Properties

Label 630.2.g.b.379.2
Level $630$
Weight $2$
Character 630.379
Analytic conductor $5.031$
Analytic rank $1$
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] = [630,2,Mod(379,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("630.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.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: \(5.03057532734\)
Analytic rank: \(1\)
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 379.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 630.379
Dual form 630.2.g.b.379.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(-2.00000 - 1.00000i) q^{10} -6.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -4.00000 q^{19} +(1.00000 - 2.00000i) q^{20} -6.00000i q^{22} +4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +2.00000 q^{26} +1.00000i q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(2.00000 + 1.00000i) q^{35} -10.0000i q^{37} -4.00000i q^{38} +(2.00000 + 1.00000i) q^{40} -6.00000 q^{41} +2.00000i q^{43} +6.00000 q^{44} -4.00000 q^{46} +2.00000i q^{47} -1.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +2.00000i q^{52} -6.00000i q^{53} +(6.00000 - 12.0000i) q^{55} -1.00000 q^{56} +2.00000i q^{58} -4.00000 q^{59} -12.0000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +(4.00000 + 2.00000i) q^{65} +10.0000i q^{67} +2.00000i q^{68} +(-1.00000 + 2.00000i) q^{70} -12.0000 q^{71} +2.00000i q^{73} +10.0000 q^{74} +4.00000 q^{76} +6.00000i q^{77} +16.0000 q^{79} +(-1.00000 + 2.00000i) q^{80} -6.00000i q^{82} +12.0000i q^{83} +(4.00000 + 2.00000i) q^{85} -2.00000 q^{86} +6.00000i q^{88} +14.0000 q^{89} -2.00000 q^{91} -4.00000i q^{92} -2.00000 q^{94} +(4.00000 - 8.00000i) q^{95} +18.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} - 4 q^{10} - 12 q^{11} + 2 q^{14} + 2 q^{16} - 8 q^{19} + 2 q^{20} - 6 q^{25} + 4 q^{26} + 4 q^{29} - 4 q^{31} + 4 q^{34} + 4 q^{35} + 4 q^{40} - 12 q^{41} + 12 q^{44} - 8 q^{46} - 2 q^{49} + 8 q^{50} + 12 q^{55} - 2 q^{56} - 8 q^{59} - 24 q^{61} - 2 q^{64} + 8 q^{65} - 2 q^{70} - 24 q^{71} + 20 q^{74} + 8 q^{76} + 32 q^{79} - 2 q^{80} + 8 q^{85} - 4 q^{86} + 28 q^{89} - 4 q^{91} - 4 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −2.00000 1.00000i −0.632456 0.316228i
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 2.00000i 0.223607 0.447214i
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 + 1.00000i 0.338062 + 0.169031i
\(36\) 0 0
\(37\) 10.0000i 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 2.00000 + 1.00000i 0.316228 + 0.158114i
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 4.00000 3.00000i 0.565685 0.424264i
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 6.00000 12.0000i 0.809040 1.61808i
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000i 0.262613i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 2.00000i 0.254000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 4.00000 + 2.00000i 0.496139 + 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) −1.00000 + 2.00000i −0.119523 + 0.239046i
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −1.00000 + 2.00000i −0.111803 + 0.223607i
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 4.00000 + 2.00000i 0.433861 + 0.216930i
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 6.00000i 0.639602i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 4.00000 8.00000i 0.410391 0.820783i
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 3.00000 + 4.00000i 0.300000 + 0.400000i
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000i 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 12.0000 + 6.00000i 1.14416 + 0.572078i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 14.0000i 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) −8.00000 4.00000i −0.746004 0.373002i
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 12.0000i 1.08643i
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 4.00000i 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −2.00000 + 4.00000i −0.175412 + 0.350823i
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −2.00000 1.00000i −0.169031 0.0845154i
\(141\) 0 0
\(142\) 12.0000i 1.00702i
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) −2.00000 + 4.00000i −0.166091 + 0.332182i
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 10.0000i 0.821995i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 2.00000 4.00000i 0.160644 0.321288i
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 0 0
\(160\) −2.00000 1.00000i −0.158114 0.0790569i
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 6.00000i 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) −2.00000 + 4.00000i −0.153393 + 0.306786i
\(171\) 0 0
\(172\) 2.00000i 0.152499i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 14.0000i 1.04934i
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 20.0000 + 10.0000i 1.47043 + 0.735215i
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 2.00000i 0.145865i
\(189\) 0 0
\(190\) 8.00000 + 4.00000i 0.580381 + 0.290191i
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −4.00000 + 3.00000i −0.282843 + 0.212132i
\(201\) 0 0
\(202\) 6.00000i 0.422159i
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 6.00000 12.0000i 0.419058 0.838116i
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −4.00000 2.00000i −0.272798 0.136399i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 14.0000i 0.948200i
\(219\) 0 0
\(220\) −6.00000 + 12.0000i −0.404520 + 0.809040i
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 8.00000i 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 24.0000i 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 4.00000 8.00000i 0.263752 0.527504i
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 22.0000i 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) 0 0
\(235\) −4.00000 2.00000i −0.260931 0.130466i
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) 1.00000 2.00000i 0.0638877 0.127775i
\(246\) 0 0
\(247\) 8.00000i 0.509028i
\(248\) 2.00000i 0.127000i
\(249\) 0 0
\(250\) 2.00000 + 11.0000i 0.126491 + 0.695701i
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) −4.00000 2.00000i −0.248069 0.124035i
\(261\) 0 0
\(262\) 20.0000i 1.23560i
\(263\) 12.0000i 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 12.0000 + 6.00000i 0.737154 + 0.368577i
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 10.0000i 0.610847i
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −30.0000 −1.82237 −0.911185 0.411997i \(-0.864831\pi\)
−0.911185 + 0.411997i \(0.864831\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 18.0000 + 24.0000i 1.08544 + 1.44725i
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 0 0
\(280\) 1.00000 2.00000i 0.0597614 0.119523i
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) −4.00000 2.00000i −0.234888 0.117444i
\(291\) 0 0
\(292\) 2.00000i 0.117041i
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 4.00000 8.00000i 0.232889 0.465778i
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 6.00000i 0.347571i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 12.0000i 0.690522i
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 12.0000 24.0000i 0.687118 1.37424i
\(306\) 0 0
\(307\) 32.0000i 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 0 0
\(310\) 4.00000 + 2.00000i 0.227185 + 0.113592i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 34.0000i 1.92179i −0.276907 0.960897i \(-0.589309\pi\)
0.276907 0.960897i \(-0.410691\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 26.0000i 1.46031i −0.683284 0.730153i \(-0.739449\pi\)
0.683284 0.730153i \(-0.260551\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 1.00000 2.00000i 0.0559017 0.111803i
\(321\) 0 0
\(322\) 4.00000i 0.222911i
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) −8.00000 + 6.00000i −0.443760 + 0.332820i
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) −20.0000 10.0000i −1.09272 0.546358i
\(336\) 0 0
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) −4.00000 2.00000i −0.216930 0.108465i
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000i 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) −3.00000 4.00000i −0.160357 0.213809i
\(351\) 0 0
\(352\) 6.00000i 0.319801i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 0 0
\(355\) 12.0000 24.0000i 0.636894 1.27379i
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 14.0000i 0.739923i
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 8.00000i 0.420471i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −4.00000 2.00000i −0.209370 0.104685i
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) −10.0000 + 20.0000i −0.519875 + 1.03975i
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 34.0000i 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.00000 + 8.00000i −0.205196 + 0.410391i
\(381\) 0 0
\(382\) 16.0000i 0.818631i
\(383\) 30.0000i 1.53293i −0.642287 0.766464i \(-0.722014\pi\)
0.642287 0.766464i \(-0.277986\pi\)
\(384\) 0 0
\(385\) −12.0000 6.00000i −0.611577 0.305788i
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) 18.0000i 0.913812i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) −16.0000 + 32.0000i −0.805047 + 1.61009i
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 18.0000i 0.902258i
\(399\) 0 0
\(400\) −3.00000 4.00000i −0.150000 0.200000i
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 60.0000i 2.97409i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 12.0000 + 6.00000i 0.592638 + 0.296319i
\(411\) 0 0
\(412\) 16.0000i 0.788263i
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) −24.0000 12.0000i −1.17811 0.589057i
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 24.0000i 1.17388i
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −8.00000 + 6.00000i −0.388057 + 0.291043i
\(426\) 0 0
\(427\) 12.0000i 0.580721i
\(428\) 4.00000i 0.193347i
\(429\) 0 0
\(430\) 2.00000 4.00000i 0.0964486 0.192897i
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −12.0000 6.00000i −0.572078 0.286039i
\(441\) 0 0
\(442\) 4.00000i 0.190261i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) −14.0000 + 28.0000i −0.663664 + 1.32733i
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 14.0000i 0.658505i
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 2.00000 4.00000i 0.0937614 0.187523i
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 8.00000i 0.373815i
\(459\) 0 0
\(460\) 8.00000 + 4.00000i 0.373002 + 0.186501i
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 20.0000i 0.925490i −0.886492 0.462745i \(-0.846865\pi\)
0.886492 0.462745i \(-0.153135\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 2.00000 4.00000i 0.0922531 0.184506i
\(471\) 0 0
\(472\) 4.00000i 0.184115i
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 12.0000 + 16.0000i 0.550598 + 0.734130i
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 18.0000i 0.819878i
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) −36.0000 18.0000i −1.63468 0.817338i
\(486\) 0 0
\(487\) 4.00000i 0.181257i −0.995885 0.0906287i \(-0.971112\pi\)
0.995885 0.0906287i \(-0.0288876\pi\)
\(488\) 12.0000i 0.543214i
\(489\) 0 0
\(490\) 2.00000 + 1.00000i 0.0903508 + 0.0451754i
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) 4.00000i 0.180151i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −11.0000 + 2.00000i −0.491935 + 0.0894427i
\(501\) 0 0
\(502\) 16.0000i 0.714115i
\(503\) 10.0000i 0.445878i −0.974832 0.222939i \(-0.928435\pi\)
0.974832 0.222939i \(-0.0715651\pi\)
\(504\) 0 0
\(505\) 6.00000 12.0000i 0.266996 0.533993i
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) 4.00000i 0.177471i
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −32.0000 16.0000i −1.41009 0.705044i
\(516\) 0 0
\(517\) 12.0000i 0.527759i
\(518\) 10.0000i 0.439375i
\(519\) 0 0
\(520\) 2.00000 4.00000i 0.0877058 0.175412i
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) −6.00000 + 12.0000i −0.260623 + 0.521247i
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 8.00000 + 4.00000i 0.345870 + 0.172935i
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 14.0000i 0.603583i
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 30.0000i 1.28861i
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 14.0000 28.0000i 0.599694 1.19939i
\(546\) 0 0
\(547\) 26.0000i 1.11168i 0.831289 + 0.555840i \(0.187603\pi\)
−0.831289 + 0.555840i \(0.812397\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 0 0
\(550\) −24.0000 + 18.0000i −1.02336 + 0.767523i
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 38.0000i 1.61011i 0.593199 + 0.805056i \(0.297865\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 2.00000 + 1.00000i 0.0845154 + 0.0422577i
\(561\) 0 0
\(562\) 12.0000i 0.506189i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 28.0000 + 14.0000i 1.17797 + 0.588984i
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 12.0000i 0.501745i
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 16.0000 12.0000i 0.667246 0.500435i
\(576\) 0 0
\(577\) 14.0000i 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 2.00000 4.00000i 0.0830455 0.166091i
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 8.00000 + 4.00000i 0.329355 + 0.164677i
\(591\) 0 0
\(592\) 10.0000i 0.410997i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 2.00000 4.00000i 0.0819920 0.163984i
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 2.00000i 0.0815139i
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) −25.0000 + 50.0000i −1.01639 + 2.03279i
\(606\) 0 0
\(607\) 8.00000i 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 24.0000 + 12.0000i 0.971732 + 0.485866i
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 26.0000i 1.04672i 0.852111 + 0.523360i \(0.175322\pi\)
−0.852111 + 0.523360i \(0.824678\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −2.00000 + 4.00000i −0.0803219 + 0.160644i
\(621\) 0 0
\(622\) 0 0
\(623\) 14.0000i 0.560898i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) 18.0000i 0.718278i
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 16.0000i 0.636446i
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) 8.00000 + 4.00000i 0.317470 + 0.158735i
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 12.0000i 0.475085i
\(639\) 0 0
\(640\) 2.00000 + 1.00000i 0.0790569 + 0.0395285i
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 42.0000i 1.65119i −0.564263 0.825595i \(-0.690840\pi\)
0.564263 0.825595i \(-0.309160\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) −6.00000 8.00000i −0.235339 0.313786i
\(651\) 0 0
\(652\) 6.00000i 0.234978i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 20.0000 40.0000i 0.781465 1.56293i
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 2.00000i 0.0779681i
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −8.00000 4.00000i −0.310227 0.155113i
\(666\) 0 0
\(667\) 8.00000i 0.309761i
\(668\) 18.0000i 0.696441i
\(669\) 0 0
\(670\) 10.0000 20.0000i 0.386334 0.772667i
\(671\) 72.0000 2.77953
\(672\) 0 0
\(673\) 16.0000i 0.616755i −0.951264 0.308377i \(-0.900214\pi\)
0.951264 0.308377i \(-0.0997859\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 8.00000i 0.307465i −0.988113 0.153732i \(-0.950871\pi\)
0.988113 0.153732i \(-0.0491294\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 2.00000 4.00000i 0.0766965 0.153393i
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) 28.0000i 1.07139i −0.844411 0.535695i \(-0.820050\pi\)
0.844411 0.535695i \(-0.179950\pi\)
\(684\) 0 0
\(685\) 4.00000 + 2.00000i 0.152832 + 0.0764161i
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 2.00000i 0.0762493i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −8.00000 + 16.0000i −0.303457 + 0.606915i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 24.0000i 0.908413i
\(699\) 0 0
\(700\) 4.00000 3.00000i 0.151186 0.113389i
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 24.0000 + 12.0000i 0.900704 + 0.450352i
\(711\) 0 0
\(712\) 14.0000i 0.524672i
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) −24.0000 12.0000i −0.897549 0.448775i
\(716\) 14.0000 0.523205
\(717\) 0 0
\(718\) 4.00000i 0.149279i
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 3.00000i 0.111648i
\(723\) 0 0
\(724\) −8.00000 −0.297318
\(725\) −6.00000 8.00000i −0.222834 0.297113i
\(726\) 0 0
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) 0 0
\(730\) 2.00000 4.00000i 0.0740233 0.148047i
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 60.0000i 2.21013i
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −20.0000 10.0000i −0.735215 0.367607i
\(741\) 0 0
\(742\) 6.00000i 0.220267i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 6.00000 12.0000i 0.219823 0.439646i
\(746\) 34.0000 1.24483
\(747\) 0 0
\(748\) 12.0000i 0.438763i
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 2.00000i 0.0729325i
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −12.0000 + 24.0000i −0.436725 + 0.873449i
\(756\) 0 0
\(757\) 30.0000i 1.09037i −0.838316 0.545184i \(-0.816460\pi\)
0.838316 0.545184i \(-0.183540\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) −8.00000 4.00000i −0.290191 0.145095i
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) 8.00000i 0.288863i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 6.00000 12.0000i 0.216225 0.432450i
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 0 0
\(775\) 6.00000 + 8.00000i 0.215526 + 0.287368i
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 72.0000 2.57636
\(782\) 8.00000i 0.286079i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −36.0000 18.0000i −1.28490 0.642448i
\(786\) 0 0
\(787\) 48.0000i 1.71102i −0.517790 0.855508i \(-0.673245\pi\)
0.517790 0.855508i \(-0.326755\pi\)
\(788\) 10.0000i 0.356235i
\(789\) 0 0
\(790\) −32.0000 16.0000i −1.13851 0.569254i
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 24.0000i 0.852265i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 40.0000i 1.41687i 0.705775 + 0.708436i \(0.250599\pi\)
−0.705775 + 0.708436i \(0.749401\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 4.00000 3.00000i 0.141421 0.106066i
\(801\) 0 0
\(802\) 24.0000i 0.847469i
\(803\) 12.0000i 0.423471i
\(804\) 0 0
\(805\) −4.00000 + 8.00000i −0.140981 + 0.281963i
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) 52.0000 1.82822 0.914111 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) −60.0000 −2.10300
\(815\) 12.0000 + 6.00000i 0.420342 + 0.210171i
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) −6.00000 + 12.0000i −0.209529 + 0.419058i
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 24.0000i 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 4.00000i 0.139094i −0.997579 0.0695468i \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 12.0000 24.0000i 0.416526 0.833052i
\(831\) 0 0
\(832\) 2.00000i 0.0693375i
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) −36.0000 18.0000i −1.24583 0.622916i
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 16.0000i 0.552711i
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000i 0.0689246i
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) −9.00000 + 18.0000i −0.309609 + 0.619219i
\(846\) 0 0
\(847\) 25.0000i 0.859010i
\(848\) 6.00000i 0.206041i
\(849\) 0 0
\(850\) −6.00000 8.00000i −0.205798 0.274398i
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 22.0000i 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 34.0000i 1.16142i −0.814111 0.580709i \(-0.802775\pi\)
0.814111 0.580709i \(-0.197225\pi\)
\(858\) 0 0
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 4.00000 + 2.00000i 0.136399 + 0.0681994i
\(861\) 0 0
\(862\) 4.00000i 0.136241i
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 2.00000i 0.0678844i
\(869\) −96.0000 −3.25658
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 14.0000i 0.474100i
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) −2.00000 11.0000i −0.0676123 0.371868i
\(876\) 0 0
\(877\) 22.0000i 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 10.0000i 0.337484i
\(879\) 0 0
\(880\) 6.00000 12.0000i 0.202260 0.404520i
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 46.0000i 1.54802i 0.633171 + 0.774012i \(0.281753\pi\)
−0.633171 + 0.774012i \(0.718247\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 42.0000i 1.41022i −0.709097 0.705111i \(-0.750897\pi\)
0.709097 0.705111i \(-0.249103\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) −28.0000 14.0000i −0.938562 0.469281i
\(891\) 0 0
\(892\) 8.00000i 0.267860i
\(893\) 8.00000i 0.267710i
\(894\) 0 0
\(895\) 14.0000 28.0000i 0.467968 0.935937i
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 20.0000i 0.667409i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 36.0000i 1.19867i
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −8.00000 + 16.0000i −0.265929 + 0.531858i
\(906\) 0 0
\(907\) 38.0000i 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 24.0000i 0.796468i
\(909\) 0 0
\(910\) 4.00000 + 2.00000i 0.132599 + 0.0662994i
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 72.0000i 2.38285i
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 20.0000i 0.660458i
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) −4.00000 + 8.00000i −0.131876 + 0.263752i
\(921\) 0 0
\(922\) 34.0000i 1.11973i
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) −40.0000 + 30.0000i −1.31519 + 0.986394i
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 2.00000i 0.0656532i
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 22.0000i 0.720634i
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) −24.0000 12.0000i −0.784884 0.392442i
\(936\) 0 0
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 10.0000i 0.326512i
\(939\) 0 0
\(940\) 4.00000 + 2.00000i 0.130466 + 0.0652328i
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.781548i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −16.0000 + 12.0000i −0.519109 + 0.389331i
\(951\) 0 0
\(952\) 2.00000i 0.0648204i
\(953\) 42.0000i 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 16.0000 32.0000i 0.517748 1.03550i
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 20.0000i 0.644826i
\(963\) 0 0
\(964\) 18.0000 0.579741
\(965\) 8.00000 + 4.00000i 0.257529 + 0.128765i
\(966\) 0 0
\(967\) 56.0000i 1.80084i −0.435023 0.900419i \(-0.643260\pi\)
0.435023 0.900419i \(-0.356740\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 0 0
\(970\) 18.0000 36.0000i 0.577945 1.15589i
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) −84.0000 −2.68465
\(980\) −1.00000 + 2.00000i −0.0319438 + 0.0638877i
\(981\) 0 0
\(982\) 6.00000i 0.191468i
\(983\) 14.0000i 0.446531i 0.974758 + 0.223265i \(0.0716716\pi\)
−0.974758 + 0.223265i \(0.928328\pi\)
\(984\) 0 0
\(985\) −20.0000 10.0000i −0.637253 0.318626i
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 2.00000i 0.0635001i
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 18.0000 36.0000i 0.570638 1.14128i
\(996\) 0 0
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 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 630.2.g.b.379.2 yes 2
3.2 odd 2 630.2.g.e.379.1 yes 2
4.3 odd 2 5040.2.t.i.1009.2 2
5.2 odd 4 3150.2.a.l.1.1 1
5.3 odd 4 3150.2.a.v.1.1 1
5.4 even 2 inner 630.2.g.b.379.1 2
12.11 even 2 5040.2.t.j.1009.1 2
15.2 even 4 3150.2.a.br.1.1 1
15.8 even 4 3150.2.a.k.1.1 1
15.14 odd 2 630.2.g.e.379.2 yes 2
20.19 odd 2 5040.2.t.i.1009.1 2
60.59 even 2 5040.2.t.j.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.g.b.379.1 2 5.4 even 2 inner
630.2.g.b.379.2 yes 2 1.1 even 1 trivial
630.2.g.e.379.1 yes 2 3.2 odd 2
630.2.g.e.379.2 yes 2 15.14 odd 2
3150.2.a.k.1.1 1 15.8 even 4
3150.2.a.l.1.1 1 5.2 odd 4
3150.2.a.v.1.1 1 5.3 odd 4
3150.2.a.br.1.1 1 15.2 even 4
5040.2.t.i.1009.1 2 20.19 odd 2
5040.2.t.i.1009.2 2 4.3 odd 2
5040.2.t.j.1009.1 2 12.11 even 2
5040.2.t.j.1009.2 2 60.59 even 2