Properties

Label 1200.3.bg.a.1057.1
Level $1200$
Weight $3$
Character 1200.1057
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(193,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1057
Dual form 1200.3.bg.a.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(-4.77526 - 4.77526i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(-4.77526 - 4.77526i) q^{7} -3.00000i q^{9} -20.6969 q^{11} +(-8.32577 + 8.32577i) q^{13} +(-1.34847 - 1.34847i) q^{17} +33.6969i q^{19} +11.6969 q^{21} +(16.0454 - 16.0454i) q^{23} +(3.67423 + 3.67423i) q^{27} +8.69694i q^{29} +49.0908 q^{31} +(25.3485 - 25.3485i) q^{33} +(-36.4949 - 36.4949i) q^{37} -20.3939i q^{39} +33.3031 q^{41} +(24.1237 - 24.1237i) q^{43} +(13.9546 + 13.9546i) q^{47} -3.39388i q^{49} +3.30306 q^{51} +(59.3939 - 59.3939i) q^{53} +(-41.2702 - 41.2702i) q^{57} +30.0000i q^{59} -47.7878 q^{61} +(-14.3258 + 14.3258i) q^{63} +(-17.0227 - 17.0227i) q^{67} +39.3031i q^{69} +9.30306 q^{71} +(22.2929 - 22.2929i) q^{73} +(98.8332 + 98.8332i) q^{77} -27.3939i q^{79} -9.00000 q^{81} +(63.4393 - 63.4393i) q^{83} +(-10.6515 - 10.6515i) q^{87} +17.3939i q^{89} +79.5153 q^{91} +(-60.1237 + 60.1237i) q^{93} +(41.8763 + 41.8763i) q^{97} +62.0908i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{7} - 24 q^{11} - 48 q^{13} + 24 q^{17} - 12 q^{21} - 24 q^{23} + 20 q^{31} + 72 q^{33} - 48 q^{37} + 192 q^{41} + 72 q^{43} + 144 q^{47} + 72 q^{51} + 120 q^{53} - 72 q^{57} + 44 q^{61} - 72 q^{63} - 24 q^{67} + 96 q^{71} - 48 q^{73} + 72 q^{77} - 36 q^{81} + 48 q^{83} - 72 q^{87} + 612 q^{91} - 216 q^{93} + 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.22474 + 1.22474i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.77526 4.77526i −0.682179 0.682179i 0.278312 0.960491i \(-0.410225\pi\)
−0.960491 + 0.278312i \(0.910225\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −20.6969 −1.88154 −0.940770 0.339046i \(-0.889896\pi\)
−0.940770 + 0.339046i \(0.889896\pi\)
\(12\) 0 0
\(13\) −8.32577 + 8.32577i −0.640443 + 0.640443i −0.950664 0.310221i \(-0.899597\pi\)
0.310221 + 0.950664i \(0.399597\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.34847 1.34847i −0.0793217 0.0793217i 0.666333 0.745654i \(-0.267863\pi\)
−0.745654 + 0.666333i \(0.767863\pi\)
\(18\) 0 0
\(19\) 33.6969i 1.77352i 0.462227 + 0.886762i \(0.347050\pi\)
−0.462227 + 0.886762i \(0.652950\pi\)
\(20\) 0 0
\(21\) 11.6969 0.556997
\(22\) 0 0
\(23\) 16.0454 16.0454i 0.697626 0.697626i −0.266272 0.963898i \(-0.585792\pi\)
0.963898 + 0.266272i \(0.0857918\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 8.69694i 0.299894i 0.988694 + 0.149947i \(0.0479104\pi\)
−0.988694 + 0.149947i \(0.952090\pi\)
\(30\) 0 0
\(31\) 49.0908 1.58357 0.791787 0.610797i \(-0.209151\pi\)
0.791787 + 0.610797i \(0.209151\pi\)
\(32\) 0 0
\(33\) 25.3485 25.3485i 0.768135 0.768135i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −36.4949 36.4949i −0.986349 0.986349i 0.0135595 0.999908i \(-0.495684\pi\)
−0.999908 + 0.0135595i \(0.995684\pi\)
\(38\) 0 0
\(39\) 20.3939i 0.522920i
\(40\) 0 0
\(41\) 33.3031 0.812270 0.406135 0.913813i \(-0.366876\pi\)
0.406135 + 0.913813i \(0.366876\pi\)
\(42\) 0 0
\(43\) 24.1237 24.1237i 0.561017 0.561017i −0.368579 0.929596i \(-0.620156\pi\)
0.929596 + 0.368579i \(0.120156\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.9546 + 13.9546i 0.296906 + 0.296906i 0.839801 0.542895i \(-0.182672\pi\)
−0.542895 + 0.839801i \(0.682672\pi\)
\(48\) 0 0
\(49\) 3.39388i 0.0692628i
\(50\) 0 0
\(51\) 3.30306 0.0647659
\(52\) 0 0
\(53\) 59.3939 59.3939i 1.12064 1.12064i 0.128994 0.991645i \(-0.458825\pi\)
0.991645 0.128994i \(-0.0411747\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −41.2702 41.2702i −0.724038 0.724038i
\(58\) 0 0
\(59\) 30.0000i 0.508475i 0.967142 + 0.254237i \(0.0818244\pi\)
−0.967142 + 0.254237i \(0.918176\pi\)
\(60\) 0 0
\(61\) −47.7878 −0.783406 −0.391703 0.920092i \(-0.628114\pi\)
−0.391703 + 0.920092i \(0.628114\pi\)
\(62\) 0 0
\(63\) −14.3258 + 14.3258i −0.227393 + 0.227393i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −17.0227 17.0227i −0.254070 0.254070i 0.568567 0.822637i \(-0.307498\pi\)
−0.822637 + 0.568567i \(0.807498\pi\)
\(68\) 0 0
\(69\) 39.3031i 0.569610i
\(70\) 0 0
\(71\) 9.30306 0.131029 0.0655145 0.997852i \(-0.479131\pi\)
0.0655145 + 0.997852i \(0.479131\pi\)
\(72\) 0 0
\(73\) 22.2929 22.2929i 0.305382 0.305382i −0.537733 0.843115i \(-0.680719\pi\)
0.843115 + 0.537733i \(0.180719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 98.8332 + 98.8332i 1.28355 + 1.28355i
\(78\) 0 0
\(79\) 27.3939i 0.346758i −0.984855 0.173379i \(-0.944531\pi\)
0.984855 0.173379i \(-0.0554685\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 63.4393 63.4393i 0.764329 0.764329i −0.212773 0.977102i \(-0.568249\pi\)
0.977102 + 0.212773i \(0.0682495\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −10.6515 10.6515i −0.122431 0.122431i
\(88\) 0 0
\(89\) 17.3939i 0.195437i 0.995214 + 0.0977184i \(0.0311545\pi\)
−0.995214 + 0.0977184i \(0.968846\pi\)
\(90\) 0 0
\(91\) 79.5153 0.873795
\(92\) 0 0
\(93\) −60.1237 + 60.1237i −0.646492 + 0.646492i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 41.8763 + 41.8763i 0.431714 + 0.431714i 0.889211 0.457497i \(-0.151254\pi\)
−0.457497 + 0.889211i \(0.651254\pi\)
\(98\) 0 0
\(99\) 62.0908i 0.627180i
\(100\) 0 0
\(101\) 162.879 1.61266 0.806330 0.591467i \(-0.201451\pi\)
0.806330 + 0.591467i \(0.201451\pi\)
\(102\) 0 0
\(103\) 36.9898 36.9898i 0.359124 0.359124i −0.504366 0.863490i \(-0.668274\pi\)
0.863490 + 0.504366i \(0.168274\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −134.091 134.091i −1.25319 1.25319i −0.954284 0.298901i \(-0.903380\pi\)
−0.298901 0.954284i \(-0.596620\pi\)
\(108\) 0 0
\(109\) 145.000i 1.33028i 0.746721 + 0.665138i \(0.231627\pi\)
−0.746721 + 0.665138i \(0.768373\pi\)
\(110\) 0 0
\(111\) 89.3939 0.805350
\(112\) 0 0
\(113\) −13.2122 + 13.2122i −0.116923 + 0.116923i −0.763147 0.646225i \(-0.776347\pi\)
0.646225 + 0.763147i \(0.276347\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 24.9773 + 24.9773i 0.213481 + 0.213481i
\(118\) 0 0
\(119\) 12.8786i 0.108223i
\(120\) 0 0
\(121\) 307.363 2.54019
\(122\) 0 0
\(123\) −40.7878 + 40.7878i −0.331608 + 0.331608i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 50.6969 + 50.6969i 0.399188 + 0.399188i 0.877947 0.478758i \(-0.158913\pi\)
−0.478758 + 0.877947i \(0.658913\pi\)
\(128\) 0 0
\(129\) 59.0908i 0.458068i
\(130\) 0 0
\(131\) 156.272 1.19292 0.596460 0.802643i \(-0.296574\pi\)
0.596460 + 0.802643i \(0.296574\pi\)
\(132\) 0 0
\(133\) 160.911 160.911i 1.20986 1.20986i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −91.3485 91.3485i −0.666777 0.666777i 0.290191 0.956969i \(-0.406281\pi\)
−0.956969 + 0.290191i \(0.906281\pi\)
\(138\) 0 0
\(139\) 35.2122i 0.253326i 0.991946 + 0.126663i \(0.0404266\pi\)
−0.991946 + 0.126663i \(0.959573\pi\)
\(140\) 0 0
\(141\) −34.1816 −0.242423
\(142\) 0 0
\(143\) 172.318 172.318i 1.20502 1.20502i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.15663 + 4.15663i 0.0282764 + 0.0282764i
\(148\) 0 0
\(149\) 4.78775i 0.0321326i −0.999871 0.0160663i \(-0.994886\pi\)
0.999871 0.0160663i \(-0.00511428\pi\)
\(150\) 0 0
\(151\) −37.8786 −0.250851 −0.125426 0.992103i \(-0.540030\pi\)
−0.125426 + 0.992103i \(0.540030\pi\)
\(152\) 0 0
\(153\) −4.04541 + 4.04541i −0.0264406 + 0.0264406i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −29.1339 29.1339i −0.185566 0.185566i 0.608210 0.793776i \(-0.291888\pi\)
−0.793776 + 0.608210i \(0.791888\pi\)
\(158\) 0 0
\(159\) 145.485i 0.914998i
\(160\) 0 0
\(161\) −153.242 −0.951813
\(162\) 0 0
\(163\) −193.856 + 193.856i −1.18930 + 1.18930i −0.212038 + 0.977261i \(0.568010\pi\)
−0.977261 + 0.212038i \(0.931990\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 167.666 + 167.666i 1.00399 + 1.00399i 0.999992 + 0.00399795i \(0.00127259\pi\)
0.00399795 + 0.999992i \(0.498727\pi\)
\(168\) 0 0
\(169\) 30.3633i 0.179664i
\(170\) 0 0
\(171\) 101.091 0.591174
\(172\) 0 0
\(173\) 74.8332 74.8332i 0.432562 0.432562i −0.456937 0.889499i \(-0.651054\pi\)
0.889499 + 0.456937i \(0.151054\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −36.7423 36.7423i −0.207584 0.207584i
\(178\) 0 0
\(179\) 229.151i 1.28017i −0.768303 0.640087i \(-0.778898\pi\)
0.768303 0.640087i \(-0.221102\pi\)
\(180\) 0 0
\(181\) −135.182 −0.746860 −0.373430 0.927658i \(-0.621818\pi\)
−0.373430 + 0.927658i \(0.621818\pi\)
\(182\) 0 0
\(183\) 58.5278 58.5278i 0.319824 0.319824i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 27.9092 + 27.9092i 0.149247 + 0.149247i
\(188\) 0 0
\(189\) 35.0908i 0.185666i
\(190\) 0 0
\(191\) 21.9092 0.114708 0.0573539 0.998354i \(-0.481734\pi\)
0.0573539 + 0.998354i \(0.481734\pi\)
\(192\) 0 0
\(193\) −98.4870 + 98.4870i −0.510295 + 0.510295i −0.914617 0.404322i \(-0.867508\pi\)
0.404322 + 0.914617i \(0.367508\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 63.4393 + 63.4393i 0.322027 + 0.322027i 0.849544 0.527517i \(-0.176877\pi\)
−0.527517 + 0.849544i \(0.676877\pi\)
\(198\) 0 0
\(199\) 181.091i 0.910004i 0.890490 + 0.455002i \(0.150361\pi\)
−0.890490 + 0.455002i \(0.849639\pi\)
\(200\) 0 0
\(201\) 41.6969 0.207447
\(202\) 0 0
\(203\) 41.5301 41.5301i 0.204582 0.204582i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −48.1362 48.1362i −0.232542 0.232542i
\(208\) 0 0
\(209\) 697.423i 3.33695i
\(210\) 0 0
\(211\) −28.7276 −0.136150 −0.0680748 0.997680i \(-0.521686\pi\)
−0.0680748 + 0.997680i \(0.521686\pi\)
\(212\) 0 0
\(213\) −11.3939 + 11.3939i −0.0534924 + 0.0534924i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −234.421 234.421i −1.08028 1.08028i
\(218\) 0 0
\(219\) 54.6061i 0.249343i
\(220\) 0 0
\(221\) 22.4541 0.101602
\(222\) 0 0
\(223\) 33.1793 33.1793i 0.148786 0.148786i −0.628789 0.777576i \(-0.716449\pi\)
0.777576 + 0.628789i \(0.216449\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 252.879 + 252.879i 1.11400 + 1.11400i 0.992604 + 0.121399i \(0.0387380\pi\)
0.121399 + 0.992604i \(0.461262\pi\)
\(228\) 0 0
\(229\) 275.000i 1.20087i 0.799672 + 0.600437i \(0.205007\pi\)
−0.799672 + 0.600437i \(0.794993\pi\)
\(230\) 0 0
\(231\) −242.091 −1.04801
\(232\) 0 0
\(233\) 223.757 223.757i 0.960331 0.960331i −0.0389116 0.999243i \(-0.512389\pi\)
0.999243 + 0.0389116i \(0.0123891\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 33.5505 + 33.5505i 0.141563 + 0.141563i
\(238\) 0 0
\(239\) 305.666i 1.27894i −0.768817 0.639469i \(-0.779154\pi\)
0.768817 0.639469i \(-0.220846\pi\)
\(240\) 0 0
\(241\) 167.000 0.692946 0.346473 0.938060i \(-0.387379\pi\)
0.346473 + 0.938060i \(0.387379\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −280.553 280.553i −1.13584 1.13584i
\(248\) 0 0
\(249\) 155.394i 0.624072i
\(250\) 0 0
\(251\) −261.576 −1.04213 −0.521067 0.853516i \(-0.674466\pi\)
−0.521067 + 0.853516i \(0.674466\pi\)
\(252\) 0 0
\(253\) −332.091 + 332.091i −1.31261 + 1.31261i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 94.5153 + 94.5153i 0.367764 + 0.367764i 0.866661 0.498897i \(-0.166262\pi\)
−0.498897 + 0.866661i \(0.666262\pi\)
\(258\) 0 0
\(259\) 348.545i 1.34573i
\(260\) 0 0
\(261\) 26.0908 0.0999648
\(262\) 0 0
\(263\) −34.1816 + 34.1816i −0.129968 + 0.129968i −0.769098 0.639130i \(-0.779294\pi\)
0.639130 + 0.769098i \(0.279294\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −21.3031 21.3031i −0.0797867 0.0797867i
\(268\) 0 0
\(269\) 389.605i 1.44835i 0.689618 + 0.724173i \(0.257778\pi\)
−0.689618 + 0.724173i \(0.742222\pi\)
\(270\) 0 0
\(271\) −156.788 −0.578553 −0.289276 0.957246i \(-0.593415\pi\)
−0.289276 + 0.957246i \(0.593415\pi\)
\(272\) 0 0
\(273\) −97.3860 + 97.3860i −0.356725 + 0.356725i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 285.588 + 285.588i 1.03100 + 1.03100i 0.999504 + 0.0314999i \(0.0100284\pi\)
0.0314999 + 0.999504i \(0.489972\pi\)
\(278\) 0 0
\(279\) 147.272i 0.527858i
\(280\) 0 0
\(281\) −20.1520 −0.0717155 −0.0358577 0.999357i \(-0.511416\pi\)
−0.0358577 + 0.999357i \(0.511416\pi\)
\(282\) 0 0
\(283\) 170.573 170.573i 0.602732 0.602732i −0.338304 0.941037i \(-0.609853\pi\)
0.941037 + 0.338304i \(0.109853\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −159.031 159.031i −0.554114 0.554114i
\(288\) 0 0
\(289\) 285.363i 0.987416i
\(290\) 0 0
\(291\) −102.576 −0.352493
\(292\) 0 0
\(293\) 259.621 259.621i 0.886078 0.886078i −0.108066 0.994144i \(-0.534466\pi\)
0.994144 + 0.108066i \(0.0344656\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −76.0454 76.0454i −0.256045 0.256045i
\(298\) 0 0
\(299\) 267.181i 0.893581i
\(300\) 0 0
\(301\) −230.394 −0.765428
\(302\) 0 0
\(303\) −199.485 + 199.485i −0.658365 + 0.658365i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −118.911 118.911i −0.387334 0.387334i 0.486402 0.873735i \(-0.338309\pi\)
−0.873735 + 0.486402i \(0.838309\pi\)
\(308\) 0 0
\(309\) 90.6061i 0.293224i
\(310\) 0 0
\(311\) 52.7878 0.169736 0.0848678 0.996392i \(-0.472953\pi\)
0.0848678 + 0.996392i \(0.472953\pi\)
\(312\) 0 0
\(313\) −238.553 + 238.553i −0.762150 + 0.762150i −0.976711 0.214561i \(-0.931168\pi\)
0.214561 + 0.976711i \(0.431168\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −48.9398 48.9398i −0.154384 0.154384i 0.625689 0.780073i \(-0.284818\pi\)
−0.780073 + 0.625689i \(0.784818\pi\)
\(318\) 0 0
\(319\) 180.000i 0.564263i
\(320\) 0 0
\(321\) 328.454 1.02322
\(322\) 0 0
\(323\) 45.4393 45.4393i 0.140679 0.140679i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −177.588 177.588i −0.543083 0.543083i
\(328\) 0 0
\(329\) 133.273i 0.405087i
\(330\) 0 0
\(331\) 291.939 0.881990 0.440995 0.897510i \(-0.354626\pi\)
0.440995 + 0.897510i \(0.354626\pi\)
\(332\) 0 0
\(333\) −109.485 + 109.485i −0.328783 + 0.328783i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −266.462 266.462i −0.790688 0.790688i 0.190918 0.981606i \(-0.438854\pi\)
−0.981606 + 0.190918i \(0.938854\pi\)
\(338\) 0 0
\(339\) 32.3633i 0.0954668i
\(340\) 0 0
\(341\) −1016.03 −2.97956
\(342\) 0 0
\(343\) −250.194 + 250.194i −0.729429 + 0.729429i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 141.773 + 141.773i 0.408568 + 0.408568i 0.881239 0.472671i \(-0.156710\pi\)
−0.472671 + 0.881239i \(0.656710\pi\)
\(348\) 0 0
\(349\) 183.939i 0.527045i −0.964653 0.263523i \(-0.915116\pi\)
0.964653 0.263523i \(-0.0848844\pi\)
\(350\) 0 0
\(351\) −61.1816 −0.174307
\(352\) 0 0
\(353\) −54.7423 + 54.7423i −0.155077 + 0.155077i −0.780381 0.625304i \(-0.784975\pi\)
0.625304 + 0.780381i \(0.284975\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.7730 15.7730i −0.0441820 0.0441820i
\(358\) 0 0
\(359\) 648.272i 1.80577i −0.429879 0.902886i \(-0.641444\pi\)
0.429879 0.902886i \(-0.358556\pi\)
\(360\) 0 0
\(361\) −774.484 −2.14538
\(362\) 0 0
\(363\) −376.442 + 376.442i −1.03703 + 1.03703i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 56.6594 + 56.6594i 0.154385 + 0.154385i 0.780073 0.625688i \(-0.215182\pi\)
−0.625688 + 0.780073i \(0.715182\pi\)
\(368\) 0 0
\(369\) 99.9092i 0.270757i
\(370\) 0 0
\(371\) −567.242 −1.52895
\(372\) 0 0
\(373\) −155.815 + 155.815i −0.417735 + 0.417735i −0.884422 0.466688i \(-0.845447\pi\)
0.466688 + 0.884422i \(0.345447\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −72.4087 72.4087i −0.192065 0.192065i
\(378\) 0 0
\(379\) 674.181i 1.77884i −0.457090 0.889420i \(-0.651108\pi\)
0.457090 0.889420i \(-0.348892\pi\)
\(380\) 0 0
\(381\) −124.182 −0.325936
\(382\) 0 0
\(383\) −244.182 + 244.182i −0.637550 + 0.637550i −0.949950 0.312401i \(-0.898867\pi\)
0.312401 + 0.949950i \(0.398867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −72.3712 72.3712i −0.187006 0.187006i
\(388\) 0 0
\(389\) 44.7582i 0.115060i 0.998344 + 0.0575298i \(0.0183224\pi\)
−0.998344 + 0.0575298i \(0.981678\pi\)
\(390\) 0 0
\(391\) −43.2735 −0.110674
\(392\) 0 0
\(393\) −191.394 + 191.394i −0.487007 + 0.487007i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 548.648 + 548.648i 1.38199 + 1.38199i 0.841090 + 0.540896i \(0.181915\pi\)
0.540896 + 0.841090i \(0.318085\pi\)
\(398\) 0 0
\(399\) 394.151i 0.987847i
\(400\) 0 0
\(401\) 273.303 0.681554 0.340777 0.940144i \(-0.389310\pi\)
0.340777 + 0.940144i \(0.389310\pi\)
\(402\) 0 0
\(403\) −408.719 + 408.719i −1.01419 + 1.01419i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 755.333 + 755.333i 1.85585 + 1.85585i
\(408\) 0 0
\(409\) 197.182i 0.482107i −0.970512 0.241053i \(-0.922507\pi\)
0.970512 0.241053i \(-0.0774929\pi\)
\(410\) 0 0
\(411\) 223.757 0.544421
\(412\) 0 0
\(413\) 143.258 143.258i 0.346871 0.346871i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −43.1260 43.1260i −0.103420 0.103420i
\(418\) 0 0
\(419\) 613.485i 1.46416i −0.681217 0.732082i \(-0.738549\pi\)
0.681217 0.732082i \(-0.261451\pi\)
\(420\) 0 0
\(421\) 128.120 0.304324 0.152162 0.988356i \(-0.451376\pi\)
0.152162 + 0.988356i \(0.451376\pi\)
\(422\) 0 0
\(423\) 41.8638 41.8638i 0.0989687 0.0989687i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 228.199 + 228.199i 0.534423 + 0.534423i
\(428\) 0 0
\(429\) 422.091i 0.983895i
\(430\) 0 0
\(431\) 393.242 0.912394 0.456197 0.889879i \(-0.349211\pi\)
0.456197 + 0.889879i \(0.349211\pi\)
\(432\) 0 0
\(433\) 221.381 221.381i 0.511273 0.511273i −0.403643 0.914917i \(-0.632256\pi\)
0.914917 + 0.403643i \(0.132256\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 540.681 + 540.681i 1.23726 + 1.23726i
\(438\) 0 0
\(439\) 361.091i 0.822530i 0.911516 + 0.411265i \(0.134913\pi\)
−0.911516 + 0.411265i \(0.865087\pi\)
\(440\) 0 0
\(441\) −10.1816 −0.0230876
\(442\) 0 0
\(443\) 440.636 440.636i 0.994663 0.994663i −0.00532284 0.999986i \(-0.501694\pi\)
0.999986 + 0.00532284i \(0.00169432\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.86378 + 5.86378i 0.0131181 + 0.0131181i
\(448\) 0 0
\(449\) 280.849i 0.625499i −0.949836 0.312749i \(-0.898750\pi\)
0.949836 0.312749i \(-0.101250\pi\)
\(450\) 0 0
\(451\) −689.271 −1.52832
\(452\) 0 0
\(453\) 46.3916 46.3916i 0.102410 0.102410i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −261.576 261.576i −0.572375 0.572375i 0.360416 0.932792i \(-0.382635\pi\)
−0.932792 + 0.360416i \(0.882635\pi\)
\(458\) 0 0
\(459\) 9.90918i 0.0215886i
\(460\) 0 0
\(461\) −9.30306 −0.0201802 −0.0100901 0.999949i \(-0.503212\pi\)
−0.0100901 + 0.999949i \(0.503212\pi\)
\(462\) 0 0
\(463\) 252.495 252.495i 0.545345 0.545345i −0.379746 0.925091i \(-0.623988\pi\)
0.925091 + 0.379746i \(0.123988\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −366.470 366.470i −0.784732 0.784732i 0.195893 0.980625i \(-0.437239\pi\)
−0.980625 + 0.195893i \(0.937239\pi\)
\(468\) 0 0
\(469\) 162.576i 0.346643i
\(470\) 0 0
\(471\) 71.3633 0.151514
\(472\) 0 0
\(473\) −499.287 + 499.287i −1.05558 + 1.05558i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −178.182 178.182i −0.373546 0.373546i
\(478\) 0 0
\(479\) 442.182i 0.923135i 0.887105 + 0.461567i \(0.152713\pi\)
−0.887105 + 0.461567i \(0.847287\pi\)
\(480\) 0 0
\(481\) 607.696 1.26340
\(482\) 0 0
\(483\) 187.682 187.682i 0.388576 0.388576i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −100.083 100.083i −0.205509 0.205509i 0.596846 0.802355i \(-0.296420\pi\)
−0.802355 + 0.596846i \(0.796420\pi\)
\(488\) 0 0
\(489\) 474.848i 0.971059i
\(490\) 0 0
\(491\) −286.788 −0.584089 −0.292045 0.956405i \(-0.594336\pi\)
−0.292045 + 0.956405i \(0.594336\pi\)
\(492\) 0 0
\(493\) 11.7276 11.7276i 0.0237881 0.0237881i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −44.4245 44.4245i −0.0893853 0.0893853i
\(498\) 0 0
\(499\) 82.3643i 0.165059i −0.996589 0.0825294i \(-0.973700\pi\)
0.996589 0.0825294i \(-0.0262998\pi\)
\(500\) 0 0
\(501\) −410.697 −0.819754
\(502\) 0 0
\(503\) −219.848 + 219.848i −0.437073 + 0.437073i −0.891026 0.453952i \(-0.850014\pi\)
0.453952 + 0.891026i \(0.350014\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −37.1872 37.1872i −0.0733476 0.0733476i
\(508\) 0 0
\(509\) 805.696i 1.58290i −0.611234 0.791450i \(-0.709327\pi\)
0.611234 0.791450i \(-0.290673\pi\)
\(510\) 0 0
\(511\) −212.908 −0.416650
\(512\) 0 0
\(513\) −123.810 + 123.810i −0.241346 + 0.241346i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −288.817 288.817i −0.558641 0.558641i
\(518\) 0 0
\(519\) 183.303i 0.353185i
\(520\) 0 0
\(521\) 372.879 0.715698 0.357849 0.933779i \(-0.383510\pi\)
0.357849 + 0.933779i \(0.383510\pi\)
\(522\) 0 0
\(523\) −56.1395 + 56.1395i −0.107341 + 0.107341i −0.758738 0.651396i \(-0.774184\pi\)
0.651396 + 0.758738i \(0.274184\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −66.1975 66.1975i −0.125612 0.125612i
\(528\) 0 0
\(529\) 14.0898i 0.0266348i
\(530\) 0 0
\(531\) 90.0000 0.169492
\(532\) 0 0
\(533\) −277.273 + 277.273i −0.520213 + 0.520213i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 280.652 + 280.652i 0.522629 + 0.522629i
\(538\) 0 0
\(539\) 70.2429i 0.130321i
\(540\) 0 0
\(541\) 633.120 1.17028 0.585139 0.810933i \(-0.301040\pi\)
0.585139 + 0.810933i \(0.301040\pi\)
\(542\) 0 0
\(543\) 165.563 165.563i 0.304904 0.304904i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −32.0408 32.0408i −0.0585755 0.0585755i 0.677212 0.735788i \(-0.263188\pi\)
−0.735788 + 0.677212i \(0.763188\pi\)
\(548\) 0 0
\(549\) 143.363i 0.261135i
\(550\) 0 0
\(551\) −293.060 −0.531870
\(552\) 0 0
\(553\) −130.813 + 130.813i −0.236551 + 0.236551i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 701.741 + 701.741i 1.25986 + 1.25986i 0.951160 + 0.308699i \(0.0998936\pi\)
0.308699 + 0.951160i \(0.400106\pi\)
\(558\) 0 0
\(559\) 401.697i 0.718599i
\(560\) 0 0
\(561\) −68.3633 −0.121860
\(562\) 0 0
\(563\) 265.621 265.621i 0.471796 0.471796i −0.430700 0.902495i \(-0.641733\pi\)
0.902495 + 0.430700i \(0.141733\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 42.9773 + 42.9773i 0.0757977 + 0.0757977i
\(568\) 0 0
\(569\) 946.999i 1.66432i −0.554534 0.832161i \(-0.687104\pi\)
0.554534 0.832161i \(-0.312896\pi\)
\(570\) 0 0
\(571\) 337.334 0.590777 0.295389 0.955377i \(-0.404551\pi\)
0.295389 + 0.955377i \(0.404551\pi\)
\(572\) 0 0
\(573\) −26.8332 + 26.8332i −0.0468293 + 0.0468293i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −325.547 325.547i −0.564207 0.564207i 0.366293 0.930500i \(-0.380627\pi\)
−0.930500 + 0.366293i \(0.880627\pi\)
\(578\) 0 0
\(579\) 241.243i 0.416654i
\(580\) 0 0
\(581\) −605.878 −1.04282
\(582\) 0 0
\(583\) −1229.27 + 1229.27i −2.10853 + 2.10853i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −381.514 381.514i −0.649939 0.649939i 0.303039 0.952978i \(-0.401999\pi\)
−0.952978 + 0.303039i \(0.901999\pi\)
\(588\) 0 0
\(589\) 1654.21i 2.80851i
\(590\) 0 0
\(591\) −155.394 −0.262934
\(592\) 0 0
\(593\) 200.697 200.697i 0.338443 0.338443i −0.517338 0.855781i \(-0.673077\pi\)
0.855781 + 0.517338i \(0.173077\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −221.790 221.790i −0.371508 0.371508i
\(598\) 0 0
\(599\) 112.182i 0.187282i 0.995606 + 0.0936408i \(0.0298505\pi\)
−0.995606 + 0.0936408i \(0.970149\pi\)
\(600\) 0 0
\(601\) −59.5449 −0.0990764 −0.0495382 0.998772i \(-0.515775\pi\)
−0.0495382 + 0.998772i \(0.515775\pi\)
\(602\) 0 0
\(603\) −51.0681 + 51.0681i −0.0846901 + 0.0846901i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 406.070 + 406.070i 0.668979 + 0.668979i 0.957480 0.288500i \(-0.0931567\pi\)
−0.288500 + 0.957480i \(0.593157\pi\)
\(608\) 0 0
\(609\) 101.728i 0.167040i
\(610\) 0 0
\(611\) −232.365 −0.380303
\(612\) 0 0
\(613\) −81.7480 + 81.7480i −0.133357 + 0.133357i −0.770635 0.637277i \(-0.780061\pi\)
0.637277 + 0.770635i \(0.280061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −734.636 734.636i −1.19066 1.19066i −0.976883 0.213775i \(-0.931424\pi\)
−0.213775 0.976883i \(-0.568576\pi\)
\(618\) 0 0
\(619\) 266.303i 0.430215i 0.976590 + 0.215107i \(0.0690102\pi\)
−0.976590 + 0.215107i \(0.930990\pi\)
\(620\) 0 0
\(621\) 117.909 0.189870
\(622\) 0 0
\(623\) 83.0602 83.0602i 0.133323 0.133323i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 854.166 + 854.166i 1.36231 + 1.36231i
\(628\) 0 0
\(629\) 98.4245i 0.156478i
\(630\) 0 0
\(631\) −456.546 −0.723528 −0.361764 0.932270i \(-0.617825\pi\)
−0.361764 + 0.932270i \(0.617825\pi\)
\(632\) 0 0
\(633\) 35.1839 35.1839i 0.0555828 0.0555828i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 28.2566 + 28.2566i 0.0443589 + 0.0443589i
\(638\) 0 0
\(639\) 27.9092i 0.0436763i
\(640\) 0 0
\(641\) 538.120 0.839501 0.419751 0.907639i \(-0.362118\pi\)
0.419751 + 0.907639i \(0.362118\pi\)
\(642\) 0 0
\(643\) −31.5459 + 31.5459i −0.0490605 + 0.0490605i −0.731211 0.682151i \(-0.761045\pi\)
0.682151 + 0.731211i \(0.261045\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 112.849 + 112.849i 0.174419 + 0.174419i 0.788918 0.614499i \(-0.210642\pi\)
−0.614499 + 0.788918i \(0.710642\pi\)
\(648\) 0 0
\(649\) 620.908i 0.956715i
\(650\) 0 0
\(651\) 574.212 0.882046
\(652\) 0 0
\(653\) 491.803 491.803i 0.753143 0.753143i −0.221921 0.975065i \(-0.571233\pi\)
0.975065 + 0.221921i \(0.0712328\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −66.8786 66.8786i −0.101794 0.101794i
\(658\) 0 0
\(659\) 471.787i 0.715913i −0.933738 0.357957i \(-0.883474\pi\)
0.933738 0.357957i \(-0.116526\pi\)
\(660\) 0 0
\(661\) −897.151 −1.35726 −0.678632 0.734479i \(-0.737427\pi\)
−0.678632 + 0.734479i \(0.737427\pi\)
\(662\) 0 0
\(663\) −27.5005 + 27.5005i −0.0414789 + 0.0414789i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 139.546 + 139.546i 0.209214 + 0.209214i
\(668\) 0 0
\(669\) 81.2724i 0.121483i
\(670\) 0 0
\(671\) 989.060 1.47401
\(672\) 0 0
\(673\) −105.526 + 105.526i −0.156799 + 0.156799i −0.781146 0.624348i \(-0.785365\pi\)
0.624348 + 0.781146i \(0.285365\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −547.015 547.015i −0.807998 0.807998i 0.176332 0.984331i \(-0.443577\pi\)
−0.984331 + 0.176332i \(0.943577\pi\)
\(678\) 0 0
\(679\) 399.940i 0.589013i
\(680\) 0 0
\(681\) −619.423 −0.909579
\(682\) 0 0
\(683\) 533.271 533.271i 0.780778 0.780778i −0.199184 0.979962i \(-0.563829\pi\)
0.979962 + 0.199184i \(0.0638292\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −336.805 336.805i −0.490254 0.490254i
\(688\) 0 0
\(689\) 988.999i 1.43541i
\(690\) 0 0
\(691\) 1191.09 1.72372 0.861859 0.507147i \(-0.169300\pi\)
0.861859 + 0.507147i \(0.169300\pi\)
\(692\) 0 0
\(693\) 296.499 296.499i 0.427849 0.427849i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −44.9082 44.9082i −0.0644306 0.0644306i
\(698\) 0 0
\(699\) 548.091i 0.784107i
\(700\) 0 0
\(701\) 717.242 1.02317 0.511585 0.859233i \(-0.329059\pi\)
0.511585 + 0.859233i \(0.329059\pi\)
\(702\) 0 0
\(703\) 1229.77 1229.77i 1.74931 1.74931i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −777.787 777.787i −1.10012 1.10012i
\(708\) 0 0
\(709\) 148.514i 0.209470i −0.994500 0.104735i \(-0.966601\pi\)
0.994500 0.104735i \(-0.0333994\pi\)
\(710\) 0 0
\(711\) −82.1816 −0.115586
\(712\) 0 0
\(713\) 787.682 787.682i 1.10474 1.10474i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 374.363 + 374.363i 0.522124 + 0.522124i
\(718\) 0 0
\(719\) 43.4847i 0.0604794i −0.999543 0.0302397i \(-0.990373\pi\)
0.999543 0.0302397i \(-0.00962706\pi\)
\(720\) 0 0
\(721\) −353.271 −0.489974
\(722\) 0 0
\(723\) −204.532 + 204.532i −0.282894 + 0.282894i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −817.608 817.608i −1.12463 1.12463i −0.991036 0.133598i \(-0.957347\pi\)
−0.133598 0.991036i \(-0.542653\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −65.0602 −0.0890016
\(732\) 0 0
\(733\) 842.030 842.030i 1.14874 1.14874i 0.161944 0.986800i \(-0.448223\pi\)
0.986800 0.161944i \(-0.0517765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 352.318 + 352.318i 0.478043 + 0.478043i
\(738\) 0 0
\(739\) 406.545i 0.550128i 0.961426 + 0.275064i \(0.0886991\pi\)
−0.961426 + 0.275064i \(0.911301\pi\)
\(740\) 0 0
\(741\) 687.211 0.927411
\(742\) 0 0
\(743\) 394.515 394.515i 0.530976 0.530976i −0.389887 0.920863i \(-0.627486\pi\)
0.920863 + 0.389887i \(0.127486\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −190.318 190.318i −0.254776 0.254776i
\(748\) 0 0
\(749\) 1280.64i 1.70979i
\(750\) 0 0
\(751\) 612.788 0.815962 0.407981 0.912990i \(-0.366233\pi\)
0.407981 + 0.912990i \(0.366233\pi\)
\(752\) 0 0
\(753\) 320.363 320.363i 0.425449 0.425449i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 181.547 + 181.547i 0.239825 + 0.239825i 0.816777 0.576953i \(-0.195758\pi\)
−0.576953 + 0.816777i \(0.695758\pi\)
\(758\) 0 0
\(759\) 813.453i 1.07174i
\(760\) 0 0
\(761\) 497.271 0.653445 0.326722 0.945120i \(-0.394056\pi\)
0.326722 + 0.945120i \(0.394056\pi\)
\(762\) 0 0
\(763\) 692.412 692.412i 0.907486 0.907486i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −249.773 249.773i −0.325649 0.325649i
\(768\) 0 0
\(769\) 220.271i 0.286439i −0.989691 0.143219i \(-0.954255\pi\)
0.989691 0.143219i \(-0.0457455\pi\)
\(770\) 0 0
\(771\) −231.514 −0.300278
\(772\) 0 0
\(773\) −917.605 + 917.605i −1.18707 + 1.18707i −0.209196 + 0.977874i \(0.567085\pi\)
−0.977874 + 0.209196i \(0.932915\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −426.879 426.879i −0.549393 0.549393i
\(778\) 0 0
\(779\) 1122.21i 1.44058i
\(780\) 0 0
\(781\) −192.545 −0.246536
\(782\) 0 0
\(783\) −31.9546 + 31.9546i −0.0408105 + 0.0408105i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 862.194 + 862.194i 1.09555 + 1.09555i 0.994925 + 0.100620i \(0.0320828\pi\)
0.100620 + 0.994925i \(0.467917\pi\)
\(788\) 0 0
\(789\) 83.7276i 0.106119i
\(790\) 0 0
\(791\) 126.184 0.159524
\(792\) 0 0
\(793\) 397.870 397.870i 0.501727 0.501727i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 175.818 + 175.818i 0.220600 + 0.220600i 0.808751 0.588151i \(-0.200144\pi\)
−0.588151 + 0.808751i \(0.700144\pi\)
\(798\) 0 0
\(799\) 37.6347i 0.0471022i
\(800\) 0 0
\(801\) 52.1816 0.0651456
\(802\) 0 0
\(803\) −461.394 + 461.394i −0.574588 + 0.574588i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −477.167 477.167i −0.591285 0.591285i
\(808\) 0 0
\(809\) 828.272i 1.02382i 0.859038 + 0.511911i \(0.171062\pi\)
−0.859038 + 0.511911i \(0.828938\pi\)
\(810\) 0 0
\(811\) 116.909 0.144154 0.0720772 0.997399i \(-0.477037\pi\)
0.0720772 + 0.997399i \(0.477037\pi\)
\(812\) 0 0
\(813\) 192.025 192.025i 0.236193 0.236193i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 812.896 + 812.896i 0.994976 + 0.994976i
\(818\) 0 0
\(819\) 238.546i 0.291265i
\(820\) 0 0
\(821\) 632.424 0.770310 0.385155 0.922852i \(-0.374148\pi\)
0.385155 + 0.922852i \(0.374148\pi\)
\(822\) 0 0
\(823\) 1028.81 1028.81i 1.25007 1.25007i 0.294385 0.955687i \(-0.404885\pi\)
0.955687 0.294385i \(-0.0951149\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 348.061 + 348.061i 0.420872 + 0.420872i 0.885504 0.464632i \(-0.153813\pi\)
−0.464632 + 0.885504i \(0.653813\pi\)
\(828\) 0 0
\(829\) 901.392i 1.08732i −0.839304 0.543662i \(-0.817037\pi\)
0.839304 0.543662i \(-0.182963\pi\)
\(830\) 0 0
\(831\) −699.545 −0.841811
\(832\) 0 0
\(833\) −4.57654 + 4.57654i −0.00549404 + 0.00549404i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 180.371 + 180.371i 0.215497 + 0.215497i
\(838\) 0 0
\(839\) 99.1806i 0.118213i 0.998252 + 0.0591064i \(0.0188251\pi\)
−0.998252 + 0.0591064i \(0.981175\pi\)
\(840\) 0 0
\(841\) 765.363 0.910063
\(842\) 0 0
\(843\) 24.6811 24.6811i 0.0292777 0.0292777i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1467.74 1467.74i −1.73287 1.73287i
\(848\) 0 0
\(849\) 417.817i 0.492129i
\(850\) 0 0
\(851\) −1171.15 −1.37621
\(852\) 0 0
\(853\) 662.612 662.612i 0.776802 0.776802i −0.202484 0.979286i \(-0.564901\pi\)
0.979286 + 0.202484i \(0.0649013\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 487.546 + 487.546i 0.568898 + 0.568898i 0.931820 0.362921i \(-0.118221\pi\)
−0.362921 + 0.931820i \(0.618221\pi\)
\(858\) 0 0
\(859\) 110.849i 0.129044i −0.997916 0.0645221i \(-0.979448\pi\)
0.997916 0.0645221i \(-0.0205523\pi\)
\(860\) 0 0
\(861\) 389.544 0.452432
\(862\) 0 0
\(863\) −917.271 + 917.271i −1.06289 + 1.06289i −0.0650018 + 0.997885i \(0.520705\pi\)
−0.997885 + 0.0650018i \(0.979295\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 349.497 + 349.497i 0.403111 + 0.403111i
\(868\) 0 0
\(869\) 566.969i 0.652439i
\(870\) 0 0
\(871\) 283.454 0.325435
\(872\) 0 0
\(873\) 125.629 125.629i 0.143905 0.143905i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −53.7538 53.7538i −0.0612928 0.0612928i 0.675796 0.737089i \(-0.263800\pi\)
−0.737089 + 0.675796i \(0.763800\pi\)
\(878\) 0 0
\(879\) 635.939i 0.723480i
\(880\) 0 0
\(881\) 648.545 0.736146 0.368073 0.929797i \(-0.380018\pi\)
0.368073 + 0.929797i \(0.380018\pi\)
\(882\) 0 0
\(883\) −191.381 + 191.381i −0.216740 + 0.216740i −0.807123 0.590383i \(-0.798977\pi\)
0.590383 + 0.807123i \(0.298977\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 468.347 + 468.347i 0.528013 + 0.528013i 0.919979 0.391967i \(-0.128205\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(888\) 0 0
\(889\) 484.182i 0.544636i
\(890\) 0 0
\(891\) 186.272 0.209060
\(892\) 0 0
\(893\) −470.227 + 470.227i −0.526570 + 0.526570i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −327.228 327.228i −0.364803 0.364803i
\(898\) 0 0
\(899\) 426.940i 0.474905i
\(900\) 0 0
\(901\) −160.182 −0.177782
\(902\) 0 0
\(903\) 282.174 282.174i 0.312485 0.312485i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 359.160 + 359.160i 0.395987 + 0.395987i 0.876815 0.480828i \(-0.159664\pi\)
−0.480828 + 0.876815i \(0.659664\pi\)
\(908\) 0 0
\(909\) 488.636i 0.537553i
\(910\) 0 0
\(911\) 1698.00 1.86389 0.931943 0.362605i \(-0.118113\pi\)
0.931943 + 0.362605i \(0.118113\pi\)
\(912\) 0 0
\(913\) −1313.00 + 1313.00i −1.43811 + 1.43811i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −746.241 746.241i −0.813785 0.813785i
\(918\) 0 0
\(919\) 316.728i 0.344644i −0.985041 0.172322i \(-0.944873\pi\)
0.985041 0.172322i \(-0.0551269\pi\)
\(920\) 0 0
\(921\) 291.272 0.316257
\(922\) 0 0
\(923\) −77.4551 + 77.4551i −0.0839167 + 0.0839167i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −110.969 110.969i −0.119708 0.119708i
\(928\) 0 0
\(929\) 123.426i 0.132858i −0.997791 0.0664292i \(-0.978839\pi\)
0.997791 0.0664292i \(-0.0211607\pi\)
\(930\) 0 0
\(931\) 114.363 0.122839
\(932\) 0 0
\(933\) −64.6515 + 64.6515i −0.0692942 + 0.0692942i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −261.513 261.513i −0.279096 0.279096i 0.553652 0.832748i \(-0.313234\pi\)
−0.832748 + 0.553652i \(0.813234\pi\)
\(938\) 0 0
\(939\) 584.333i 0.622292i
\(940\) 0 0
\(941\) −942.422 −1.00151 −0.500756 0.865589i \(-0.666945\pi\)
−0.500756 + 0.865589i \(0.666945\pi\)
\(942\) 0 0
\(943\) 534.361 534.361i 0.566661 0.566661i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −986.983 986.983i −1.04222 1.04222i −0.999069 0.0431523i \(-0.986260\pi\)
−0.0431523 0.999069i \(-0.513740\pi\)
\(948\) 0 0
\(949\) 371.210i 0.391159i
\(950\) 0 0
\(951\) 119.878 0.126054
\(952\) 0 0
\(953\) 112.454 112.454i 0.118000 0.118000i −0.645641 0.763641i \(-0.723410\pi\)
0.763641 + 0.645641i \(0.223410\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 220.454 + 220.454i 0.230360 + 0.230360i
\(958\) 0 0
\(959\) 872.424i 0.909723i
\(960\) 0 0
\(961\) 1448.91 1.50771
\(962\) 0 0
\(963\) −402.272 + 402.272i −0.417728 + 0.417728i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 700.009 + 700.009i 0.723898 + 0.723898i 0.969397 0.245499i \(-0.0789518\pi\)
−0.245499 + 0.969397i \(0.578952\pi\)
\(968\) 0 0
\(969\) 111.303i 0.114864i
\(970\) 0 0
\(971\) 1450.18 1.49349 0.746746 0.665109i \(-0.231615\pi\)
0.746746 + 0.665109i \(0.231615\pi\)
\(972\) 0 0
\(973\) 168.147 168.147i 0.172813 0.172813i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1146.14 1146.14i −1.17312 1.17312i −0.981462 0.191656i \(-0.938614\pi\)
−0.191656 0.981462i \(-0.561386\pi\)
\(978\) 0 0
\(979\) 360.000i 0.367722i
\(980\) 0 0
\(981\) 435.000 0.443425
\(982\) 0 0
\(983\) −760.332 + 760.332i −0.773481 + 0.773481i −0.978713 0.205232i \(-0.934205\pi\)
0.205232 + 0.978713i \(0.434205\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 163.226 + 163.226i 0.165376 + 0.165376i
\(988\) 0 0
\(989\) 774.150i 0.782760i
\(990\) 0 0
\(991\) 222.605 0.224627 0.112313 0.993673i \(-0.464174\pi\)
0.112313 + 0.993673i \(0.464174\pi\)
\(992\) 0 0
\(993\) −357.551 + 357.551i −0.360071 + 0.360071i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −467.183 467.183i −0.468588 0.468588i 0.432869 0.901457i \(-0.357501\pi\)
−0.901457 + 0.432869i \(0.857501\pi\)
\(998\) 0 0
\(999\) 268.182i 0.268450i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.3.bg.a.1057.1 4
4.3 odd 2 150.3.f.c.7.2 yes 4
5.2 odd 4 1200.3.bg.p.193.2 4
5.3 odd 4 inner 1200.3.bg.a.193.1 4
5.4 even 2 1200.3.bg.p.1057.2 4
12.11 even 2 450.3.g.g.307.1 4
20.3 even 4 150.3.f.c.43.2 yes 4
20.7 even 4 150.3.f.a.43.1 yes 4
20.19 odd 2 150.3.f.a.7.1 4
60.23 odd 4 450.3.g.g.343.1 4
60.47 odd 4 450.3.g.h.343.2 4
60.59 even 2 450.3.g.h.307.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.3.f.a.7.1 4 20.19 odd 2
150.3.f.a.43.1 yes 4 20.7 even 4
150.3.f.c.7.2 yes 4 4.3 odd 2
150.3.f.c.43.2 yes 4 20.3 even 4
450.3.g.g.307.1 4 12.11 even 2
450.3.g.g.343.1 4 60.23 odd 4
450.3.g.h.307.2 4 60.59 even 2
450.3.g.h.343.2 4 60.47 odd 4
1200.3.bg.a.193.1 4 5.3 odd 4 inner
1200.3.bg.a.1057.1 4 1.1 even 1 trivial
1200.3.bg.p.193.2 4 5.2 odd 4
1200.3.bg.p.1057.2 4 5.4 even 2