000i 0.863064i
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −41.0000 −1.76273 −0.881364 0.472438i \(-0.843374\pi\)
−0.881364 + 0.472438i \(0.843374\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 30.0000 + 15.0000i 1.28506 + 0.642529i
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 11.0000i 0.467768i
\(554\) 0 0
\(555\) 4.00000 + 2.00000i 0.169791 + 0.0848953i
\(556\) 0 0
\(557\) 36.0000i 1.52537i −0.646771 0.762684i \(-0.723881\pi\)
0.646771 0.762684i \(-0.276119\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) 0 0
\(563\) 24.0000i 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) 14.0000 28.0000i 0.588984 1.17797i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) 3.00000i 0.125327i
\(574\) 0 0
\(575\) 8.00000 6.00000i 0.333623 0.250217i
\(576\) 0 0
\(577\) 31.0000i 1.29055i 0.763952 + 0.645273i \(0.223257\pi\)
−0.763952 + 0.645273i \(0.776743\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 14.0000i 0.579821i
\(584\) 0 0
\(585\) −2.00000 + 4.00000i −0.0826898 + 0.165380i
\(586\) 0 0
\(587\) 20.0000i 0.825488i −0.910847 0.412744i \(-0.864570\pi\)
0.910847 0.412744i \(-0.135430\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 19.0000i 0.780236i −0.920765 0.390118i \(-0.872434\pi\)
0.920765 0.390118i \(-0.127566\pi\)
\(594\) 0 0
\(595\) −6.00000 3.00000i −0.245976 0.122988i
\(596\) 0 0
\(597\) 10.0000i 0.409273i
\(598\) 0 0
\(599\) −7.00000 −0.286012 −0.143006 0.989722i \(-0.545677\pi\)
−0.143006 + 0.989722i \(0.545677\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 20.0000i 0.814463i
\(604\) 0 0
\(605\) −20.0000 10.0000i −0.813116 0.406558i
\(606\) 0 0
\(607\) 1.00000i 0.0405887i −0.999794 0.0202944i \(-0.993540\pi\)
0.999794 0.0202944i \(-0.00646034\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 4.00000i 0.161558i 0.996732 + 0.0807792i \(0.0257409\pi\)
−0.996732 + 0.0807792i \(0.974259\pi\)
\(614\) 0 0
\(615\) −10.0000 + 20.0000i −0.403239 + 0.806478i
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) −10.0000 −0.401286
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 0 0
\(633\) 1.00000i 0.0397464i
\(634\) 0 0
\(635\) −18.0000 + 36.0000i −0.714308 + 1.42862i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) −32.0000 −1.26590
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 15.0000i 0.591542i 0.955259 + 0.295771i \(0.0955766\pi\)
−0.955259 + 0.295771i \(0.904423\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000i 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 12.0000 + 6.00000i 0.468879 + 0.234439i
\(656\) 0 0
\(657\) 20.0000i 0.780274i
\(658\) 0 0
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 0 0
\(663\) 3.00000i 0.116510i
\(664\) 0 0
\(665\) 4.00000 8.00000i 0.155113 0.310227i
\(666\) 0 0
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) 15.0000 0.579934
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 20.0000i 0.770943i 0.922720 + 0.385472i \(0.125961\pi\)
−0.922720 + 0.385472i \(0.874039\pi\)
\(674\) 0 0
\(675\) 20.0000 15.0000i 0.769800 0.577350i
\(676\) 0 0
\(677\) 3.00000i 0.115299i 0.998337 + 0.0576497i \(0.0183606\pi\)
−0.998337 + 0.0576497i \(0.981639\pi\)
\(678\) 0 0
\(679\) −19.0000 −0.729153
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 8.00000 16.0000i 0.305664 0.611329i
\(686\) 0 0
\(687\) 18.0000i 0.686743i
\(688\) 0 0
\(689\) −14.0000 −0.533358
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 28.0000 + 14.0000i 1.06210 + 0.531050i
\(696\) 0 0
\(697\) 30.0000i 1.13633i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −41.0000 −1.54855 −0.774274 0.632850i \(-0.781885\pi\)
−0.774274 + 0.632850i \(0.781885\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 18.0000 + 9.00000i 0.677919 + 0.338960i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 22.0000 0.825064
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 1.00000 2.00000i 0.0373979 0.0747958i
\(716\) 0 0
\(717\) 17.0000i 0.634877i
\(718\) 0 0
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 3.00000 + 4.00000i 0.111417 + 0.148556i
\(726\) 0 0
\(727\) 36.0000i 1.33517i −0.744535 0.667583i \(-0.767329\pi\)
0.744535 0.667583i \(-0.232671\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 9.00000i 0.332423i 0.986090 + 0.166211i \(0.0531534\pi\)
−0.986090 + 0.166211i \(0.946847\pi\)
\(734\) 0 0
\(735\) −1.00000 + 2.00000i −0.0368856 + 0.0737711i
\(736\) 0 0
\(737\) 10.0000i 0.368355i
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 42.0000i 1.54083i −0.637542 0.770415i \(-0.720049\pi\)
0.637542 0.770415i \(-0.279951\pi\)
\(744\) 0 0
\(745\) 12.0000 + 6.00000i 0.439646 + 0.219823i
\(746\) 0 0
\(747\) 8.00000i 0.292705i
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) 0 0
\(753\) 10.0000i 0.364420i
\(754\) 0 0
\(755\) 26.0000 + 13.0000i 0.946237 + 0.473118i
\(756\) 0 0
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 0 0
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 15.0000i 0.543036i
\(764\) 0 0
\(765\) 6.00000 12.0000i 0.216930 0.433861i
\(766\) 0 0
\(767\) 6.00000i 0.216647i
\(768\) 0 0
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 13.0000i 0.467578i 0.972287 + 0.233789i \(0.0751124\pi\)
−0.972287 + 0.233789i \(0.924888\pi\)
\(774\) 0 0
\(775\) −18.0000 24.0000i −0.646579 0.862105i
\(776\) 0 0
\(777\) 2.00000i 0.0717496i
\(778\) 0 0
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 5.00000i 0.178685i
\(784\) 0 0
\(785\) 14.0000 28.0000i 0.499681 0.999363i
\(786\) 0 0
\(787\) 31.0000i 1.10503i −0.833503 0.552515i \(-0.813668\pi\)
0.833503 0.552515i \(-0.186332\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) 28.0000 + 14.0000i 0.993058 + 0.496529i
\(796\) 0 0
\(797\) 27.0000i 0.956389i −0.878254 0.478195i \(-0.841291\pi\)
0.878254 0.478195i \(-0.158709\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) −24.0000 −0.847998
\(802\) 0 0
\(803\) 10.0000i 0.352892i
\(804\) 0 0
\(805\) −4.00000 2.00000i −0.140981 0.0704907i
\(806\) 0 0
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) 51.0000 1.79306 0.896532 0.442978i \(-0.146078\pi\)
0.896532 + 0.442978i \(0.146078\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) 0 0
\(813\) 4.00000i 0.140286i
\(814\) 0 0
\(815\) 10.0000 20.0000i 0.350285 0.700569i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 0 0
\(823\) 10.0000i 0.348578i 0.984695 + 0.174289i \(0.0557627\pi\)
−0.984695 + 0.174289i \(0.944237\pi\)
\(824\) 0 0
\(825\) −4.00000 + 3.00000i −0.139262 + 0.104447i
\(826\) 0 0
\(827\) 42.0000i 1.46048i −0.683189 0.730242i \(-0.739408\pi\)
0.683189 0.730242i \(-0.260592\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 3.00000i 0.103944i
\(834\) 0 0
\(835\) 15.0000 30.0000i 0.519096 1.03819i
\(836\) 0 0
\(837\) 30.0000i 1.03695i
\(838\) 0 0
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 15.0000i 0.516627i
\(844\) 0 0
\(845\) 24.0000 + 12.0000i 0.825625 + 0.412813i
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) 21.0000 0.720718
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 14.0000i 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) 16.0000 + 8.00000i 0.547188 + 0.273594i
\(856\) 0 0
\(857\) 10.0000i 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) 6.00000i 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) −15.0000 + 30.0000i −0.510015 + 1.02003i
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 38.0000i 1.28611i
\(874\) 0 0
\(875\) 11.0000 2.00000i 0.371868 0.0676123i
\(876\) 0 0
\(877\) 52.0000i 1.75592i 0.478738 + 0.877958i \(0.341094\pi\)
−0.478738 + 0.877958i \(0.658906\pi\)
\(878\) 0 0
\(879\) 31.0000 1.04560
\(880\) 0 0
\(881\) 16.0000 0.539054 0.269527 0.962993i \(-0.413133\pi\)
0.269527 + 0.962993i \(0.413133\pi\)
\(882\) 0 0
\(883\) 16.0000i 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 0 0
\(885\) −6.00000 + 12.0000i −0.201688 + 0.403376i
\(886\) 0 0
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 36.0000i 1.20469i
\(894\) 0 0
\(895\) −40.0000 20.0000i −1.33705 0.668526i
\(896\) 0 0
\(897\) 2.00000i 0.0667781i
\(898\) 0 0
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 42.0000 1.39922
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 2.00000i −0.132964 0.0664822i
\(906\) 0 0
\(907\) 14.0000i 0.464862i 0.972613 + 0.232431i \(0.0746680\pi\)
−0.972613 + 0.232431i \(0.925332\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 4.00000i 0.132381i
\(914\) 0 0
\(915\) −4.00000 + 8.00000i −0.132236 + 0.264472i
\(916\) 0 0
\(917\) 6.00000i 0.198137i
\(918\) 0 0
\(919\) 27.0000 0.890648 0.445324 0.895370i \(-0.353089\pi\)
0.445324 + 0.895370i \(0.353089\pi\)
\(920\) 0 0
\(921\) −25.0000 −0.823778
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) −8.00000 + 6.00000i −0.263038 + 0.197279i
\(926\) 0 0
\(927\) 2.00000i 0.0656886i
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.00000 + 6.00000i −0.0981105 + 0.196221i
\(936\) 0 0
\(937\) 1.00000i 0.0326686i 0.999867 + 0.0163343i \(0.00519960\pi\)
−0.999867 + 0.0163343i \(0.994800\pi\)
\(938\) 0 0
\(939\) −7.00000 −0.228436
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 0 0
\(943\) 20.0000i 0.651290i
\(944\) 0 0
\(945\) −10.0000 5.00000i −0.325300 0.162650i
\(946\) 0 0
\(947\) 30.0000i 0.974869i 0.873160 + 0.487435i \(0.162067\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 30.0000i 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 0 0
\(955\) 6.00000 + 3.00000i 0.194155 + 0.0970777i
\(956\) 0 0
\(957\) 1.00000i 0.0323254i
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 32.0000i 1.03119i
\(964\) 0 0
\(965\) 4.00000 8.00000i 0.128765 0.257529i
\(966\) 0 0
\(967\) 32.0000i 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) 0 0
\(975\) 3.00000 + 4.00000i 0.0960769 + 0.128103i
\(976\) 0 0
\(977\) 40.0000i 1.27971i 0.768494 + 0.639857i \(0.221006\pi\)
−0.768494 + 0.639857i \(0.778994\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) 1.00000i 0.0318950i 0.999873 + 0.0159475i \(0.00507647\pi\)
−0.999873 + 0.0159475i \(0.994924\pi\)
\(984\) 0 0
\(985\) 6.00000 12.0000i 0.191176 0.382352i
\(986\) 0 0
\(987\) 9.00000i 0.286473i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) −20.0000 10.0000i −0.634043 0.317021i
\(996\) 0 0
\(997\) 27.0000i 0.855099i −0.903992 0.427549i \(-0.859377\pi\)
0.903992 0.427549i \(-0.140623\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 280.2.g.a.169.1 2
3.2 odd 2 2520.2.t.a.1009.1 2
4.3 odd 2 560.2.g.d.449.2 2
5.2 odd 4 1400.2.a.d.1.1 1
5.3 odd 4 1400.2.a.j.1.1 1
5.4 even 2 inner 280.2.g.a.169.2 yes 2
7.6 odd 2 1960.2.g.a.1569.2 2
8.3 odd 2 2240.2.g.a.449.1 2
8.5 even 2 2240.2.g.b.449.2 2
12.11 even 2 5040.2.t.a.1009.1 2
15.14 odd 2 2520.2.t.a.1009.2 2
20.3 even 4 2800.2.a.k.1.1 1
20.7 even 4 2800.2.a.u.1.1 1
20.19 odd 2 560.2.g.d.449.1 2
35.13 even 4 9800.2.a.p.1.1 1
35.27 even 4 9800.2.a.bb.1.1 1
35.34 odd 2 1960.2.g.a.1569.1 2
40.19 odd 2 2240.2.g.a.449.2 2
40.29 even 2 2240.2.g.b.449.1 2
60.59 even 2 5040.2.t.a.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.a.169.1 2 1.1 even 1 trivial
280.2.g.a.169.2 yes 2 5.4 even 2 inner
560.2.g.d.449.1 2 20.19 odd 2
560.2.g.d.449.2 2 4.3 odd 2
1400.2.a.d.1.1 1 5.2 odd 4
1400.2.a.j.1.1 1 5.3 odd 4
1960.2.g.a.1569.1 2 35.34 odd 2
1960.2.g.a.1569.2 2 7.6 odd 2
2240.2.g.a.449.1 2 8.3 odd 2
2240.2.g.a.449.2 2 40.19 odd 2
2240.2.g.b.449.1 2 40.29 even 2
2240.2.g.b.449.2 2 8.5 even 2
2520.2.t.a.1009.1 2 3.2 odd 2
2520.2.t.a.1009.2 2 15.14 odd 2
2800.2.a.k.1.1 1 20.3 even 4
2800.2.a.u.1.1 1 20.7 even 4
5040.2.t.a.1009.1 2 12.11 even 2
5040.2.t.a.1009.2 2 60.59 even 2
9800.2.a.p.1.1 1 35.13 even 4
9800.2.a.bb.1.1 1 35.27 even 4