Properties

Label 240.2.v.b.17.2
Level $240$
Weight $2$
Character 240.17
Analytic conductor $1.916$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,2,Mod(17,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 240.17
Dual form 240.2.v.b.113.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.618034 + 1.61803i) q^{3} +2.23607 q^{5} +(1.00000 + 1.00000i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\) \(q+(0.618034 + 1.61803i) q^{3} +2.23607 q^{5} +(1.00000 + 1.00000i) q^{7} +(-2.23607 + 2.00000i) q^{9} -4.47214i q^{11} +(-3.00000 + 3.00000i) q^{13} +(1.38197 + 3.61803i) q^{15} +(2.23607 - 2.23607i) q^{17} +2.00000i q^{19} +(-1.00000 + 2.23607i) q^{21} +(2.23607 + 2.23607i) q^{23} +5.00000 q^{25} +(-4.61803 - 2.38197i) q^{27} -4.47214 q^{29} -4.00000 q^{31} +(7.23607 - 2.76393i) q^{33} +(2.23607 + 2.23607i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(-6.70820 - 3.00000i) q^{39} -8.94427i q^{41} +(3.00000 - 3.00000i) q^{43} +(-5.00000 + 4.47214i) q^{45} +(6.70820 - 6.70820i) q^{47} -5.00000i q^{49} +(5.00000 + 2.23607i) q^{51} +(2.23607 + 2.23607i) q^{53} -10.0000i q^{55} +(-3.23607 + 1.23607i) q^{57} -8.94427 q^{59} -6.00000 q^{61} +(-4.23607 - 0.236068i) q^{63} +(-6.70820 + 6.70820i) q^{65} +(1.00000 + 1.00000i) q^{67} +(-2.23607 + 5.00000i) q^{69} +4.47214i q^{71} +(1.00000 - 1.00000i) q^{73} +(3.09017 + 8.09017i) q^{75} +(4.47214 - 4.47214i) q^{77} +6.00000i q^{79} +(1.00000 - 8.94427i) q^{81} +(-6.70820 - 6.70820i) q^{83} +(5.00000 - 5.00000i) q^{85} +(-2.76393 - 7.23607i) q^{87} +4.47214 q^{89} -6.00000 q^{91} +(-2.47214 - 6.47214i) q^{93} +4.47214i q^{95} +(9.00000 + 9.00000i) q^{97} +(8.94427 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{7} - 12 q^{13} + 10 q^{15} - 4 q^{21} + 20 q^{25} - 14 q^{27} - 16 q^{31} + 20 q^{33} - 12 q^{37} + 12 q^{43} - 20 q^{45} + 20 q^{51} - 4 q^{57} - 24 q^{61} - 8 q^{63} + 4 q^{67} + 4 q^{73} - 10 q^{75} + 4 q^{81} + 20 q^{85} - 20 q^{87} - 24 q^{91} + 8 q^{93} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.618034 + 1.61803i 0.356822 + 0.934172i
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 1.00000 + 1.00000i 0.377964 + 0.377964i 0.870367 0.492403i \(-0.163881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 0 0
\(9\) −2.23607 + 2.00000i −0.745356 + 0.666667i
\(10\) 0 0
\(11\) 4.47214i 1.34840i −0.738549 0.674200i \(-0.764489\pi\)
0.738549 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) −3.00000 + 3.00000i −0.832050 + 0.832050i −0.987797 0.155747i \(-0.950222\pi\)
0.155747 + 0.987797i \(0.450222\pi\)
\(14\) 0 0
\(15\) 1.38197 + 3.61803i 0.356822 + 0.934172i
\(16\) 0 0
\(17\) 2.23607 2.23607i 0.542326 0.542326i −0.381884 0.924210i \(-0.624725\pi\)
0.924210 + 0.381884i \(0.124725\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) −1.00000 + 2.23607i −0.218218 + 0.487950i
\(22\) 0 0
\(23\) 2.23607 + 2.23607i 0.466252 + 0.466252i 0.900698 0.434446i \(-0.143056\pi\)
−0.434446 + 0.900698i \(0.643056\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) −4.61803 2.38197i −0.888741 0.458410i
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 7.23607 2.76393i 1.25964 0.481139i
\(34\) 0 0
\(35\) 2.23607 + 2.23607i 0.377964 + 0.377964i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.493197 0.493197i 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) −6.70820 3.00000i −1.07417 0.480384i
\(40\) 0 0
\(41\) 8.94427i 1.39686i −0.715678 0.698430i \(-0.753882\pi\)
0.715678 0.698430i \(-0.246118\pi\)
\(42\) 0 0
\(43\) 3.00000 3.00000i 0.457496 0.457496i −0.440337 0.897833i \(-0.645141\pi\)
0.897833 + 0.440337i \(0.145141\pi\)
\(44\) 0 0
\(45\) −5.00000 + 4.47214i −0.745356 + 0.666667i
\(46\) 0 0
\(47\) 6.70820 6.70820i 0.978492 0.978492i −0.0212814 0.999774i \(-0.506775\pi\)
0.999774 + 0.0212814i \(0.00677460\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 5.00000 + 2.23607i 0.700140 + 0.313112i
\(52\) 0 0
\(53\) 2.23607 + 2.23607i 0.307148 + 0.307148i 0.843802 0.536655i \(-0.180312\pi\)
−0.536655 + 0.843802i \(0.680312\pi\)
\(54\) 0 0
\(55\) 10.0000i 1.34840i
\(56\) 0 0
\(57\) −3.23607 + 1.23607i −0.428628 + 0.163721i
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −4.23607 0.236068i −0.533694 0.0297418i
\(64\) 0 0
\(65\) −6.70820 + 6.70820i −0.832050 + 0.832050i
\(66\) 0 0
\(67\) 1.00000 + 1.00000i 0.122169 + 0.122169i 0.765548 0.643379i \(-0.222468\pi\)
−0.643379 + 0.765548i \(0.722468\pi\)
\(68\) 0 0
\(69\) −2.23607 + 5.00000i −0.269191 + 0.601929i
\(70\) 0 0
\(71\) 4.47214i 0.530745i 0.964146 + 0.265372i \(0.0854949\pi\)
−0.964146 + 0.265372i \(0.914505\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000i 0.117041 0.117041i −0.646160 0.763202i \(-0.723626\pi\)
0.763202 + 0.646160i \(0.223626\pi\)
\(74\) 0 0
\(75\) 3.09017 + 8.09017i 0.356822 + 0.934172i
\(76\) 0 0
\(77\) 4.47214 4.47214i 0.509647 0.509647i
\(78\) 0 0
\(79\) 6.00000i 0.675053i 0.941316 + 0.337526i \(0.109590\pi\)
−0.941316 + 0.337526i \(0.890410\pi\)
\(80\) 0 0
\(81\) 1.00000 8.94427i 0.111111 0.993808i
\(82\) 0 0
\(83\) −6.70820 6.70820i −0.736321 0.736321i 0.235543 0.971864i \(-0.424313\pi\)
−0.971864 + 0.235543i \(0.924313\pi\)
\(84\) 0 0
\(85\) 5.00000 5.00000i 0.542326 0.542326i
\(86\) 0 0
\(87\) −2.76393 7.23607i −0.296325 0.775788i
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −2.47214 6.47214i −0.256349 0.671129i
\(94\) 0 0
\(95\) 4.47214i 0.458831i
\(96\) 0 0
\(97\) 9.00000 + 9.00000i 0.913812 + 0.913812i 0.996570 0.0827581i \(-0.0263729\pi\)
−0.0827581 + 0.996570i \(0.526373\pi\)
\(98\) 0 0
\(99\) 8.94427 + 10.0000i 0.898933 + 1.00504i
\(100\) 0 0
\(101\) 8.94427i 0.889988i 0.895533 + 0.444994i \(0.146794\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −1.00000 + 1.00000i −0.0985329 + 0.0985329i −0.754655 0.656122i \(-0.772196\pi\)
0.656122 + 0.754655i \(0.272196\pi\)
\(104\) 0 0
\(105\) −2.23607 + 5.00000i −0.218218 + 0.487950i
\(106\) 0 0
\(107\) −2.23607 + 2.23607i −0.216169 + 0.216169i −0.806882 0.590713i \(-0.798847\pi\)
0.590713 + 0.806882i \(0.298847\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) 3.00000 6.70820i 0.284747 0.636715i
\(112\) 0 0
\(113\) 2.23607 + 2.23607i 0.210352 + 0.210352i 0.804417 0.594065i \(-0.202478\pi\)
−0.594065 + 0.804417i \(0.702478\pi\)
\(114\) 0 0
\(115\) 5.00000 + 5.00000i 0.466252 + 0.466252i
\(116\) 0 0
\(117\) 0.708204 12.7082i 0.0654735 1.17487i
\(118\) 0 0
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 14.4721 5.52786i 1.30491 0.498431i
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 13.0000 + 13.0000i 1.15356 + 1.15356i 0.985833 + 0.167731i \(0.0536439\pi\)
0.167731 + 0.985833i \(0.446356\pi\)
\(128\) 0 0
\(129\) 6.70820 + 3.00000i 0.590624 + 0.264135i
\(130\) 0 0
\(131\) 4.47214i 0.390732i −0.980730 0.195366i \(-0.937410\pi\)
0.980730 0.195366i \(-0.0625895\pi\)
\(132\) 0 0
\(133\) −2.00000 + 2.00000i −0.173422 + 0.173422i
\(134\) 0 0
\(135\) −10.3262 5.32624i −0.888741 0.458410i
\(136\) 0 0
\(137\) −6.70820 + 6.70820i −0.573121 + 0.573121i −0.932999 0.359879i \(-0.882818\pi\)
0.359879 + 0.932999i \(0.382818\pi\)
\(138\) 0 0
\(139\) 14.0000i 1.18746i −0.804663 0.593732i \(-0.797654\pi\)
0.804663 0.593732i \(-0.202346\pi\)
\(140\) 0 0
\(141\) 15.0000 + 6.70820i 1.26323 + 0.564933i
\(142\) 0 0
\(143\) 13.4164 + 13.4164i 1.12194 + 1.12194i
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 8.09017 3.09017i 0.667266 0.254873i
\(148\) 0 0
\(149\) −13.4164 −1.09911 −0.549557 0.835456i \(-0.685204\pi\)
−0.549557 + 0.835456i \(0.685204\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −0.527864 + 9.47214i −0.0426753 + 0.765777i
\(154\) 0 0
\(155\) −8.94427 −0.718421
\(156\) 0 0
\(157\) 9.00000 + 9.00000i 0.718278 + 0.718278i 0.968252 0.249974i \(-0.0804222\pi\)
−0.249974 + 0.968252i \(0.580422\pi\)
\(158\) 0 0
\(159\) −2.23607 + 5.00000i −0.177332 + 0.396526i
\(160\) 0 0
\(161\) 4.47214i 0.352454i
\(162\) 0 0
\(163\) −17.0000 + 17.0000i −1.33154 + 1.33154i −0.427552 + 0.903991i \(0.640624\pi\)
−0.903991 + 0.427552i \(0.859376\pi\)
\(164\) 0 0
\(165\) 16.1803 6.18034i 1.25964 0.481139i
\(166\) 0 0
\(167\) −11.1803 + 11.1803i −0.865161 + 0.865161i −0.991932 0.126771i \(-0.959539\pi\)
0.126771 + 0.991932i \(0.459539\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) −4.00000 4.47214i −0.305888 0.341993i
\(172\) 0 0
\(173\) −15.6525 15.6525i −1.19004 1.19004i −0.977057 0.212979i \(-0.931683\pi\)
−0.212979 0.977057i \(-0.568317\pi\)
\(174\) 0 0
\(175\) 5.00000 + 5.00000i 0.377964 + 0.377964i
\(176\) 0 0
\(177\) −5.52786 14.4721i −0.415500 1.08779i
\(178\) 0 0
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −3.70820 9.70820i −0.274118 0.717651i
\(184\) 0 0
\(185\) −6.70820 6.70820i −0.493197 0.493197i
\(186\) 0 0
\(187\) −10.0000 10.0000i −0.731272 0.731272i
\(188\) 0 0
\(189\) −2.23607 7.00000i −0.162650 0.509175i
\(190\) 0 0
\(191\) 4.47214i 0.323592i 0.986824 + 0.161796i \(0.0517287\pi\)
−0.986824 + 0.161796i \(0.948271\pi\)
\(192\) 0 0
\(193\) −3.00000 + 3.00000i −0.215945 + 0.215945i −0.806787 0.590842i \(-0.798796\pi\)
0.590842 + 0.806787i \(0.298796\pi\)
\(194\) 0 0
\(195\) −15.0000 6.70820i −1.07417 0.480384i
\(196\) 0 0
\(197\) −6.70820 + 6.70820i −0.477940 + 0.477940i −0.904472 0.426532i \(-0.859735\pi\)
0.426532 + 0.904472i \(0.359735\pi\)
\(198\) 0 0
\(199\) 18.0000i 1.27599i −0.770042 0.637993i \(-0.779765\pi\)
0.770042 0.637993i \(-0.220235\pi\)
\(200\) 0 0
\(201\) −1.00000 + 2.23607i −0.0705346 + 0.157720i
\(202\) 0 0
\(203\) −4.47214 4.47214i −0.313882 0.313882i
\(204\) 0 0
\(205\) 20.0000i 1.39686i
\(206\) 0 0
\(207\) −9.47214 0.527864i −0.658359 0.0366891i
\(208\) 0 0
\(209\) 8.94427 0.618688
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −7.23607 + 2.76393i −0.495807 + 0.189382i
\(214\) 0 0
\(215\) 6.70820 6.70820i 0.457496 0.457496i
\(216\) 0 0
\(217\) −4.00000 4.00000i −0.271538 0.271538i
\(218\) 0 0
\(219\) 2.23607 + 1.00000i 0.151099 + 0.0675737i
\(220\) 0 0
\(221\) 13.4164i 0.902485i
\(222\) 0 0
\(223\) 3.00000 3.00000i 0.200895 0.200895i −0.599489 0.800383i \(-0.704629\pi\)
0.800383 + 0.599489i \(0.204629\pi\)
\(224\) 0 0
\(225\) −11.1803 + 10.0000i −0.745356 + 0.666667i
\(226\) 0 0
\(227\) −11.1803 + 11.1803i −0.742065 + 0.742065i −0.972975 0.230910i \(-0.925830\pi\)
0.230910 + 0.972975i \(0.425830\pi\)
\(228\) 0 0
\(229\) 12.0000i 0.792982i −0.918039 0.396491i \(-0.870228\pi\)
0.918039 0.396491i \(-0.129772\pi\)
\(230\) 0 0
\(231\) 10.0000 + 4.47214i 0.657952 + 0.294245i
\(232\) 0 0
\(233\) 20.1246 + 20.1246i 1.31841 + 1.31841i 0.915037 + 0.403370i \(0.132161\pi\)
0.403370 + 0.915037i \(0.367839\pi\)
\(234\) 0 0
\(235\) 15.0000 15.0000i 0.978492 0.978492i
\(236\) 0 0
\(237\) −9.70820 + 3.70820i −0.630616 + 0.240874i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 15.0902 3.90983i 0.968035 0.250816i
\(244\) 0 0
\(245\) 11.1803i 0.714286i
\(246\) 0 0
\(247\) −6.00000 6.00000i −0.381771 0.381771i
\(248\) 0 0
\(249\) 6.70820 15.0000i 0.425115 0.950586i
\(250\) 0 0
\(251\) 13.4164i 0.846836i 0.905934 + 0.423418i \(0.139170\pi\)
−0.905934 + 0.423418i \(0.860830\pi\)
\(252\) 0 0
\(253\) 10.0000 10.0000i 0.628695 0.628695i
\(254\) 0 0
\(255\) 11.1803 + 5.00000i 0.700140 + 0.313112i
\(256\) 0 0
\(257\) −15.6525 + 15.6525i −0.976375 + 0.976375i −0.999727 0.0233527i \(-0.992566\pi\)
0.0233527 + 0.999727i \(0.492566\pi\)
\(258\) 0 0
\(259\) 6.00000i 0.372822i
\(260\) 0 0
\(261\) 10.0000 8.94427i 0.618984 0.553637i
\(262\) 0 0
\(263\) −15.6525 15.6525i −0.965173 0.965173i 0.0342406 0.999414i \(-0.489099\pi\)
−0.999414 + 0.0342406i \(0.989099\pi\)
\(264\) 0 0
\(265\) 5.00000 + 5.00000i 0.307148 + 0.307148i
\(266\) 0 0
\(267\) 2.76393 + 7.23607i 0.169150 + 0.442840i
\(268\) 0 0
\(269\) 22.3607 1.36335 0.681677 0.731653i \(-0.261251\pi\)
0.681677 + 0.731653i \(0.261251\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) −3.70820 9.70820i −0.224431 0.587567i
\(274\) 0 0
\(275\) 22.3607i 1.34840i
\(276\) 0 0
\(277\) 9.00000 + 9.00000i 0.540758 + 0.540758i 0.923751 0.382993i \(-0.125107\pi\)
−0.382993 + 0.923751i \(0.625107\pi\)
\(278\) 0 0
\(279\) 8.94427 8.00000i 0.535480 0.478947i
\(280\) 0 0
\(281\) 26.8328i 1.60071i −0.599525 0.800356i \(-0.704644\pi\)
0.599525 0.800356i \(-0.295356\pi\)
\(282\) 0 0
\(283\) −17.0000 + 17.0000i −1.01055 + 1.01055i −0.0106013 + 0.999944i \(0.503375\pi\)
−0.999944 + 0.0106013i \(0.996625\pi\)
\(284\) 0 0
\(285\) −7.23607 + 2.76393i −0.428628 + 0.163721i
\(286\) 0 0
\(287\) 8.94427 8.94427i 0.527964 0.527964i
\(288\) 0 0
\(289\) 7.00000i 0.411765i
\(290\) 0 0
\(291\) −9.00000 + 20.1246i −0.527589 + 1.17973i
\(292\) 0 0
\(293\) −6.70820 6.70820i −0.391897 0.391897i 0.483466 0.875363i \(-0.339378\pi\)
−0.875363 + 0.483466i \(0.839378\pi\)
\(294\) 0 0
\(295\) −20.0000 −1.16445
\(296\) 0 0
\(297\) −10.6525 + 20.6525i −0.618119 + 1.19838i
\(298\) 0 0
\(299\) −13.4164 −0.775891
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) −14.4721 + 5.52786i −0.831402 + 0.317567i
\(304\) 0 0
\(305\) −13.4164 −0.768221
\(306\) 0 0
\(307\) 21.0000 + 21.0000i 1.19853 + 1.19853i 0.974606 + 0.223928i \(0.0718879\pi\)
0.223928 + 0.974606i \(0.428112\pi\)
\(308\) 0 0
\(309\) −2.23607 1.00000i −0.127205 0.0568880i
\(310\) 0 0
\(311\) 4.47214i 0.253592i 0.991929 + 0.126796i \(0.0404693\pi\)
−0.991929 + 0.126796i \(0.959531\pi\)
\(312\) 0 0
\(313\) −3.00000 + 3.00000i −0.169570 + 0.169570i −0.786790 0.617220i \(-0.788259\pi\)
0.617220 + 0.786790i \(0.288259\pi\)
\(314\) 0 0
\(315\) −9.47214 0.527864i −0.533694 0.0297418i
\(316\) 0 0
\(317\) 20.1246 20.1246i 1.13031 1.13031i 0.140186 0.990125i \(-0.455230\pi\)
0.990125 0.140186i \(-0.0447699\pi\)
\(318\) 0 0
\(319\) 20.0000i 1.11979i
\(320\) 0 0
\(321\) −5.00000 2.23607i −0.279073 0.124805i
\(322\) 0 0
\(323\) 4.47214 + 4.47214i 0.248836 + 0.248836i
\(324\) 0 0
\(325\) −15.0000 + 15.0000i −0.832050 + 0.832050i
\(326\) 0 0
\(327\) −12.9443 + 4.94427i −0.715820 + 0.273419i
\(328\) 0 0
\(329\) 13.4164 0.739671
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 12.7082 + 0.708204i 0.696405 + 0.0388093i
\(334\) 0 0
\(335\) 2.23607 + 2.23607i 0.122169 + 0.122169i
\(336\) 0 0
\(337\) 9.00000 + 9.00000i 0.490261 + 0.490261i 0.908388 0.418127i \(-0.137313\pi\)
−0.418127 + 0.908388i \(0.637313\pi\)
\(338\) 0 0
\(339\) −2.23607 + 5.00000i −0.121447 + 0.271563i
\(340\) 0 0
\(341\) 17.8885i 0.968719i
\(342\) 0 0
\(343\) 12.0000 12.0000i 0.647939 0.647939i
\(344\) 0 0
\(345\) −5.00000 + 11.1803i −0.269191 + 0.601929i
\(346\) 0 0
\(347\) 24.5967 24.5967i 1.32042 1.32042i 0.406990 0.913433i \(-0.366578\pi\)
0.913433 0.406990i \(-0.133422\pi\)
\(348\) 0 0
\(349\) 4.00000i 0.214115i 0.994253 + 0.107058i \(0.0341429\pi\)
−0.994253 + 0.107058i \(0.965857\pi\)
\(350\) 0 0
\(351\) 21.0000 6.70820i 1.12090 0.358057i
\(352\) 0 0
\(353\) −6.70820 6.70820i −0.357042 0.357042i 0.505680 0.862721i \(-0.331242\pi\)
−0.862721 + 0.505680i \(0.831242\pi\)
\(354\) 0 0
\(355\) 10.0000i 0.530745i
\(356\) 0 0
\(357\) 2.76393 + 7.23607i 0.146283 + 0.382973i
\(358\) 0 0
\(359\) 35.7771 1.88824 0.944121 0.329598i \(-0.106913\pi\)
0.944121 + 0.329598i \(0.106913\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) −5.56231 14.5623i −0.291945 0.764323i
\(364\) 0 0
\(365\) 2.23607 2.23607i 0.117041 0.117041i
\(366\) 0 0
\(367\) 1.00000 + 1.00000i 0.0521996 + 0.0521996i 0.732725 0.680525i \(-0.238248\pi\)
−0.680525 + 0.732725i \(0.738248\pi\)
\(368\) 0 0
\(369\) 17.8885 + 20.0000i 0.931240 + 1.04116i
\(370\) 0 0
\(371\) 4.47214i 0.232182i
\(372\) 0 0
\(373\) −23.0000 + 23.0000i −1.19089 + 1.19089i −0.214078 + 0.976816i \(0.568675\pi\)
−0.976816 + 0.214078i \(0.931325\pi\)
\(374\) 0 0
\(375\) 6.90983 + 18.0902i 0.356822 + 0.934172i
\(376\) 0 0
\(377\) 13.4164 13.4164i 0.690980 0.690980i
\(378\) 0 0
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) 0 0
\(381\) −13.0000 + 29.0689i −0.666010 + 1.48924i
\(382\) 0 0
\(383\) −6.70820 6.70820i −0.342773 0.342773i 0.514636 0.857409i \(-0.327927\pi\)
−0.857409 + 0.514636i \(0.827927\pi\)
\(384\) 0 0
\(385\) 10.0000 10.0000i 0.509647 0.509647i
\(386\) 0 0
\(387\) −0.708204 + 12.7082i −0.0360000 + 0.645994i
\(388\) 0 0
\(389\) −13.4164 −0.680239 −0.340119 0.940382i \(-0.610468\pi\)
−0.340119 + 0.940382i \(0.610468\pi\)
\(390\) 0 0
\(391\) 10.0000 0.505722
\(392\) 0 0
\(393\) 7.23607 2.76393i 0.365011 0.139422i
\(394\) 0 0
\(395\) 13.4164i 0.675053i
\(396\) 0 0
\(397\) −11.0000 11.0000i −0.552074 0.552074i 0.374965 0.927039i \(-0.377655\pi\)
−0.927039 + 0.374965i \(0.877655\pi\)
\(398\) 0 0
\(399\) −4.47214 2.00000i −0.223887 0.100125i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 12.0000 12.0000i 0.597763 0.597763i
\(404\) 0 0
\(405\) 2.23607 20.0000i 0.111111 0.993808i
\(406\) 0 0
\(407\) −13.4164 + 13.4164i −0.665027 + 0.665027i
\(408\) 0 0
\(409\) 12.0000i 0.593362i −0.954977 0.296681i \(-0.904120\pi\)
0.954977 0.296681i \(-0.0958798\pi\)
\(410\) 0 0
\(411\) −15.0000 6.70820i −0.739895 0.330891i
\(412\) 0 0
\(413\) −8.94427 8.94427i −0.440119 0.440119i
\(414\) 0 0
\(415\) −15.0000 15.0000i −0.736321 0.736321i
\(416\) 0 0
\(417\) 22.6525 8.65248i 1.10930 0.423714i
\(418\) 0 0
\(419\) 26.8328 1.31087 0.655434 0.755252i \(-0.272486\pi\)
0.655434 + 0.755252i \(0.272486\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −1.58359 + 28.4164i −0.0769969 + 1.38165i
\(424\) 0 0
\(425\) 11.1803 11.1803i 0.542326 0.542326i
\(426\) 0 0
\(427\) −6.00000 6.00000i −0.290360 0.290360i
\(428\) 0 0
\(429\) −13.4164 + 30.0000i −0.647750 + 1.44841i
\(430\) 0 0
\(431\) 31.3050i 1.50791i −0.656928 0.753953i \(-0.728145\pi\)
0.656928 0.753953i \(-0.271855\pi\)
\(432\) 0 0
\(433\) 17.0000 17.0000i 0.816968 0.816968i −0.168700 0.985668i \(-0.553957\pi\)
0.985668 + 0.168700i \(0.0539568\pi\)
\(434\) 0 0
\(435\) −6.18034 16.1803i −0.296325 0.775788i
\(436\) 0 0
\(437\) −4.47214 + 4.47214i −0.213931 + 0.213931i
\(438\) 0 0
\(439\) 22.0000i 1.05000i 0.851101 + 0.525001i \(0.175935\pi\)
−0.851101 + 0.525001i \(0.824065\pi\)
\(440\) 0 0
\(441\) 10.0000 + 11.1803i 0.476190 + 0.532397i
\(442\) 0 0
\(443\) 20.1246 + 20.1246i 0.956149 + 0.956149i 0.999078 0.0429290i \(-0.0136689\pi\)
−0.0429290 + 0.999078i \(0.513669\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) −8.29180 21.7082i −0.392188 1.02676i
\(448\) 0 0
\(449\) −4.47214 −0.211053 −0.105527 0.994416i \(-0.533653\pi\)
−0.105527 + 0.994416i \(0.533653\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 0 0
\(453\) −2.47214 6.47214i −0.116151 0.304087i
\(454\) 0 0
\(455\) −13.4164 −0.628971
\(456\) 0 0
\(457\) −3.00000 3.00000i −0.140334 0.140334i 0.633450 0.773784i \(-0.281638\pi\)
−0.773784 + 0.633450i \(0.781638\pi\)
\(458\) 0 0
\(459\) −15.6525 + 5.00000i −0.730595 + 0.233380i
\(460\) 0 0
\(461\) 8.94427i 0.416576i −0.978068 0.208288i \(-0.933211\pi\)
0.978068 0.208288i \(-0.0667892\pi\)
\(462\) 0 0
\(463\) 19.0000 19.0000i 0.883005 0.883005i −0.110834 0.993839i \(-0.535352\pi\)
0.993839 + 0.110834i \(0.0353522\pi\)
\(464\) 0 0
\(465\) −5.52786 14.4721i −0.256349 0.671129i
\(466\) 0 0
\(467\) −2.23607 + 2.23607i −0.103473 + 0.103473i −0.756948 0.653475i \(-0.773310\pi\)
0.653475 + 0.756948i \(0.273310\pi\)
\(468\) 0 0
\(469\) 2.00000i 0.0923514i
\(470\) 0 0
\(471\) −9.00000 + 20.1246i −0.414698 + 0.927293i
\(472\) 0 0
\(473\) −13.4164 13.4164i −0.616887 0.616887i
\(474\) 0 0
\(475\) 10.0000i 0.458831i
\(476\) 0 0
\(477\) −9.47214 0.527864i −0.433699 0.0241692i
\(478\) 0 0
\(479\) −35.7771 −1.63470 −0.817348 0.576144i \(-0.804557\pi\)
−0.817348 + 0.576144i \(0.804557\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) −7.23607 + 2.76393i −0.329252 + 0.125763i
\(484\) 0 0
\(485\) 20.1246 + 20.1246i 0.913812 + 0.913812i
\(486\) 0 0
\(487\) 21.0000 + 21.0000i 0.951601 + 0.951601i 0.998882 0.0472808i \(-0.0150556\pi\)
−0.0472808 + 0.998882i \(0.515056\pi\)
\(488\) 0 0
\(489\) −38.0132 17.0000i −1.71901 0.768767i
\(490\) 0 0
\(491\) 22.3607i 1.00912i −0.863376 0.504562i \(-0.831654\pi\)
0.863376 0.504562i \(-0.168346\pi\)
\(492\) 0 0
\(493\) −10.0000 + 10.0000i −0.450377 + 0.450377i
\(494\) 0 0
\(495\) 20.0000 + 22.3607i 0.898933 + 1.00504i
\(496\) 0 0
\(497\) −4.47214 + 4.47214i −0.200603 + 0.200603i
\(498\) 0 0
\(499\) 38.0000i 1.70111i −0.525883 0.850557i \(-0.676265\pi\)
0.525883 0.850557i \(-0.323735\pi\)
\(500\) 0 0
\(501\) −25.0000 11.1803i −1.11692 0.499501i
\(502\) 0 0
\(503\) −6.70820 6.70820i −0.299104 0.299104i 0.541559 0.840663i \(-0.317834\pi\)
−0.840663 + 0.541559i \(0.817834\pi\)
\(504\) 0 0
\(505\) 20.0000i 0.889988i
\(506\) 0 0
\(507\) 8.09017 3.09017i 0.359297 0.137239i
\(508\) 0 0
\(509\) −4.47214 −0.198224 −0.0991120 0.995076i \(-0.531600\pi\)
−0.0991120 + 0.995076i \(0.531600\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 4.76393 9.23607i 0.210333 0.407782i
\(514\) 0 0
\(515\) −2.23607 + 2.23607i −0.0985329 + 0.0985329i
\(516\) 0 0
\(517\) −30.0000 30.0000i −1.31940 1.31940i
\(518\) 0 0
\(519\) 15.6525 35.0000i 0.687067 1.53633i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 3.00000 3.00000i 0.131181 0.131181i −0.638468 0.769649i \(-0.720431\pi\)
0.769649 + 0.638468i \(0.220431\pi\)
\(524\) 0 0
\(525\) −5.00000 + 11.1803i −0.218218 + 0.487950i
\(526\) 0 0
\(527\) −8.94427 + 8.94427i −0.389619 + 0.389619i
\(528\) 0 0
\(529\) 13.0000i 0.565217i
\(530\) 0 0
\(531\) 20.0000 17.8885i 0.867926 0.776297i
\(532\) 0 0
\(533\) 26.8328 + 26.8328i 1.16226 + 1.16226i
\(534\) 0 0
\(535\) −5.00000 + 5.00000i −0.216169 + 0.216169i
\(536\) 0 0
\(537\) 5.52786 + 14.4721i 0.238545 + 0.624519i
\(538\) 0 0
\(539\) −22.3607 −0.963143
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 8.65248 + 22.6525i 0.371313 + 0.972111i
\(544\) 0 0
\(545\) 17.8885i 0.766261i
\(546\) 0 0
\(547\) −27.0000 27.0000i −1.15444 1.15444i −0.985653 0.168783i \(-0.946016\pi\)
−0.168783 0.985653i \(-0.553984\pi\)
\(548\) 0 0
\(549\) 13.4164 12.0000i 0.572598 0.512148i
\(550\) 0 0
\(551\) 8.94427i 0.381039i
\(552\) 0 0
\(553\) −6.00000 + 6.00000i −0.255146 + 0.255146i
\(554\) 0 0
\(555\) 6.70820 15.0000i 0.284747 0.636715i
\(556\) 0 0
\(557\) −6.70820 + 6.70820i −0.284236 + 0.284236i −0.834796 0.550560i \(-0.814414\pi\)
0.550560 + 0.834796i \(0.314414\pi\)
\(558\) 0 0
\(559\) 18.0000i 0.761319i
\(560\) 0 0
\(561\) 10.0000 22.3607i 0.422200 0.944069i
\(562\) 0 0
\(563\) −6.70820 6.70820i −0.282717 0.282717i 0.551475 0.834192i \(-0.314065\pi\)
−0.834192 + 0.551475i \(0.814065\pi\)
\(564\) 0 0
\(565\) 5.00000 + 5.00000i 0.210352 + 0.210352i
\(566\) 0 0
\(567\) 9.94427 7.94427i 0.417620 0.333628i
\(568\) 0 0
\(569\) −22.3607 −0.937408 −0.468704 0.883355i \(-0.655279\pi\)
−0.468704 + 0.883355i \(0.655279\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −7.23607 + 2.76393i −0.302291 + 0.115465i
\(574\) 0 0
\(575\) 11.1803 + 11.1803i 0.466252 + 0.466252i
\(576\) 0 0
\(577\) −11.0000 11.0000i −0.457936 0.457936i 0.440041 0.897977i \(-0.354964\pi\)
−0.897977 + 0.440041i \(0.854964\pi\)
\(578\) 0 0
\(579\) −6.70820 3.00000i −0.278783 0.124676i
\(580\) 0 0
\(581\) 13.4164i 0.556606i
\(582\) 0 0
\(583\) 10.0000 10.0000i 0.414158 0.414158i
\(584\) 0 0
\(585\) 1.58359 28.4164i 0.0654735 1.17487i
\(586\) 0 0
\(587\) 6.70820 6.70820i 0.276877 0.276877i −0.554984 0.831861i \(-0.687276\pi\)
0.831861 + 0.554984i \(0.187276\pi\)
\(588\) 0 0
\(589\) 8.00000i 0.329634i
\(590\) 0 0
\(591\) −15.0000 6.70820i −0.617018 0.275939i
\(592\) 0 0
\(593\) 11.1803 + 11.1803i 0.459122 + 0.459122i 0.898367 0.439246i \(-0.144754\pi\)
−0.439246 + 0.898367i \(0.644754\pi\)
\(594\) 0 0
\(595\) 10.0000 0.409960
\(596\) 0 0
\(597\) 29.1246 11.1246i 1.19199 0.455300i
\(598\) 0 0
\(599\) −35.7771 −1.46181 −0.730906 0.682478i \(-0.760902\pi\)
−0.730906 + 0.682478i \(0.760902\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −4.23607 0.236068i −0.172506 0.00961343i
\(604\) 0 0
\(605\) −20.1246 −0.818182
\(606\) 0 0
\(607\) −27.0000 27.0000i −1.09590 1.09590i −0.994885 0.101011i \(-0.967792\pi\)
−0.101011 0.994885i \(-0.532208\pi\)
\(608\) 0 0
\(609\) 4.47214 10.0000i 0.181220 0.405220i
\(610\) 0 0
\(611\) 40.2492i 1.62831i
\(612\) 0 0
\(613\) 21.0000 21.0000i 0.848182 0.848182i −0.141724 0.989906i \(-0.545265\pi\)
0.989906 + 0.141724i \(0.0452646\pi\)
\(614\) 0 0
\(615\) 32.3607 12.3607i 1.30491 0.498431i
\(616\) 0 0
\(617\) 20.1246 20.1246i 0.810186 0.810186i −0.174475 0.984662i \(-0.555823\pi\)
0.984662 + 0.174475i \(0.0558229\pi\)
\(618\) 0 0
\(619\) 42.0000i 1.68812i 0.536247 + 0.844061i \(0.319842\pi\)
−0.536247 + 0.844061i \(0.680158\pi\)
\(620\) 0 0
\(621\) −5.00000 15.6525i −0.200643 0.628112i
\(622\) 0 0
\(623\) 4.47214 + 4.47214i 0.179172 + 0.179172i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 5.52786 + 14.4721i 0.220762 + 0.577961i
\(628\) 0 0
\(629\) −13.4164 −0.534947
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 9.88854 + 25.8885i 0.393034 + 1.02898i
\(634\) 0 0
\(635\) 29.0689 + 29.0689i 1.15356 + 1.15356i
\(636\) 0 0
\(637\) 15.0000 + 15.0000i 0.594322 + 0.594322i
\(638\) 0 0
\(639\) −8.94427 10.0000i −0.353830 0.395594i
\(640\) 0 0
\(641\) 8.94427i 0.353278i 0.984276 + 0.176639i \(0.0565224\pi\)
−0.984276 + 0.176639i \(0.943478\pi\)
\(642\) 0 0
\(643\) −21.0000 + 21.0000i −0.828159 + 0.828159i −0.987262 0.159103i \(-0.949140\pi\)
0.159103 + 0.987262i \(0.449140\pi\)
\(644\) 0 0
\(645\) 15.0000 + 6.70820i 0.590624 + 0.264135i
\(646\) 0 0
\(647\) 15.6525 15.6525i 0.615362 0.615362i −0.328976 0.944338i \(-0.606704\pi\)
0.944338 + 0.328976i \(0.106704\pi\)
\(648\) 0 0
\(649\) 40.0000i 1.57014i
\(650\) 0 0
\(651\) 4.00000 8.94427i 0.156772 0.350554i
\(652\) 0 0
\(653\) 20.1246 + 20.1246i 0.787537 + 0.787537i 0.981090 0.193553i \(-0.0620011\pi\)
−0.193553 + 0.981090i \(0.562001\pi\)
\(654\) 0 0
\(655\) 10.0000i 0.390732i
\(656\) 0 0
\(657\) −0.236068 + 4.23607i −0.00920989 + 0.165265i
\(658\) 0 0
\(659\) 8.94427 0.348419 0.174210 0.984709i \(-0.444263\pi\)
0.174210 + 0.984709i \(0.444263\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) −21.7082 + 8.29180i −0.843077 + 0.322027i
\(664\) 0 0
\(665\) −4.47214 + 4.47214i −0.173422 + 0.173422i
\(666\) 0 0
\(667\) −10.0000 10.0000i −0.387202 0.387202i
\(668\) 0 0
\(669\) 6.70820 + 3.00000i 0.259354 + 0.115987i
\(670\) 0 0
\(671\) 26.8328i 1.03587i
\(672\) 0 0
\(673\) 1.00000 1.00000i 0.0385472 0.0385472i −0.687570 0.726118i \(-0.741323\pi\)
0.726118 + 0.687570i \(0.241323\pi\)
\(674\) 0 0
\(675\) −23.0902 11.9098i −0.888741 0.458410i
\(676\) 0 0
\(677\) −24.5967 + 24.5967i −0.945330 + 0.945330i −0.998581 0.0532513i \(-0.983042\pi\)
0.0532513 + 0.998581i \(0.483042\pi\)
\(678\) 0 0
\(679\) 18.0000i 0.690777i
\(680\) 0 0
\(681\) −25.0000 11.1803i −0.958002 0.428432i
\(682\) 0 0
\(683\) 20.1246 + 20.1246i 0.770047 + 0.770047i 0.978114 0.208068i \(-0.0667174\pi\)
−0.208068 + 0.978114i \(0.566717\pi\)
\(684\) 0 0
\(685\) −15.0000 + 15.0000i −0.573121 + 0.573121i
\(686\) 0 0
\(687\) 19.4164 7.41641i 0.740782 0.282954i
\(688\) 0 0
\(689\) −13.4164 −0.511124
\(690\) 0 0
\(691\) 48.0000 1.82601 0.913003 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) 0 0
\(693\) −1.05573 + 18.9443i −0.0401038 + 0.719633i
\(694\) 0 0
\(695\) 31.3050i 1.18746i
\(696\) 0 0
\(697\) −20.0000 20.0000i −0.757554 0.757554i
\(698\) 0 0
\(699\) −20.1246 + 45.0000i −0.761183 + 1.70206i
\(700\) 0 0
\(701\) 35.7771i 1.35128i −0.737231 0.675641i \(-0.763867\pi\)
0.737231 0.675641i \(-0.236133\pi\)
\(702\) 0 0
\(703\) 6.00000 6.00000i 0.226294 0.226294i
\(704\) 0 0
\(705\) 33.5410 + 15.0000i 1.26323 + 0.564933i
\(706\) 0 0
\(707\) −8.94427 + 8.94427i −0.336384 + 0.336384i
\(708\) 0 0
\(709\) 28.0000i 1.05156i 0.850620 + 0.525781i \(0.176227\pi\)
−0.850620 + 0.525781i \(0.823773\pi\)
\(710\) 0 0
\(711\) −12.0000 13.4164i −0.450035 0.503155i
\(712\) 0 0
\(713\) −8.94427 8.94427i −0.334966 0.334966i
\(714\) 0 0
\(715\) 30.0000 + 30.0000i 1.12194 + 1.12194i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −35.7771 −1.33426 −0.667130 0.744941i \(-0.732478\pi\)
−0.667130 + 0.744941i \(0.732478\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) −16.0689 42.0689i −0.597608 1.56456i
\(724\) 0 0
\(725\) −22.3607 −0.830455
\(726\) 0 0
\(727\) 21.0000 + 21.0000i 0.778847 + 0.778847i 0.979635 0.200788i \(-0.0643502\pi\)
−0.200788 + 0.979635i \(0.564350\pi\)
\(728\) 0 0
\(729\) 15.6525 + 22.0000i 0.579721 + 0.814815i
\(730\) 0 0
\(731\) 13.4164i 0.496224i
\(732\) 0 0
\(733\) 1.00000 1.00000i 0.0369358 0.0369358i −0.688398 0.725333i \(-0.741686\pi\)
0.725333 + 0.688398i \(0.241686\pi\)
\(734\) 0 0
\(735\) 18.0902 6.90983i 0.667266 0.254873i
\(736\) 0 0
\(737\) 4.47214 4.47214i 0.164733 0.164733i
\(738\) 0 0
\(739\) 14.0000i 0.514998i −0.966279 0.257499i \(-0.917102\pi\)
0.966279 0.257499i \(-0.0828985\pi\)
\(740\) 0 0
\(741\) 6.00000 13.4164i 0.220416 0.492864i
\(742\) 0 0
\(743\) 11.1803 + 11.1803i 0.410167 + 0.410167i 0.881797 0.471630i \(-0.156334\pi\)
−0.471630 + 0.881797i \(0.656334\pi\)
\(744\) 0 0
\(745\) −30.0000 −1.09911
\(746\) 0 0
\(747\) 28.4164 + 1.58359i 1.03970 + 0.0579406i
\(748\) 0 0
\(749\) −4.47214 −0.163408
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −21.7082 + 8.29180i −0.791091 + 0.302170i
\(754\) 0 0
\(755\) −8.94427 −0.325515
\(756\) 0 0
\(757\) 9.00000 + 9.00000i 0.327111 + 0.327111i 0.851487 0.524376i \(-0.175701\pi\)
−0.524376 + 0.851487i \(0.675701\pi\)
\(758\) 0 0
\(759\) 22.3607 + 10.0000i 0.811641 + 0.362977i
\(760\) 0 0
\(761\) 17.8885i 0.648459i 0.945978 + 0.324230i \(0.105105\pi\)
−0.945978 + 0.324230i \(0.894895\pi\)
\(762\) 0 0
\(763\) −8.00000 + 8.00000i −0.289619 + 0.289619i
\(764\) 0 0
\(765\) −1.18034 + 21.1803i −0.0426753 + 0.765777i
\(766\) 0 0
\(767\) 26.8328 26.8328i 0.968877 0.968877i
\(768\) 0 0
\(769\) 8.00000i 0.288487i 0.989542 + 0.144244i \(0.0460749\pi\)
−0.989542 + 0.144244i \(0.953925\pi\)
\(770\) 0 0
\(771\) −35.0000 15.6525i −1.26049 0.563710i
\(772\) 0 0
\(773\) −6.70820 6.70820i −0.241277 0.241277i 0.576101 0.817378i \(-0.304573\pi\)
−0.817378 + 0.576101i \(0.804573\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 9.70820 3.70820i 0.348280 0.133031i
\(778\) 0 0
\(779\) 17.8885 0.640924
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 0 0
\(783\) 20.6525 + 10.6525i 0.738059 + 0.380688i
\(784\) 0 0
\(785\) 20.1246 + 20.1246i 0.718278 + 0.718278i
\(786\) 0 0
\(787\) −19.0000 19.0000i −0.677277 0.677277i 0.282106 0.959383i \(-0.408967\pi\)
−0.959383 + 0.282106i \(0.908967\pi\)
\(788\) 0 0
\(789\) 15.6525 35.0000i 0.557243 1.24603i
\(790\) 0 0
\(791\) 4.47214i 0.159011i
\(792\) 0 0
\(793\) 18.0000 18.0000i 0.639199 0.639199i
\(794\) 0 0
\(795\) −5.00000 + 11.1803i −0.177332 + 0.396526i
\(796\) 0 0
\(797\) 20.1246 20.1246i 0.712850 0.712850i −0.254280 0.967131i \(-0.581839\pi\)
0.967131 + 0.254280i \(0.0818386\pi\)
\(798\) 0 0
\(799\) 30.0000i 1.06132i
\(800\) 0 0
\(801\) −10.0000 + 8.94427i −0.353333 + 0.316030i
\(802\) 0 0
\(803\) −4.47214 4.47214i −0.157818 0.157818i
\(804\) 0 0
\(805\) 10.0000i 0.352454i
\(806\) 0 0
\(807\) 13.8197 + 36.1803i 0.486475 + 1.27361i
\(808\) 0 0
\(809\) 4.47214 0.157232 0.0786160 0.996905i \(-0.474950\pi\)
0.0786160 + 0.996905i \(0.474950\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −7.41641 19.4164i −0.260105 0.680963i
\(814\) 0 0
\(815\) −38.0132 + 38.0132i −1.33154 + 1.33154i
\(816\) 0 0
\(817\) 6.00000 + 6.00000i 0.209913 + 0.209913i
\(818\) 0 0
\(819\) 13.4164 12.0000i 0.468807 0.419314i
\(820\) 0 0
\(821\) 17.8885i 0.624314i 0.950030 + 0.312157i \(0.101052\pi\)
−0.950030 + 0.312157i \(0.898948\pi\)
\(822\) 0 0
\(823\) −17.0000 + 17.0000i −0.592583 + 0.592583i −0.938328 0.345746i \(-0.887626\pi\)
0.345746 + 0.938328i \(0.387626\pi\)
\(824\) 0 0
\(825\) 36.1803 13.8197i 1.25964 0.481139i
\(826\) 0 0
\(827\) −20.1246 + 20.1246i −0.699801 + 0.699801i −0.964368 0.264566i \(-0.914771\pi\)
0.264566 + 0.964368i \(0.414771\pi\)
\(828\) 0 0
\(829\) 16.0000i 0.555703i −0.960624 0.277851i \(-0.910378\pi\)
0.960624 0.277851i \(-0.0896223\pi\)
\(830\) 0 0
\(831\) −9.00000 + 20.1246i −0.312207 + 0.698115i
\(832\) 0 0
\(833\) −11.1803 11.1803i −0.387376 0.387376i
\(834\) 0 0
\(835\) −25.0000 + 25.0000i −0.865161 + 0.865161i
\(836\) 0 0
\(837\) 18.4721 + 9.52786i 0.638490 + 0.329331i
\(838\) 0 0
\(839\) 35.7771 1.23516 0.617581 0.786507i \(-0.288113\pi\)
0.617581 + 0.786507i \(0.288113\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 43.4164 16.5836i 1.49534 0.571169i
\(844\) 0 0
\(845\) 11.1803i 0.384615i
\(846\) 0 0
\(847\) −9.00000 9.00000i −0.309244 0.309244i
\(848\) 0 0
\(849\) −38.0132 17.0000i −1.30461 0.583438i
\(850\) 0 0
\(851\) 13.4164i 0.459909i
\(852\) 0 0
\(853\) 37.0000 37.0000i 1.26686 1.26686i 0.319152 0.947703i \(-0.396602\pi\)
0.947703 0.319152i \(-0.103398\pi\)
\(854\) 0 0
\(855\) −8.94427 10.0000i −0.305888 0.341993i
\(856\) 0 0
\(857\) −6.70820 + 6.70820i −0.229148 + 0.229148i −0.812337 0.583189i \(-0.801805\pi\)
0.583189 + 0.812337i \(0.301805\pi\)
\(858\) 0 0
\(859\) 14.0000i 0.477674i −0.971060 0.238837i \(-0.923234\pi\)
0.971060 0.238837i \(-0.0767661\pi\)
\(860\) 0 0
\(861\) 20.0000 + 8.94427i 0.681598 + 0.304820i
\(862\) 0 0
\(863\) −24.5967 24.5967i −0.837283 0.837283i 0.151217 0.988501i \(-0.451681\pi\)
−0.988501 + 0.151217i \(0.951681\pi\)
\(864\) 0 0
\(865\) −35.0000 35.0000i −1.19004 1.19004i
\(866\) 0 0
\(867\) −11.3262 + 4.32624i −0.384659 + 0.146927i
\(868\) 0 0
\(869\) 26.8328 0.910241
\(870\) 0 0
\(871\) −6.00000 −0.203302
\(872\) 0 0
\(873\) −38.1246 2.12461i −1.29032 0.0719072i
\(874\) 0 0
\(875\) 11.1803 + 11.1803i 0.377964 + 0.377964i
\(876\) 0 0
\(877\) 9.00000 + 9.00000i 0.303908 + 0.303908i 0.842541 0.538632i \(-0.181059\pi\)
−0.538632 + 0.842541i \(0.681059\pi\)
\(878\) 0 0
\(879\) 6.70820 15.0000i 0.226262 0.505937i
\(880\) 0 0
\(881\) 44.7214i 1.50670i 0.657619 + 0.753350i \(0.271564\pi\)
−0.657619 + 0.753350i \(0.728436\pi\)
\(882\) 0 0
\(883\) 23.0000 23.0000i 0.774012 0.774012i −0.204794 0.978805i \(-0.565652\pi\)
0.978805 + 0.204794i \(0.0656523\pi\)
\(884\) 0 0
\(885\) −12.3607 32.3607i −0.415500 1.08779i
\(886\) 0 0
\(887\) −20.1246 + 20.1246i −0.675718 + 0.675718i −0.959028 0.283310i \(-0.908567\pi\)
0.283310 + 0.959028i \(0.408567\pi\)
\(888\) 0 0
\(889\) 26.0000i 0.872012i
\(890\) 0 0
\(891\) −40.0000 4.47214i −1.34005 0.149822i
\(892\) 0 0
\(893\) 13.4164 + 13.4164i 0.448963 + 0.448963i
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) −8.29180 21.7082i −0.276855 0.724816i
\(898\) 0 0
\(899\) 17.8885 0.596616
\(900\) 0 0
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 3.70820 + 9.70820i 0.123401 + 0.323069i
\(904\) 0 0
\(905\) 31.3050 1.04061
\(906\) 0 0
\(907\) −7.00000 7.00000i −0.232431 0.232431i 0.581276 0.813707i \(-0.302554\pi\)
−0.813707 + 0.581276i \(0.802554\pi\)
\(908\) 0 0
\(909\) −17.8885 20.0000i −0.593326 0.663358i
\(910\) 0 0
\(911\) 49.1935i 1.62985i −0.579564 0.814927i \(-0.696777\pi\)
0.579564 0.814927i \(-0.303223\pi\)
\(912\) 0 0
\(913\) −30.0000 + 30.0000i −0.992855 + 0.992855i
\(914\) 0 0
\(915\) −8.29180 21.7082i −0.274118 0.717651i
\(916\) 0 0
\(917\) 4.47214 4.47214i 0.147683 0.147683i
\(918\) 0 0
\(919\) 22.0000i 0.725713i 0.931845 + 0.362857i \(0.118198\pi\)
−0.931845 + 0.362857i \(0.881802\pi\)
\(920\) 0 0
\(921\) −21.0000 + 46.9574i −0.691974 + 1.54730i
\(922\) 0 0
\(923\) −13.4164 13.4164i −0.441606 0.441606i
\(924\) 0 0
\(925\) −15.0000 15.0000i −0.493197 0.493197i
\(926\) 0 0
\(927\) 0.236068 4.23607i 0.00775349 0.139131i
\(928\) 0 0
\(929\) −4.47214 −0.146726 −0.0733630 0.997305i \(-0.523373\pi\)
−0.0733630 + 0.997305i \(0.523373\pi\)
\(930\) 0 0
\(931\) 10.0000 0.327737
\(932\) 0 0
\(933\) −7.23607 + 2.76393i −0.236898 + 0.0904871i
\(934\) 0 0
\(935\) −22.3607 22.3607i −0.731272 0.731272i
\(936\) 0 0
\(937\) −11.0000 11.0000i −0.359354 0.359354i 0.504221 0.863575i \(-0.331780\pi\)
−0.863575 + 0.504221i \(0.831780\pi\)
\(938\) 0 0
\(939\) −6.70820 3.00000i −0.218914 0.0979013i
\(940\) 0 0
\(941\) 26.8328i 0.874725i −0.899285 0.437362i \(-0.855913\pi\)
0.899285 0.437362i \(-0.144087\pi\)
\(942\) 0 0
\(943\) 20.0000 20.0000i 0.651290 0.651290i
\(944\) 0 0
\(945\) −5.00000 15.6525i −0.162650 0.509175i
\(946\) 0 0
\(947\) −11.1803 + 11.1803i −0.363312 + 0.363312i −0.865031 0.501719i \(-0.832701\pi\)
0.501719 + 0.865031i \(0.332701\pi\)
\(948\) 0 0
\(949\) 6.00000i 0.194768i
\(950\) 0 0
\(951\) 45.0000 + 20.1246i 1.45922 + 0.652585i
\(952\) 0 0
\(953\) −33.5410 33.5410i −1.08650 1.08650i −0.995886 0.0906141i \(-0.971117\pi\)
−0.0906141 0.995886i \(-0.528883\pi\)
\(954\) 0 0
\(955\) 10.0000i 0.323592i
\(956\) 0 0
\(957\) −32.3607 + 12.3607i −1.04607 + 0.399564i
\(958\) 0 0
\(959\) −13.4164 −0.433238
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0.527864 9.47214i 0.0170102 0.305235i
\(964\) 0 0
\(965\) −6.70820 + 6.70820i −0.215945 + 0.215945i
\(966\) 0 0
\(967\) −7.00000 7.00000i −0.225105 0.225105i 0.585539 0.810644i \(-0.300883\pi\)
−0.810644 + 0.585539i \(0.800883\pi\)
\(968\) 0 0
\(969\) −4.47214 + 10.0000i −0.143666 + 0.321246i
\(970\) 0 0
\(971\) 22.3607i 0.717588i −0.933417 0.358794i \(-0.883188\pi\)
0.933417 0.358794i \(-0.116812\pi\)
\(972\) 0 0
\(973\) 14.0000 14.0000i 0.448819 0.448819i
\(974\) 0 0
\(975\) −33.5410 15.0000i −1.07417 0.480384i
\(976\) 0 0
\(977\) −6.70820 + 6.70820i −0.214614 + 0.214614i −0.806224 0.591610i \(-0.798493\pi\)
0.591610 + 0.806224i \(0.298493\pi\)
\(978\) 0 0
\(979\) 20.0000i 0.639203i
\(980\) 0 0
\(981\) −16.0000 17.8885i −0.510841 0.571137i
\(982\) 0 0
\(983\) 11.1803 + 11.1803i 0.356597 + 0.356597i 0.862557 0.505960i \(-0.168862\pi\)
−0.505960 + 0.862557i \(0.668862\pi\)
\(984\) 0 0
\(985\) −15.0000 + 15.0000i −0.477940 + 0.477940i
\(986\) 0 0
\(987\) 8.29180 + 21.7082i 0.263931 + 0.690980i
\(988\) 0 0
\(989\) 13.4164 0.426617
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 4.94427 + 12.9443i 0.156902 + 0.410774i
\(994\) 0 0
\(995\) 40.2492i 1.27599i
\(996\) 0 0
\(997\) −23.0000 23.0000i −0.728417 0.728417i 0.241887 0.970304i \(-0.422234\pi\)
−0.970304 + 0.241887i \(0.922234\pi\)
\(998\) 0 0
\(999\) 6.70820 + 21.0000i 0.212238 + 0.664411i
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 240.2.v.b.17.2 4
3.2 odd 2 inner 240.2.v.b.17.1 4
4.3 odd 2 60.2.i.a.17.1 4
5.2 odd 4 1200.2.v.i.593.2 4
5.3 odd 4 inner 240.2.v.b.113.1 4
5.4 even 2 1200.2.v.i.257.1 4
8.3 odd 2 960.2.v.e.257.2 4
8.5 even 2 960.2.v.h.257.1 4
12.11 even 2 60.2.i.a.17.2 yes 4
15.2 even 4 1200.2.v.i.593.1 4
15.8 even 4 inner 240.2.v.b.113.2 4
15.14 odd 2 1200.2.v.i.257.2 4
20.3 even 4 60.2.i.a.53.2 yes 4
20.7 even 4 300.2.i.a.293.1 4
20.19 odd 2 300.2.i.a.257.2 4
24.5 odd 2 960.2.v.h.257.2 4
24.11 even 2 960.2.v.e.257.1 4
36.7 odd 6 1620.2.x.b.377.1 8
36.11 even 6 1620.2.x.b.377.2 8
36.23 even 6 1620.2.x.b.917.2 8
36.31 odd 6 1620.2.x.b.917.1 8
40.3 even 4 960.2.v.e.833.1 4
40.13 odd 4 960.2.v.h.833.2 4
60.23 odd 4 60.2.i.a.53.1 yes 4
60.47 odd 4 300.2.i.a.293.2 4
60.59 even 2 300.2.i.a.257.1 4
120.53 even 4 960.2.v.h.833.1 4
120.83 odd 4 960.2.v.e.833.2 4
180.23 odd 12 1620.2.x.b.593.1 8
180.43 even 12 1620.2.x.b.53.2 8
180.83 odd 12 1620.2.x.b.53.1 8
180.103 even 12 1620.2.x.b.593.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.i.a.17.1 4 4.3 odd 2
60.2.i.a.17.2 yes 4 12.11 even 2
60.2.i.a.53.1 yes 4 60.23 odd 4
60.2.i.a.53.2 yes 4 20.3 even 4
240.2.v.b.17.1 4 3.2 odd 2 inner
240.2.v.b.17.2 4 1.1 even 1 trivial
240.2.v.b.113.1 4 5.3 odd 4 inner
240.2.v.b.113.2 4 15.8 even 4 inner
300.2.i.a.257.1 4 60.59 even 2
300.2.i.a.257.2 4 20.19 odd 2
300.2.i.a.293.1 4 20.7 even 4
300.2.i.a.293.2 4 60.47 odd 4
960.2.v.e.257.1 4 24.11 even 2
960.2.v.e.257.2 4 8.3 odd 2
960.2.v.e.833.1 4 40.3 even 4
960.2.v.e.833.2 4 120.83 odd 4
960.2.v.h.257.1 4 8.5 even 2
960.2.v.h.257.2 4 24.5 odd 2
960.2.v.h.833.1 4 120.53 even 4
960.2.v.h.833.2 4 40.13 odd 4
1200.2.v.i.257.1 4 5.4 even 2
1200.2.v.i.257.2 4 15.14 odd 2
1200.2.v.i.593.1 4 15.2 even 4
1200.2.v.i.593.2 4 5.2 odd 4
1620.2.x.b.53.1 8 180.83 odd 12
1620.2.x.b.53.2 8 180.43 even 12
1620.2.x.b.377.1 8 36.7 odd 6
1620.2.x.b.377.2 8 36.11 even 6
1620.2.x.b.593.1 8 180.23 odd 12
1620.2.x.b.593.2 8 180.103 even 12
1620.2.x.b.917.1 8 36.31 odd 6
1620.2.x.b.917.2 8 36.23 even 6