Properties

Label 156.3.j.a.73.2
Level $156$
Weight $3$
Character 156.73
Analytic conductor $4.251$
Analytic rank $0$
Dimension $8$
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] = [156,3,Mod(73,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1579585536.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 73.2
Root \(2.22833 + 1.32913i\) of defining polynomial
Character \(\chi\) \(=\) 156.73
Dual form 156.3.j.a.109.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +(1.92621 + 1.92621i) q^{5} +(-3.58447 + 3.58447i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(1.92621 + 1.92621i) q^{5} +(-3.58447 + 3.58447i) q^{7} +3.00000 q^{9} +(-6.83954 + 6.83954i) q^{11} +(4.86475 + 12.0555i) q^{13} +(-3.33629 - 3.33629i) q^{15} +22.4992i q^{17} +(1.26106 + 1.26106i) q^{19} +(6.20849 - 6.20849i) q^{21} +35.8702i q^{23} -17.5794i q^{25} -5.19615 q^{27} +5.29185 q^{29} +(-22.8901 - 22.8901i) q^{31} +(11.8464 - 11.8464i) q^{33} -13.8089 q^{35} +(27.7791 - 27.7791i) q^{37} +(-8.42600 - 20.8807i) q^{39} +(-2.72861 - 2.72861i) q^{41} -53.4988i q^{43} +(5.77863 + 5.77863i) q^{45} +(24.5386 - 24.5386i) q^{47} +23.3031i q^{49} -38.9698i q^{51} +20.8175 q^{53} -26.3488 q^{55} +(-2.18422 - 2.18422i) q^{57} +(-38.5988 + 38.5988i) q^{59} +19.7505 q^{61} +(-10.7534 + 10.7534i) q^{63} +(-13.8508 + 32.5919i) q^{65} +(-6.69940 - 6.69940i) q^{67} -62.1290i q^{69} +(35.7577 + 35.7577i) q^{71} +(68.0975 - 68.0975i) q^{73} +30.4485i q^{75} -49.0323i q^{77} +3.68606 q^{79} +9.00000 q^{81} +(-40.8170 - 40.8170i) q^{83} +(-43.3383 + 43.3383i) q^{85} -9.16576 q^{87} +(-70.1454 + 70.1454i) q^{89} +(-60.6500 - 25.7749i) q^{91} +(39.6468 + 39.6468i) q^{93} +4.85813i q^{95} +(115.441 + 115.441i) q^{97} +(-20.5186 + 20.5186i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{5} - 8 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{5} - 8 q^{7} + 24 q^{9} + 12 q^{11} + 24 q^{13} + 12 q^{15} + 88 q^{19} + 24 q^{29} - 16 q^{31} - 36 q^{33} - 216 q^{35} + 32 q^{37} + 72 q^{39} - 180 q^{41} + 36 q^{45} + 36 q^{47} - 72 q^{53} - 240 q^{55} + 24 q^{57} - 228 q^{59} - 192 q^{61} - 24 q^{63} + 132 q^{65} + 16 q^{67} + 36 q^{71} + 160 q^{73} + 48 q^{79} + 72 q^{81} + 12 q^{83} + 24 q^{85} - 120 q^{87} + 60 q^{89} + 112 q^{91} + 120 q^{93} + 416 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 1.92621 + 1.92621i 0.385242 + 0.385242i 0.872986 0.487744i \(-0.162180\pi\)
−0.487744 + 0.872986i \(0.662180\pi\)
\(6\) 0 0
\(7\) −3.58447 + 3.58447i −0.512067 + 0.512067i −0.915159 0.403092i \(-0.867935\pi\)
0.403092 + 0.915159i \(0.367935\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −6.83954 + 6.83954i −0.621776 + 0.621776i −0.945985 0.324209i \(-0.894902\pi\)
0.324209 + 0.945985i \(0.394902\pi\)
\(12\) 0 0
\(13\) 4.86475 + 12.0555i 0.374212 + 0.927343i
\(14\) 0 0
\(15\) −3.33629 3.33629i −0.222420 0.222420i
\(16\) 0 0
\(17\) 22.4992i 1.32348i 0.749731 + 0.661742i \(0.230183\pi\)
−0.749731 + 0.661742i \(0.769817\pi\)
\(18\) 0 0
\(19\) 1.26106 + 1.26106i 0.0663716 + 0.0663716i 0.739513 0.673142i \(-0.235056\pi\)
−0.673142 + 0.739513i \(0.735056\pi\)
\(20\) 0 0
\(21\) 6.20849 6.20849i 0.295642 0.295642i
\(22\) 0 0
\(23\) 35.8702i 1.55957i 0.626045 + 0.779787i \(0.284672\pi\)
−0.626045 + 0.779787i \(0.715328\pi\)
\(24\) 0 0
\(25\) 17.5794i 0.703177i
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 5.29185 0.182478 0.0912389 0.995829i \(-0.470917\pi\)
0.0912389 + 0.995829i \(0.470917\pi\)
\(30\) 0 0
\(31\) −22.8901 22.8901i −0.738390 0.738390i 0.233876 0.972266i \(-0.424859\pi\)
−0.972266 + 0.233876i \(0.924859\pi\)
\(32\) 0 0
\(33\) 11.8464 11.8464i 0.358983 0.358983i
\(34\) 0 0
\(35\) −13.8089 −0.394540
\(36\) 0 0
\(37\) 27.7791 27.7791i 0.750787 0.750787i −0.223839 0.974626i \(-0.571859\pi\)
0.974626 + 0.223839i \(0.0718590\pi\)
\(38\) 0 0
\(39\) −8.42600 20.8807i −0.216051 0.535402i
\(40\) 0 0
\(41\) −2.72861 2.72861i −0.0665516 0.0665516i 0.673048 0.739599i \(-0.264985\pi\)
−0.739599 + 0.673048i \(0.764985\pi\)
\(42\) 0 0
\(43\) 53.4988i 1.24416i −0.782954 0.622079i \(-0.786288\pi\)
0.782954 0.622079i \(-0.213712\pi\)
\(44\) 0 0
\(45\) 5.77863 + 5.77863i 0.128414 + 0.128414i
\(46\) 0 0
\(47\) 24.5386 24.5386i 0.522097 0.522097i −0.396107 0.918204i \(-0.629639\pi\)
0.918204 + 0.396107i \(0.129639\pi\)
\(48\) 0 0
\(49\) 23.3031i 0.475574i
\(50\) 0 0
\(51\) 38.9698i 0.764114i
\(52\) 0 0
\(53\) 20.8175 0.392784 0.196392 0.980525i \(-0.437078\pi\)
0.196392 + 0.980525i \(0.437078\pi\)
\(54\) 0 0
\(55\) −26.3488 −0.479069
\(56\) 0 0
\(57\) −2.18422 2.18422i −0.0383197 0.0383197i
\(58\) 0 0
\(59\) −38.5988 + 38.5988i −0.654217 + 0.654217i −0.954006 0.299789i \(-0.903084\pi\)
0.299789 + 0.954006i \(0.403084\pi\)
\(60\) 0 0
\(61\) 19.7505 0.323778 0.161889 0.986809i \(-0.448241\pi\)
0.161889 + 0.986809i \(0.448241\pi\)
\(62\) 0 0
\(63\) −10.7534 + 10.7534i −0.170689 + 0.170689i
\(64\) 0 0
\(65\) −13.8508 + 32.5919i −0.213089 + 0.501414i
\(66\) 0 0
\(67\) −6.69940 6.69940i −0.0999910 0.0999910i 0.655342 0.755333i \(-0.272525\pi\)
−0.755333 + 0.655342i \(0.772525\pi\)
\(68\) 0 0
\(69\) 62.1290i 0.900420i
\(70\) 0 0
\(71\) 35.7577 + 35.7577i 0.503629 + 0.503629i 0.912564 0.408935i \(-0.134100\pi\)
−0.408935 + 0.912564i \(0.634100\pi\)
\(72\) 0 0
\(73\) 68.0975 68.0975i 0.932843 0.932843i −0.0650398 0.997883i \(-0.520717\pi\)
0.997883 + 0.0650398i \(0.0207174\pi\)
\(74\) 0 0
\(75\) 30.4485i 0.405980i
\(76\) 0 0
\(77\) 49.0323i 0.636783i
\(78\) 0 0
\(79\) 3.68606 0.0466590 0.0233295 0.999728i \(-0.492573\pi\)
0.0233295 + 0.999728i \(0.492573\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −40.8170 40.8170i −0.491772 0.491772i 0.417093 0.908864i \(-0.363049\pi\)
−0.908864 + 0.417093i \(0.863049\pi\)
\(84\) 0 0
\(85\) −43.3383 + 43.3383i −0.509862 + 0.509862i
\(86\) 0 0
\(87\) −9.16576 −0.105354
\(88\) 0 0
\(89\) −70.1454 + 70.1454i −0.788151 + 0.788151i −0.981191 0.193040i \(-0.938165\pi\)
0.193040 + 0.981191i \(0.438165\pi\)
\(90\) 0 0
\(91\) −60.6500 25.7749i −0.666484 0.283240i
\(92\) 0 0
\(93\) 39.6468 + 39.6468i 0.426310 + 0.426310i
\(94\) 0 0
\(95\) 4.85813i 0.0511383i
\(96\) 0 0
\(97\) 115.441 + 115.441i 1.19011 + 1.19011i 0.977035 + 0.213080i \(0.0683494\pi\)
0.213080 + 0.977035i \(0.431651\pi\)
\(98\) 0 0
\(99\) −20.5186 + 20.5186i −0.207259 + 0.207259i
\(100\) 0 0
\(101\) 36.6504i 0.362875i 0.983402 + 0.181438i \(0.0580750\pi\)
−0.983402 + 0.181438i \(0.941925\pi\)
\(102\) 0 0
\(103\) 130.389i 1.26592i 0.774186 + 0.632958i \(0.218159\pi\)
−0.774186 + 0.632958i \(0.781841\pi\)
\(104\) 0 0
\(105\) 23.9177 0.227788
\(106\) 0 0
\(107\) 162.941 1.52281 0.761407 0.648274i \(-0.224509\pi\)
0.761407 + 0.648274i \(0.224509\pi\)
\(108\) 0 0
\(109\) −98.1123 98.1123i −0.900113 0.900113i 0.0953329 0.995445i \(-0.469608\pi\)
−0.995445 + 0.0953329i \(0.969608\pi\)
\(110\) 0 0
\(111\) −48.1149 + 48.1149i −0.433467 + 0.433467i
\(112\) 0 0
\(113\) −26.8171 −0.237320 −0.118660 0.992935i \(-0.537860\pi\)
−0.118660 + 0.992935i \(0.537860\pi\)
\(114\) 0 0
\(115\) −69.0935 + 69.0935i −0.600813 + 0.600813i
\(116\) 0 0
\(117\) 14.5943 + 36.1664i 0.124737 + 0.309114i
\(118\) 0 0
\(119\) −80.6479 80.6479i −0.677713 0.677713i
\(120\) 0 0
\(121\) 27.4414i 0.226789i
\(122\) 0 0
\(123\) 4.72610 + 4.72610i 0.0384236 + 0.0384236i
\(124\) 0 0
\(125\) 82.0169 82.0169i 0.656135 0.656135i
\(126\) 0 0
\(127\) 124.496i 0.980282i 0.871643 + 0.490141i \(0.163055\pi\)
−0.871643 + 0.490141i \(0.836945\pi\)
\(128\) 0 0
\(129\) 92.6627i 0.718315i
\(130\) 0 0
\(131\) 246.156 1.87906 0.939529 0.342471i \(-0.111264\pi\)
0.939529 + 0.342471i \(0.111264\pi\)
\(132\) 0 0
\(133\) −9.04047 −0.0679734
\(134\) 0 0
\(135\) −10.0089 10.0089i −0.0741399 0.0741399i
\(136\) 0 0
\(137\) −106.209 + 106.209i −0.775248 + 0.775248i −0.979019 0.203771i \(-0.934680\pi\)
0.203771 + 0.979019i \(0.434680\pi\)
\(138\) 0 0
\(139\) −54.0949 −0.389172 −0.194586 0.980885i \(-0.562336\pi\)
−0.194586 + 0.980885i \(0.562336\pi\)
\(140\) 0 0
\(141\) −42.5021 + 42.5021i −0.301433 + 0.301433i
\(142\) 0 0
\(143\) −115.726 49.1811i −0.809276 0.343924i
\(144\) 0 0
\(145\) 10.1932 + 10.1932i 0.0702981 + 0.0702981i
\(146\) 0 0
\(147\) 40.3622i 0.274573i
\(148\) 0 0
\(149\) −159.415 159.415i −1.06990 1.06990i −0.997366 0.0725343i \(-0.976891\pi\)
−0.0725343 0.997366i \(-0.523109\pi\)
\(150\) 0 0
\(151\) 18.8748 18.8748i 0.124999 0.124999i −0.641840 0.766839i \(-0.721829\pi\)
0.766839 + 0.641840i \(0.221829\pi\)
\(152\) 0 0
\(153\) 67.4977i 0.441162i
\(154\) 0 0
\(155\) 88.1823i 0.568918i
\(156\) 0 0
\(157\) 265.777 1.69285 0.846423 0.532510i \(-0.178751\pi\)
0.846423 + 0.532510i \(0.178751\pi\)
\(158\) 0 0
\(159\) −36.0570 −0.226774
\(160\) 0 0
\(161\) −128.576 128.576i −0.798607 0.798607i
\(162\) 0 0
\(163\) 114.497 114.497i 0.702437 0.702437i −0.262496 0.964933i \(-0.584546\pi\)
0.964933 + 0.262496i \(0.0845456\pi\)
\(164\) 0 0
\(165\) 45.6374 0.276590
\(166\) 0 0
\(167\) 152.956 152.956i 0.915905 0.915905i −0.0808233 0.996728i \(-0.525755\pi\)
0.996728 + 0.0808233i \(0.0257550\pi\)
\(168\) 0 0
\(169\) −121.668 + 117.294i −0.719931 + 0.694046i
\(170\) 0 0
\(171\) 3.78318 + 3.78318i 0.0221239 + 0.0221239i
\(172\) 0 0
\(173\) 243.771i 1.40908i −0.709664 0.704540i \(-0.751153\pi\)
0.709664 0.704540i \(-0.248847\pi\)
\(174\) 0 0
\(175\) 63.0130 + 63.0130i 0.360074 + 0.360074i
\(176\) 0 0
\(177\) 66.8551 66.8551i 0.377712 0.377712i
\(178\) 0 0
\(179\) 86.8648i 0.485278i 0.970117 + 0.242639i \(0.0780131\pi\)
−0.970117 + 0.242639i \(0.921987\pi\)
\(180\) 0 0
\(181\) 168.352i 0.930121i −0.885279 0.465061i \(-0.846032\pi\)
0.885279 0.465061i \(-0.153968\pi\)
\(182\) 0 0
\(183\) −34.2088 −0.186933
\(184\) 0 0
\(185\) 107.017 0.578470
\(186\) 0 0
\(187\) −153.884 153.884i −0.822911 0.822911i
\(188\) 0 0
\(189\) 18.6255 18.6255i 0.0985474 0.0985474i
\(190\) 0 0
\(191\) 325.900 1.70628 0.853141 0.521680i \(-0.174695\pi\)
0.853141 + 0.521680i \(0.174695\pi\)
\(192\) 0 0
\(193\) −231.123 + 231.123i −1.19753 + 1.19753i −0.222625 + 0.974904i \(0.571462\pi\)
−0.974904 + 0.222625i \(0.928538\pi\)
\(194\) 0 0
\(195\) 23.9903 56.4508i 0.123027 0.289491i
\(196\) 0 0
\(197\) −248.700 248.700i −1.26244 1.26244i −0.949909 0.312527i \(-0.898825\pi\)
−0.312527 0.949909i \(-0.601175\pi\)
\(198\) 0 0
\(199\) 244.539i 1.22884i −0.788980 0.614419i \(-0.789391\pi\)
0.788980 0.614419i \(-0.210609\pi\)
\(200\) 0 0
\(201\) 11.6037 + 11.6037i 0.0577298 + 0.0577298i
\(202\) 0 0
\(203\) −18.9685 + 18.9685i −0.0934409 + 0.0934409i
\(204\) 0 0
\(205\) 10.5118i 0.0512769i
\(206\) 0 0
\(207\) 107.611i 0.519858i
\(208\) 0 0
\(209\) −17.2501 −0.0825365
\(210\) 0 0
\(211\) −23.6759 −0.112208 −0.0561041 0.998425i \(-0.517868\pi\)
−0.0561041 + 0.998425i \(0.517868\pi\)
\(212\) 0 0
\(213\) −61.9341 61.9341i −0.290770 0.290770i
\(214\) 0 0
\(215\) 103.050 103.050i 0.479302 0.479302i
\(216\) 0 0
\(217\) 164.098 0.756211
\(218\) 0 0
\(219\) −117.948 + 117.948i −0.538577 + 0.538577i
\(220\) 0 0
\(221\) −271.239 + 109.453i −1.22732 + 0.495264i
\(222\) 0 0
\(223\) 136.609 + 136.609i 0.612595 + 0.612595i 0.943622 0.331026i \(-0.107395\pi\)
−0.331026 + 0.943622i \(0.607395\pi\)
\(224\) 0 0
\(225\) 52.7383i 0.234392i
\(226\) 0 0
\(227\) 218.098 + 218.098i 0.960782 + 0.960782i 0.999259 0.0384775i \(-0.0122508\pi\)
−0.0384775 + 0.999259i \(0.512251\pi\)
\(228\) 0 0
\(229\) 35.0435 35.0435i 0.153028 0.153028i −0.626441 0.779469i \(-0.715489\pi\)
0.779469 + 0.626441i \(0.215489\pi\)
\(230\) 0 0
\(231\) 84.9264i 0.367647i
\(232\) 0 0
\(233\) 44.4323i 0.190697i 0.995444 + 0.0953483i \(0.0303965\pi\)
−0.995444 + 0.0953483i \(0.969604\pi\)
\(234\) 0 0
\(235\) 94.5329 0.402268
\(236\) 0 0
\(237\) −6.38444 −0.0269386
\(238\) 0 0
\(239\) −133.935 133.935i −0.560396 0.560396i 0.369024 0.929420i \(-0.379692\pi\)
−0.929420 + 0.369024i \(0.879692\pi\)
\(240\) 0 0
\(241\) −115.524 + 115.524i −0.479351 + 0.479351i −0.904924 0.425573i \(-0.860073\pi\)
0.425573 + 0.904924i \(0.360073\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) −44.8867 + 44.8867i −0.183211 + 0.183211i
\(246\) 0 0
\(247\) −9.06791 + 21.3374i −0.0367122 + 0.0863863i
\(248\) 0 0
\(249\) 70.6972 + 70.6972i 0.283924 + 0.283924i
\(250\) 0 0
\(251\) 380.503i 1.51595i 0.652284 + 0.757975i \(0.273811\pi\)
−0.652284 + 0.757975i \(0.726189\pi\)
\(252\) 0 0
\(253\) −245.335 245.335i −0.969705 0.969705i
\(254\) 0 0
\(255\) 75.0641 75.0641i 0.294369 0.294369i
\(256\) 0 0
\(257\) 293.869i 1.14346i 0.820442 + 0.571730i \(0.193727\pi\)
−0.820442 + 0.571730i \(0.806273\pi\)
\(258\) 0 0
\(259\) 199.147i 0.768907i
\(260\) 0 0
\(261\) 15.8756 0.0608259
\(262\) 0 0
\(263\) 157.450 0.598671 0.299335 0.954148i \(-0.403235\pi\)
0.299335 + 0.954148i \(0.403235\pi\)
\(264\) 0 0
\(265\) 40.0990 + 40.0990i 0.151317 + 0.151317i
\(266\) 0 0
\(267\) 121.495 121.495i 0.455039 0.455039i
\(268\) 0 0
\(269\) 174.739 0.649586 0.324793 0.945785i \(-0.394705\pi\)
0.324793 + 0.945785i \(0.394705\pi\)
\(270\) 0 0
\(271\) 15.8688 15.8688i 0.0585564 0.0585564i −0.677222 0.735779i \(-0.736816\pi\)
0.735779 + 0.677222i \(0.236816\pi\)
\(272\) 0 0
\(273\) 105.049 + 44.6434i 0.384795 + 0.163529i
\(274\) 0 0
\(275\) 120.235 + 120.235i 0.437219 + 0.437219i
\(276\) 0 0
\(277\) 241.083i 0.870335i −0.900350 0.435167i \(-0.856689\pi\)
0.900350 0.435167i \(-0.143311\pi\)
\(278\) 0 0
\(279\) −68.6703 68.6703i −0.246130 0.246130i
\(280\) 0 0
\(281\) −149.037 + 149.037i −0.530381 + 0.530381i −0.920686 0.390305i \(-0.872370\pi\)
0.390305 + 0.920686i \(0.372370\pi\)
\(282\) 0 0
\(283\) 350.794i 1.23956i 0.784777 + 0.619778i \(0.212777\pi\)
−0.784777 + 0.619778i \(0.787223\pi\)
\(284\) 0 0
\(285\) 8.41453i 0.0295247i
\(286\) 0 0
\(287\) 19.5613 0.0681578
\(288\) 0 0
\(289\) −217.216 −0.751612
\(290\) 0 0
\(291\) −199.950 199.950i −0.687113 0.687113i
\(292\) 0 0
\(293\) −349.661 + 349.661i −1.19338 + 1.19338i −0.217270 + 0.976112i \(0.569715\pi\)
−0.976112 + 0.217270i \(0.930285\pi\)
\(294\) 0 0
\(295\) −148.699 −0.504064
\(296\) 0 0
\(297\) 35.5393 35.5393i 0.119661 0.119661i
\(298\) 0 0
\(299\) −432.432 + 174.500i −1.44626 + 0.583611i
\(300\) 0 0
\(301\) 191.765 + 191.765i 0.637093 + 0.637093i
\(302\) 0 0
\(303\) 63.4803i 0.209506i
\(304\) 0 0
\(305\) 38.0435 + 38.0435i 0.124733 + 0.124733i
\(306\) 0 0
\(307\) 397.145 397.145i 1.29363 1.29363i 0.361109 0.932524i \(-0.382398\pi\)
0.932524 0.361109i \(-0.117602\pi\)
\(308\) 0 0
\(309\) 225.841i 0.730877i
\(310\) 0 0
\(311\) 157.872i 0.507626i −0.967253 0.253813i \(-0.918315\pi\)
0.967253 0.253813i \(-0.0816847\pi\)
\(312\) 0 0
\(313\) 87.3756 0.279155 0.139578 0.990211i \(-0.455426\pi\)
0.139578 + 0.990211i \(0.455426\pi\)
\(314\) 0 0
\(315\) −41.4267 −0.131513
\(316\) 0 0
\(317\) 351.590 + 351.590i 1.10912 + 1.10912i 0.993267 + 0.115850i \(0.0369591\pi\)
0.115850 + 0.993267i \(0.463041\pi\)
\(318\) 0 0
\(319\) −36.1938 + 36.1938i −0.113460 + 0.113460i
\(320\) 0 0
\(321\) −282.222 −0.879197
\(322\) 0 0
\(323\) −28.3729 + 28.3729i −0.0878418 + 0.0878418i
\(324\) 0 0
\(325\) 211.928 85.5196i 0.652087 0.263137i
\(326\) 0 0
\(327\) 169.935 + 169.935i 0.519680 + 0.519680i
\(328\) 0 0
\(329\) 175.916i 0.534698i
\(330\) 0 0
\(331\) −67.6157 67.6157i −0.204277 0.204277i 0.597553 0.801830i \(-0.296140\pi\)
−0.801830 + 0.597553i \(0.796140\pi\)
\(332\) 0 0
\(333\) 83.3374 83.3374i 0.250262 0.250262i
\(334\) 0 0
\(335\) 25.8089i 0.0770415i
\(336\) 0 0
\(337\) 205.231i 0.608995i 0.952513 + 0.304497i \(0.0984885\pi\)
−0.952513 + 0.304497i \(0.901512\pi\)
\(338\) 0 0
\(339\) 46.4486 0.137017
\(340\) 0 0
\(341\) 313.115 0.918227
\(342\) 0 0
\(343\) −259.168 259.168i −0.755593 0.755593i
\(344\) 0 0
\(345\) 119.673 119.673i 0.346880 0.346880i
\(346\) 0 0
\(347\) −216.750 −0.624639 −0.312319 0.949977i \(-0.601106\pi\)
−0.312319 + 0.949977i \(0.601106\pi\)
\(348\) 0 0
\(349\) −82.0098 + 82.0098i −0.234985 + 0.234985i −0.814770 0.579785i \(-0.803137\pi\)
0.579785 + 0.814770i \(0.303137\pi\)
\(350\) 0 0
\(351\) −25.2780 62.6420i −0.0720171 0.178467i
\(352\) 0 0
\(353\) −193.665 193.665i −0.548627 0.548627i 0.377417 0.926044i \(-0.376812\pi\)
−0.926044 + 0.377417i \(0.876812\pi\)
\(354\) 0 0
\(355\) 137.754i 0.388038i
\(356\) 0 0
\(357\) 139.686 + 139.686i 0.391278 + 0.391278i
\(358\) 0 0
\(359\) −170.887 + 170.887i −0.476010 + 0.476010i −0.903853 0.427843i \(-0.859274\pi\)
0.427843 + 0.903853i \(0.359274\pi\)
\(360\) 0 0
\(361\) 357.819i 0.991190i
\(362\) 0 0
\(363\) 47.5300i 0.130937i
\(364\) 0 0
\(365\) 262.340 0.718741
\(366\) 0 0
\(367\) 517.394 1.40979 0.704896 0.709310i \(-0.250994\pi\)
0.704896 + 0.709310i \(0.250994\pi\)
\(368\) 0 0
\(369\) −8.18584 8.18584i −0.0221839 0.0221839i
\(370\) 0 0
\(371\) −74.6199 + 74.6199i −0.201132 + 0.201132i
\(372\) 0 0
\(373\) −25.9133 −0.0694727 −0.0347364 0.999397i \(-0.511059\pi\)
−0.0347364 + 0.999397i \(0.511059\pi\)
\(374\) 0 0
\(375\) −142.057 + 142.057i −0.378820 + 0.378820i
\(376\) 0 0
\(377\) 25.7436 + 63.7957i 0.0682853 + 0.169219i
\(378\) 0 0
\(379\) −351.980 351.980i −0.928708 0.928708i 0.0689146 0.997623i \(-0.478046\pi\)
−0.997623 + 0.0689146i \(0.978046\pi\)
\(380\) 0 0
\(381\) 215.633i 0.565966i
\(382\) 0 0
\(383\) −3.97886 3.97886i −0.0103887 0.0103887i 0.701893 0.712282i \(-0.252338\pi\)
−0.712282 + 0.701893i \(0.752338\pi\)
\(384\) 0 0
\(385\) 94.4464 94.4464i 0.245315 0.245315i
\(386\) 0 0
\(387\) 160.496i 0.414720i
\(388\) 0 0
\(389\) 420.725i 1.08156i −0.841165 0.540778i \(-0.818130\pi\)
0.841165 0.540778i \(-0.181870\pi\)
\(390\) 0 0
\(391\) −807.052 −2.06407
\(392\) 0 0
\(393\) −426.356 −1.08487
\(394\) 0 0
\(395\) 7.10013 + 7.10013i 0.0179750 + 0.0179750i
\(396\) 0 0
\(397\) 251.922 251.922i 0.634563 0.634563i −0.314646 0.949209i \(-0.601886\pi\)
0.949209 + 0.314646i \(0.101886\pi\)
\(398\) 0 0
\(399\) 15.6585 0.0392445
\(400\) 0 0
\(401\) 544.359 544.359i 1.35750 1.35750i 0.480521 0.876983i \(-0.340448\pi\)
0.876983 0.480521i \(-0.159552\pi\)
\(402\) 0 0
\(403\) 164.596 387.306i 0.408427 0.961056i
\(404\) 0 0
\(405\) 17.3359 + 17.3359i 0.0428047 + 0.0428047i
\(406\) 0 0
\(407\) 379.993i 0.933643i
\(408\) 0 0
\(409\) 210.815 + 210.815i 0.515439 + 0.515439i 0.916188 0.400749i \(-0.131250\pi\)
−0.400749 + 0.916188i \(0.631250\pi\)
\(410\) 0 0
\(411\) 183.959 183.959i 0.447589 0.447589i
\(412\) 0 0
\(413\) 276.713i 0.670006i
\(414\) 0 0
\(415\) 157.244i 0.378902i
\(416\) 0 0
\(417\) 93.6951 0.224688
\(418\) 0 0
\(419\) 75.7366 0.180756 0.0903778 0.995908i \(-0.471193\pi\)
0.0903778 + 0.995908i \(0.471193\pi\)
\(420\) 0 0
\(421\) 37.9020 + 37.9020i 0.0900286 + 0.0900286i 0.750687 0.660658i \(-0.229723\pi\)
−0.660658 + 0.750687i \(0.729723\pi\)
\(422\) 0 0
\(423\) 73.6157 73.6157i 0.174032 0.174032i
\(424\) 0 0
\(425\) 395.524 0.930644
\(426\) 0 0
\(427\) −70.7949 + 70.7949i −0.165796 + 0.165796i
\(428\) 0 0
\(429\) 200.444 + 85.1842i 0.467236 + 0.198565i
\(430\) 0 0
\(431\) −534.376 534.376i −1.23985 1.23985i −0.960061 0.279789i \(-0.909735\pi\)
−0.279789 0.960061i \(-0.590265\pi\)
\(432\) 0 0
\(433\) 700.849i 1.61859i −0.587402 0.809295i \(-0.699849\pi\)
0.587402 0.809295i \(-0.300151\pi\)
\(434\) 0 0
\(435\) −17.6552 17.6552i −0.0405866 0.0405866i
\(436\) 0 0
\(437\) −45.2345 + 45.2345i −0.103511 + 0.103511i
\(438\) 0 0
\(439\) 97.5970i 0.222317i 0.993803 + 0.111158i \(0.0354561\pi\)
−0.993803 + 0.111158i \(0.964544\pi\)
\(440\) 0 0
\(441\) 69.9094i 0.158525i
\(442\) 0 0
\(443\) −264.575 −0.597234 −0.298617 0.954373i \(-0.596525\pi\)
−0.298617 + 0.954373i \(0.596525\pi\)
\(444\) 0 0
\(445\) −270.230 −0.607258
\(446\) 0 0
\(447\) 276.115 + 276.115i 0.617707 + 0.617707i
\(448\) 0 0
\(449\) −416.268 + 416.268i −0.927101 + 0.927101i −0.997518 0.0704170i \(-0.977567\pi\)
0.0704170 + 0.997518i \(0.477567\pi\)
\(450\) 0 0
\(451\) 37.3249 0.0827603
\(452\) 0 0
\(453\) −32.6921 + 32.6921i −0.0721680 + 0.0721680i
\(454\) 0 0
\(455\) −67.1769 166.473i −0.147641 0.365874i
\(456\) 0 0
\(457\) 431.567 + 431.567i 0.944347 + 0.944347i 0.998531 0.0541838i \(-0.0172557\pi\)
−0.0541838 + 0.998531i \(0.517256\pi\)
\(458\) 0 0
\(459\) 116.909i 0.254705i
\(460\) 0 0
\(461\) 18.4845 + 18.4845i 0.0400965 + 0.0400965i 0.726871 0.686774i \(-0.240974\pi\)
−0.686774 + 0.726871i \(0.740974\pi\)
\(462\) 0 0
\(463\) −547.447 + 547.447i −1.18239 + 1.18239i −0.203267 + 0.979123i \(0.565156\pi\)
−0.979123 + 0.203267i \(0.934844\pi\)
\(464\) 0 0
\(465\) 152.736i 0.328465i
\(466\) 0 0
\(467\) 155.958i 0.333957i −0.985961 0.166979i \(-0.946599\pi\)
0.985961 0.166979i \(-0.0534011\pi\)
\(468\) 0 0
\(469\) 48.0276 0.102404
\(470\) 0 0
\(471\) −460.339 −0.977366
\(472\) 0 0
\(473\) 365.907 + 365.907i 0.773588 + 0.773588i
\(474\) 0 0
\(475\) 22.1687 22.1687i 0.0466710 0.0466710i
\(476\) 0 0
\(477\) 62.4526 0.130928
\(478\) 0 0
\(479\) 406.516 406.516i 0.848677 0.848677i −0.141291 0.989968i \(-0.545125\pi\)
0.989968 + 0.141291i \(0.0451253\pi\)
\(480\) 0 0
\(481\) 470.029 + 199.752i 0.977191 + 0.415284i
\(482\) 0 0
\(483\) 222.700 + 222.700i 0.461076 + 0.461076i
\(484\) 0 0
\(485\) 444.728i 0.916964i
\(486\) 0 0
\(487\) 183.739 + 183.739i 0.377288 + 0.377288i 0.870123 0.492835i \(-0.164039\pi\)
−0.492835 + 0.870123i \(0.664039\pi\)
\(488\) 0 0
\(489\) −198.315 + 198.315i −0.405552 + 0.405552i
\(490\) 0 0
\(491\) 100.574i 0.204836i −0.994741 0.102418i \(-0.967342\pi\)
0.994741 0.102418i \(-0.0326579\pi\)
\(492\) 0 0
\(493\) 119.063i 0.241507i
\(494\) 0 0
\(495\) −79.0463 −0.159690
\(496\) 0 0
\(497\) −256.345 −0.515784
\(498\) 0 0
\(499\) 344.530 + 344.530i 0.690441 + 0.690441i 0.962329 0.271888i \(-0.0876480\pi\)
−0.271888 + 0.962329i \(0.587648\pi\)
\(500\) 0 0
\(501\) −264.928 + 264.928i −0.528798 + 0.528798i
\(502\) 0 0
\(503\) −592.672 −1.17827 −0.589137 0.808033i \(-0.700532\pi\)
−0.589137 + 0.808033i \(0.700532\pi\)
\(504\) 0 0
\(505\) −70.5963 + 70.5963i −0.139795 + 0.139795i
\(506\) 0 0
\(507\) 210.736 203.159i 0.415652 0.400707i
\(508\) 0 0
\(509\) −338.778 338.778i −0.665575 0.665575i 0.291113 0.956689i \(-0.405974\pi\)
−0.956689 + 0.291113i \(0.905974\pi\)
\(510\) 0 0
\(511\) 488.187i 0.955357i
\(512\) 0 0
\(513\) −6.55266 6.55266i −0.0127732 0.0127732i
\(514\) 0 0
\(515\) −251.157 + 251.157i −0.487684 + 0.487684i
\(516\) 0 0
\(517\) 335.665i 0.649255i
\(518\) 0 0
\(519\) 422.224i 0.813533i
\(520\) 0 0
\(521\) 816.440 1.56706 0.783531 0.621352i \(-0.213416\pi\)
0.783531 + 0.621352i \(0.213416\pi\)
\(522\) 0 0
\(523\) −146.348 −0.279825 −0.139912 0.990164i \(-0.544682\pi\)
−0.139912 + 0.990164i \(0.544682\pi\)
\(524\) 0 0
\(525\) −109.142 109.142i −0.207889 0.207889i
\(526\) 0 0
\(527\) 515.010 515.010i 0.977249 0.977249i
\(528\) 0 0
\(529\) −757.670 −1.43227
\(530\) 0 0
\(531\) −115.796 + 115.796i −0.218072 + 0.218072i
\(532\) 0 0
\(533\) 19.6207 46.1687i 0.0368118 0.0866205i
\(534\) 0 0
\(535\) 313.859 + 313.859i 0.586652 + 0.586652i
\(536\) 0 0
\(537\) 150.454i 0.280176i
\(538\) 0 0
\(539\) −159.383 159.383i −0.295701 0.295701i
\(540\) 0 0
\(541\) −587.359 + 587.359i −1.08569 + 1.08569i −0.0897247 + 0.995967i \(0.528599\pi\)
−0.995967 + 0.0897247i \(0.971401\pi\)
\(542\) 0 0
\(543\) 291.594i 0.537006i
\(544\) 0 0
\(545\) 377.970i 0.693522i
\(546\) 0 0
\(547\) −304.970 −0.557531 −0.278766 0.960359i \(-0.589925\pi\)
−0.278766 + 0.960359i \(0.589925\pi\)
\(548\) 0 0
\(549\) 59.2514 0.107926
\(550\) 0 0
\(551\) 6.67335 + 6.67335i 0.0121113 + 0.0121113i
\(552\) 0 0
\(553\) −13.2126 + 13.2126i −0.0238925 + 0.0238925i
\(554\) 0 0
\(555\) −185.359 −0.333980
\(556\) 0 0
\(557\) 414.483 414.483i 0.744135 0.744135i −0.229236 0.973371i \(-0.573623\pi\)
0.973371 + 0.229236i \(0.0736228\pi\)
\(558\) 0 0
\(559\) 644.953 260.259i 1.15376 0.465579i
\(560\) 0 0
\(561\) 266.536 + 266.536i 0.475108 + 0.475108i
\(562\) 0 0
\(563\) 1016.06i 1.80472i 0.430985 + 0.902359i \(0.358166\pi\)
−0.430985 + 0.902359i \(0.641834\pi\)
\(564\) 0 0
\(565\) −51.6554 51.6554i −0.0914256 0.0914256i
\(566\) 0 0
\(567\) −32.2602 + 32.2602i −0.0568964 + 0.0568964i
\(568\) 0 0
\(569\) 563.561i 0.990442i −0.868767 0.495221i \(-0.835087\pi\)
0.868767 0.495221i \(-0.164913\pi\)
\(570\) 0 0
\(571\) 402.182i 0.704347i 0.935935 + 0.352174i \(0.114557\pi\)
−0.935935 + 0.352174i \(0.885443\pi\)
\(572\) 0 0
\(573\) −564.475 −0.985122
\(574\) 0 0
\(575\) 630.577 1.09666
\(576\) 0 0
\(577\) 338.309 + 338.309i 0.586325 + 0.586325i 0.936634 0.350309i \(-0.113924\pi\)
−0.350309 + 0.936634i \(0.613924\pi\)
\(578\) 0 0
\(579\) 400.317 400.317i 0.691394 0.691394i
\(580\) 0 0
\(581\) 292.615 0.503640
\(582\) 0 0
\(583\) −142.382 + 142.382i −0.244224 + 0.244224i
\(584\) 0 0
\(585\) −41.5524 + 97.7757i −0.0710298 + 0.167138i
\(586\) 0 0
\(587\) −617.569 617.569i −1.05208 1.05208i −0.998567 0.0535098i \(-0.982959\pi\)
−0.0535098 0.998567i \(-0.517041\pi\)
\(588\) 0 0
\(589\) 57.7316i 0.0980163i
\(590\) 0 0
\(591\) 430.761 + 430.761i 0.728868 + 0.728868i
\(592\) 0 0
\(593\) 24.7807 24.7807i 0.0417887 0.0417887i −0.685904 0.727692i \(-0.740593\pi\)
0.727692 + 0.685904i \(0.240593\pi\)
\(594\) 0 0
\(595\) 310.690i 0.522167i
\(596\) 0 0
\(597\) 423.553i 0.709470i
\(598\) 0 0
\(599\) −1084.86 −1.81112 −0.905559 0.424221i \(-0.860548\pi\)
−0.905559 + 0.424221i \(0.860548\pi\)
\(600\) 0 0
\(601\) −1048.67 −1.74488 −0.872439 0.488723i \(-0.837463\pi\)
−0.872439 + 0.488723i \(0.837463\pi\)
\(602\) 0 0
\(603\) −20.0982 20.0982i −0.0333303 0.0333303i
\(604\) 0 0
\(605\) −52.8580 + 52.8580i −0.0873686 + 0.0873686i
\(606\) 0 0
\(607\) 362.335 0.596927 0.298464 0.954421i \(-0.403526\pi\)
0.298464 + 0.954421i \(0.403526\pi\)
\(608\) 0 0
\(609\) 32.8544 32.8544i 0.0539481 0.0539481i
\(610\) 0 0
\(611\) 415.198 + 176.450i 0.679538 + 0.288788i
\(612\) 0 0
\(613\) 79.3587 + 79.3587i 0.129459 + 0.129459i 0.768868 0.639408i \(-0.220821\pi\)
−0.639408 + 0.768868i \(0.720821\pi\)
\(614\) 0 0
\(615\) 18.2069i 0.0296047i
\(616\) 0 0
\(617\) −663.590 663.590i −1.07551 1.07551i −0.996906 0.0786054i \(-0.974953\pi\)
−0.0786054 0.996906i \(-0.525047\pi\)
\(618\) 0 0
\(619\) 656.750 656.750i 1.06098 1.06098i 0.0629695 0.998015i \(-0.479943\pi\)
0.998015 0.0629695i \(-0.0200571\pi\)
\(620\) 0 0
\(621\) 186.387i 0.300140i
\(622\) 0 0
\(623\) 502.869i 0.807173i
\(624\) 0 0
\(625\) −123.522 −0.197635
\(626\) 0 0
\(627\) 29.8781 0.0476525
\(628\) 0 0
\(629\) 625.009 + 625.009i 0.993655 + 0.993655i
\(630\) 0 0
\(631\) 567.649 567.649i 0.899602 0.899602i −0.0957989 0.995401i \(-0.530541\pi\)
0.995401 + 0.0957989i \(0.0305406\pi\)
\(632\) 0 0
\(633\) 41.0079 0.0647835
\(634\) 0 0
\(635\) −239.805 + 239.805i −0.377646 + 0.377646i
\(636\) 0 0
\(637\) −280.930 + 113.364i −0.441020 + 0.177965i
\(638\) 0 0
\(639\) 107.273 + 107.273i 0.167876 + 0.167876i
\(640\) 0 0
\(641\) 157.873i 0.246292i 0.992389 + 0.123146i \(0.0392984\pi\)
−0.992389 + 0.123146i \(0.960702\pi\)
\(642\) 0 0
\(643\) −288.261 288.261i −0.448306 0.448306i 0.446485 0.894791i \(-0.352676\pi\)
−0.894791 + 0.446485i \(0.852676\pi\)
\(644\) 0 0
\(645\) −178.488 + 178.488i −0.276725 + 0.276725i
\(646\) 0 0
\(647\) 1079.96i 1.66918i −0.550875 0.834588i \(-0.685706\pi\)
0.550875 0.834588i \(-0.314294\pi\)
\(648\) 0 0
\(649\) 527.996i 0.813553i
\(650\) 0 0
\(651\) −284.226 −0.436599
\(652\) 0 0
\(653\) 757.451 1.15996 0.579978 0.814632i \(-0.303061\pi\)
0.579978 + 0.814632i \(0.303061\pi\)
\(654\) 0 0
\(655\) 474.149 + 474.149i 0.723892 + 0.723892i
\(656\) 0 0
\(657\) 204.293 204.293i 0.310948 0.310948i
\(658\) 0 0
\(659\) −19.7089 −0.0299072 −0.0149536 0.999888i \(-0.504760\pi\)
−0.0149536 + 0.999888i \(0.504760\pi\)
\(660\) 0 0
\(661\) 381.567 381.567i 0.577258 0.577258i −0.356889 0.934147i \(-0.616163\pi\)
0.934147 + 0.356889i \(0.116163\pi\)
\(662\) 0 0
\(663\) 469.799 189.579i 0.708596 0.285941i
\(664\) 0 0
\(665\) −17.4138 17.4138i −0.0261862 0.0261862i
\(666\) 0 0
\(667\) 189.820i 0.284587i
\(668\) 0 0
\(669\) −236.613 236.613i −0.353682 0.353682i
\(670\) 0 0
\(671\) −135.084 + 135.084i −0.201317 + 0.201317i
\(672\) 0 0
\(673\) 984.504i 1.46286i −0.681917 0.731429i \(-0.738854\pi\)
0.681917 0.731429i \(-0.261146\pi\)
\(674\) 0 0
\(675\) 91.3454i 0.135327i
\(676\) 0 0
\(677\) −473.501 −0.699411 −0.349705 0.936860i \(-0.613718\pi\)
−0.349705 + 0.936860i \(0.613718\pi\)
\(678\) 0 0
\(679\) −827.591 −1.21884
\(680\) 0 0
\(681\) −377.756 377.756i −0.554708 0.554708i
\(682\) 0 0
\(683\) −266.970 + 266.970i −0.390879 + 0.390879i −0.875001 0.484122i \(-0.839139\pi\)
0.484122 + 0.875001i \(0.339139\pi\)
\(684\) 0 0
\(685\) −409.161 −0.597316
\(686\) 0 0
\(687\) −60.6971 + 60.6971i −0.0883509 + 0.0883509i
\(688\) 0 0
\(689\) 101.272 + 250.965i 0.146984 + 0.364245i
\(690\) 0 0
\(691\) 353.770 + 353.770i 0.511968 + 0.511968i 0.915129 0.403161i \(-0.132088\pi\)
−0.403161 + 0.915129i \(0.632088\pi\)
\(692\) 0 0
\(693\) 147.097i 0.212261i
\(694\) 0 0
\(695\) −104.198 104.198i −0.149925 0.149925i
\(696\) 0 0
\(697\) 61.3917 61.3917i 0.0880800 0.0880800i
\(698\) 0 0
\(699\) 76.9590i 0.110099i
\(700\) 0 0
\(701\) 59.3419i 0.0846532i 0.999104 + 0.0423266i \(0.0134770\pi\)
−0.999104 + 0.0423266i \(0.986523\pi\)
\(702\) 0 0
\(703\) 70.0623 0.0996619
\(704\) 0 0
\(705\) −163.736 −0.232249
\(706\) 0 0
\(707\) −131.372 131.372i −0.185816 0.185816i
\(708\) 0 0
\(709\) −442.641 + 442.641i −0.624317 + 0.624317i −0.946632 0.322315i \(-0.895539\pi\)
0.322315 + 0.946632i \(0.395539\pi\)
\(710\) 0 0
\(711\) 11.0582 0.0155530
\(712\) 0 0
\(713\) 821.072 821.072i 1.15157 1.15157i
\(714\) 0 0
\(715\) −128.180 317.647i −0.179273 0.444261i
\(716\) 0 0
\(717\) 231.982 + 231.982i 0.323545 + 0.323545i
\(718\) 0 0
\(719\) 567.348i 0.789079i 0.918879 + 0.394539i \(0.129096\pi\)
−0.918879 + 0.394539i \(0.870904\pi\)
\(720\) 0 0
\(721\) −467.377 467.377i −0.648234 0.648234i
\(722\) 0 0
\(723\) 200.093 200.093i 0.276753 0.276753i
\(724\) 0 0
\(725\) 93.0278i 0.128314i
\(726\) 0 0
\(727\) 714.512i 0.982823i 0.870928 + 0.491411i \(0.163519\pi\)
−0.870928 + 0.491411i \(0.836481\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1203.68 1.64663
\(732\) 0 0
\(733\) 592.536 + 592.536i 0.808371 + 0.808371i 0.984387 0.176016i \(-0.0563210\pi\)
−0.176016 + 0.984387i \(0.556321\pi\)
\(734\) 0 0
\(735\) 77.7461 77.7461i 0.105777 0.105777i
\(736\) 0 0
\(737\) 91.6415 0.124344
\(738\) 0 0
\(739\) 116.248 116.248i 0.157304 0.157304i −0.624067 0.781371i \(-0.714521\pi\)
0.781371 + 0.624067i \(0.214521\pi\)
\(740\) 0 0
\(741\) 15.7061 36.9575i 0.0211958 0.0498751i
\(742\) 0 0
\(743\) −207.488 207.488i −0.279257 0.279257i 0.553555 0.832812i \(-0.313271\pi\)
−0.832812 + 0.553555i \(0.813271\pi\)
\(744\) 0 0
\(745\) 614.134i 0.824341i
\(746\) 0 0
\(747\) −122.451 122.451i −0.163924 0.163924i
\(748\) 0 0
\(749\) −584.058 + 584.058i −0.779784 + 0.779784i
\(750\) 0 0
\(751\) 1278.85i 1.70286i −0.524470 0.851429i \(-0.675736\pi\)
0.524470 0.851429i \(-0.324264\pi\)
\(752\) 0 0
\(753\) 659.051i 0.875234i
\(754\) 0 0
\(755\) 72.7137 0.0963095
\(756\) 0 0
\(757\) −271.889 −0.359166 −0.179583 0.983743i \(-0.557475\pi\)
−0.179583 + 0.983743i \(0.557475\pi\)
\(758\) 0 0
\(759\) 424.934 + 424.934i 0.559860 + 0.559860i
\(760\) 0 0
\(761\) −62.5106 + 62.5106i −0.0821427 + 0.0821427i −0.746984 0.664842i \(-0.768499\pi\)
0.664842 + 0.746984i \(0.268499\pi\)
\(762\) 0 0
\(763\) 703.361 0.921836
\(764\) 0 0
\(765\) −130.015 + 130.015i −0.169954 + 0.169954i
\(766\) 0 0
\(767\) −653.100 277.553i −0.851499 0.361868i
\(768\) 0 0
\(769\) 595.694 + 595.694i 0.774635 + 0.774635i 0.978913 0.204278i \(-0.0654847\pi\)
−0.204278 + 0.978913i \(0.565485\pi\)
\(770\) 0 0
\(771\) 508.996i 0.660177i
\(772\) 0 0
\(773\) −583.166 583.166i −0.754419 0.754419i 0.220881 0.975301i \(-0.429107\pi\)
−0.975301 + 0.220881i \(0.929107\pi\)
\(774\) 0 0
\(775\) −402.395 + 402.395i −0.519219 + 0.519219i
\(776\) 0 0
\(777\) 344.933i 0.443929i
\(778\) 0 0
\(779\) 6.88189i 0.00883426i
\(780\) 0 0
\(781\) −489.132 −0.626289
\(782\) 0 0
\(783\) −27.4973 −0.0351179
\(784\) 0 0
\(785\) 511.942 + 511.942i 0.652156 + 0.652156i
\(786\) 0 0
\(787\) 969.877 969.877i 1.23237 1.23237i 0.269322 0.963050i \(-0.413200\pi\)
0.963050 0.269322i \(-0.0867997\pi\)
\(788\) 0 0
\(789\) −272.712 −0.345643
\(790\) 0 0
\(791\) 96.1253 96.1253i 0.121524 0.121524i
\(792\) 0 0
\(793\) 96.0811 + 238.101i 0.121162 + 0.300253i
\(794\) 0 0
\(795\) −69.4534 69.4534i −0.0873628 0.0873628i
\(796\) 0 0
\(797\) 1347.04i 1.69014i 0.534659 + 0.845068i \(0.320440\pi\)
−0.534659 + 0.845068i \(0.679560\pi\)
\(798\) 0 0
\(799\) 552.099 + 552.099i 0.690988 + 0.690988i
\(800\) 0 0
\(801\) −210.436 + 210.436i −0.262717 + 0.262717i
\(802\) 0 0
\(803\) 931.511i 1.16004i
\(804\) 0 0
\(805\) 495.327i 0.615314i
\(806\) 0 0
\(807\) −302.656 −0.375039
\(808\) 0 0
\(809\) −1087.83 −1.34465 −0.672327 0.740254i \(-0.734705\pi\)
−0.672327 + 0.740254i \(0.734705\pi\)
\(810\) 0 0
\(811\) −703.042 703.042i −0.866882 0.866882i 0.125244 0.992126i \(-0.460029\pi\)
−0.992126 + 0.125244i \(0.960029\pi\)
\(812\) 0 0
\(813\) −27.4856 + 27.4856i −0.0338076 + 0.0338076i
\(814\) 0 0
\(815\) 441.092 0.541217
\(816\) 0 0
\(817\) 67.4652 67.4652i 0.0825768 0.0825768i
\(818\) 0 0
\(819\) −181.950 77.3247i −0.222161 0.0944135i
\(820\) 0 0
\(821\) 695.520 + 695.520i 0.847162 + 0.847162i 0.989778 0.142616i \(-0.0455515\pi\)
−0.142616 + 0.989778i \(0.545552\pi\)
\(822\) 0 0
\(823\) 770.965i 0.936774i 0.883523 + 0.468387i \(0.155165\pi\)
−0.883523 + 0.468387i \(0.844835\pi\)
\(824\) 0 0
\(825\) −208.253 208.253i −0.252428 0.252428i
\(826\) 0 0
\(827\) −1073.08 + 1073.08i −1.29756 + 1.29756i −0.367555 + 0.930002i \(0.619805\pi\)
−0.930002 + 0.367555i \(0.880195\pi\)
\(828\) 0 0
\(829\) 718.597i 0.866824i 0.901196 + 0.433412i \(0.142691\pi\)
−0.901196 + 0.433412i \(0.857309\pi\)
\(830\) 0 0
\(831\) 417.567i 0.502488i
\(832\) 0 0
\(833\) −524.303 −0.629415
\(834\) 0 0
\(835\) 589.251 0.705690
\(836\) 0 0
\(837\) 118.940 + 118.940i 0.142103 + 0.142103i
\(838\) 0 0
\(839\) 77.4498 77.4498i 0.0923121 0.0923121i −0.659443 0.751755i \(-0.729208\pi\)
0.751755 + 0.659443i \(0.229208\pi\)
\(840\) 0 0
\(841\) −812.996 −0.966702
\(842\) 0 0
\(843\) 258.140 258.140i 0.306216 0.306216i
\(844\) 0 0
\(845\) −460.291 8.42640i −0.544723 0.00997207i
\(846\) 0 0
\(847\) −98.3631 98.3631i −0.116131 0.116131i
\(848\) 0 0
\(849\) 607.594i 0.715658i
\(850\) 0 0
\(851\) 996.442 + 996.442i 1.17091 + 1.17091i
\(852\) 0 0
\(853\) −294.965 + 294.965i −0.345797 + 0.345797i −0.858541 0.512744i \(-0.828629\pi\)
0.512744 + 0.858541i \(0.328629\pi\)
\(854\) 0 0
\(855\) 14.5744i 0.0170461i
\(856\) 0 0
\(857\) 1291.93i 1.50750i 0.657161 + 0.753751i \(0.271757\pi\)
−0.657161 + 0.753751i \(0.728243\pi\)
\(858\) 0 0
\(859\) −770.932 −0.897476 −0.448738 0.893663i \(-0.648126\pi\)
−0.448738 + 0.893663i \(0.648126\pi\)
\(860\) 0 0
\(861\) −33.8811 −0.0393509
\(862\) 0 0
\(863\) 854.917 + 854.917i 0.990634 + 0.990634i 0.999957 0.00932255i \(-0.00296750\pi\)
−0.00932255 + 0.999957i \(0.502968\pi\)
\(864\) 0 0
\(865\) 469.554 469.554i 0.542837 0.542837i
\(866\) 0 0
\(867\) 376.229 0.433944
\(868\) 0 0
\(869\) −25.2109 + 25.2109i −0.0290114 + 0.0290114i
\(870\) 0 0
\(871\) 48.1734 113.355i 0.0553081 0.130144i
\(872\) 0 0
\(873\) 346.323 + 346.323i 0.396705 + 0.396705i
\(874\) 0 0
\(875\) 587.975i 0.671971i
\(876\) 0 0
\(877\) −333.435 333.435i −0.380199 0.380199i 0.490975 0.871174i \(-0.336641\pi\)
−0.871174 + 0.490975i \(0.836641\pi\)
\(878\) 0 0
\(879\) 605.630 605.630i 0.688999 0.688999i
\(880\) 0 0
\(881\) 297.062i 0.337187i 0.985686 + 0.168593i \(0.0539225\pi\)
−0.985686 + 0.168593i \(0.946078\pi\)
\(882\) 0 0
\(883\) 1112.56i 1.25998i 0.776603 + 0.629990i \(0.216941\pi\)
−0.776603 + 0.629990i \(0.783059\pi\)
\(884\) 0 0
\(885\) 257.554 0.291021
\(886\) 0 0
\(887\) −27.2311 −0.0307003 −0.0153501 0.999882i \(-0.504886\pi\)
−0.0153501 + 0.999882i \(0.504886\pi\)
\(888\) 0 0
\(889\) −446.252 446.252i −0.501971 0.501971i
\(890\) 0 0
\(891\) −61.5558 + 61.5558i −0.0690862 + 0.0690862i
\(892\) 0 0
\(893\) 61.8892 0.0693048
\(894\) 0 0
\(895\) −167.320 + 167.320i −0.186950 + 0.186950i
\(896\) 0 0
\(897\) 748.994 302.242i 0.834998 0.336948i
\(898\) 0 0
\(899\) −121.131 121.131i −0.134740 0.134740i
\(900\) 0 0
\(901\) 468.379i 0.519843i
\(902\) 0 0
\(903\) −332.147 332.147i −0.367826 0.367826i
\(904\) 0 0
\(905\) 324.281 324.281i 0.358322 0.358322i
\(906\) 0 0
\(907\) 183.412i 0.202219i −0.994875 0.101109i \(-0.967761\pi\)
0.994875 0.101109i \(-0.0322392\pi\)
\(908\) 0 0
\(909\) 109.951i 0.120958i
\(910\) 0 0
\(911\) 506.343 0.555810 0.277905 0.960609i \(-0.410360\pi\)
0.277905 + 0.960609i \(0.410360\pi\)
\(912\) 0 0
\(913\) 558.339 0.611544
\(914\) 0 0
\(915\) −65.8933 65.8933i −0.0720146 0.0720146i
\(916\) 0 0
\(917\) −882.341 + 882.341i −0.962204 + 0.962204i
\(918\) 0 0
\(919\) 195.184 0.212388 0.106194 0.994345i \(-0.466134\pi\)
0.106194 + 0.994345i \(0.466134\pi\)
\(920\) 0 0
\(921\) −687.876 + 687.876i −0.746879 + 0.746879i
\(922\) 0 0
\(923\) −257.123 + 605.027i −0.278573 + 0.655501i
\(924\) 0 0
\(925\) −488.341 488.341i −0.527936 0.527936i
\(926\) 0 0
\(927\) 391.168i 0.421972i
\(928\) 0 0
\(929\) −636.021 636.021i −0.684630 0.684630i 0.276410 0.961040i \(-0.410855\pi\)
−0.961040 + 0.276410i \(0.910855\pi\)
\(930\) 0 0
\(931\) −29.3866 + 29.3866i −0.0315646 + 0.0315646i
\(932\) 0 0
\(933\) 273.442i 0.293078i
\(934\) 0 0
\(935\) 592.827i 0.634040i
\(936\) 0 0
\(937\) 159.058 0.169752 0.0848761 0.996392i \(-0.472951\pi\)
0.0848761 + 0.996392i \(0.472951\pi\)
\(938\) 0 0
\(939\) −151.339 −0.161170
\(940\) 0 0
\(941\) 667.783 + 667.783i 0.709652 + 0.709652i 0.966462 0.256810i \(-0.0826714\pi\)
−0.256810 + 0.966462i \(0.582671\pi\)
\(942\) 0 0
\(943\) 97.8759 97.8759i 0.103792 0.103792i
\(944\) 0 0
\(945\) 71.7531 0.0759292
\(946\) 0 0
\(947\) −370.339 + 370.339i −0.391065 + 0.391065i −0.875067 0.484002i \(-0.839183\pi\)
0.484002 + 0.875067i \(0.339183\pi\)
\(948\) 0 0
\(949\) 1152.22 + 489.669i 1.21415 + 0.515985i
\(950\) 0 0
\(951\) −608.972 608.972i −0.640349 0.640349i
\(952\) 0 0
\(953\) 1343.91i 1.41019i −0.709113 0.705095i \(-0.750904\pi\)
0.709113 0.705095i \(-0.249096\pi\)
\(954\) 0 0
\(955\) 627.752 + 627.752i 0.657332 + 0.657332i
\(956\) 0 0
\(957\) 62.6896 62.6896i 0.0655063 0.0655063i
\(958\) 0 0
\(959\) 761.406i 0.793958i
\(960\) 0 0
\(961\) 86.9137i 0.0904409i
\(962\) 0 0
\(963\) 488.823 0.507605
\(964\) 0 0
\(965\) −890.383 −0.922677
\(966\) 0 0
\(967\) −629.403 629.403i −0.650882 0.650882i 0.302323 0.953205i \(-0.402238\pi\)
−0.953205 + 0.302323i \(0.902238\pi\)
\(968\) 0 0
\(969\) 49.1433 49.1433i 0.0507155 0.0507155i
\(970\) 0 0
\(971\) −14.3706 −0.0147998 −0.00739990 0.999973i \(-0.502355\pi\)
−0.00739990 + 0.999973i \(0.502355\pi\)
\(972\) 0 0
\(973\) 193.902 193.902i 0.199282 0.199282i
\(974\) 0 0
\(975\) −367.070 + 148.124i −0.376482 + 0.151922i
\(976\) 0 0
\(977\) −674.932 674.932i −0.690821 0.690821i 0.271592 0.962413i \(-0.412450\pi\)
−0.962413 + 0.271592i \(0.912450\pi\)
\(978\) 0 0
\(979\) 959.525i 0.980107i
\(980\) 0 0
\(981\) −294.337 294.337i −0.300038 0.300038i
\(982\) 0 0
\(983\) −255.689 + 255.689i −0.260110 + 0.260110i −0.825099 0.564988i \(-0.808881\pi\)
0.564988 + 0.825099i \(0.308881\pi\)
\(984\) 0 0
\(985\) 958.096i 0.972687i
\(986\) 0 0
\(987\) 304.695i 0.308708i
\(988\) 0 0
\(989\) 1919.01 1.94036
\(990\) 0 0
\(991\) −785.926 −0.793063 −0.396532 0.918021i \(-0.629786\pi\)
−0.396532 + 0.918021i \(0.629786\pi\)
\(992\) 0 0
\(993\) 117.114 + 117.114i 0.117939 + 0.117939i
\(994\) 0 0
\(995\) 471.033 471.033i 0.473400 0.473400i
\(996\) 0 0
\(997\) 361.986 0.363075 0.181538 0.983384i \(-0.441893\pi\)
0.181538 + 0.983384i \(0.441893\pi\)
\(998\) 0 0
\(999\) −144.345 + 144.345i −0.144489 + 0.144489i
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 156.3.j.a.73.2 8
3.2 odd 2 468.3.m.d.73.2 8
4.3 odd 2 624.3.ba.d.385.4 8
13.5 odd 4 inner 156.3.j.a.109.2 yes 8
39.5 even 4 468.3.m.d.109.2 8
52.31 even 4 624.3.ba.d.577.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.3.j.a.73.2 8 1.1 even 1 trivial
156.3.j.a.109.2 yes 8 13.5 odd 4 inner
468.3.m.d.73.2 8 3.2 odd 2
468.3.m.d.109.2 8 39.5 even 4
624.3.ba.d.385.4 8 4.3 odd 2
624.3.ba.d.577.4 8 52.31 even 4