Properties

Label 1275.2.d.d.424.1
Level $1275$
Weight $2$
Character 1275.424
Analytic conductor $10.181$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1275,2,Mod(424,1275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1275.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.d (of order \(2\), degree \(1\), 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: \(10.1809262577\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 424.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1275.424
Dual form 1275.2.d.d.424.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.00000 q^{7} -3.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.00000 q^{7} -3.00000i q^{8} +1.00000 q^{9} -4.00000i q^{11} +1.00000 q^{12} +2.00000i q^{13} -4.00000i q^{14} -1.00000 q^{16} +(-4.00000 - 1.00000i) q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -3.00000i q^{24} +2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +4.00000i q^{31} -5.00000i q^{32} -4.00000i q^{33} +(-1.00000 + 4.00000i) q^{34} +1.00000 q^{36} -8.00000 q^{37} -4.00000i q^{38} +2.00000i q^{39} +8.00000i q^{41} -4.00000i q^{42} -4.00000i q^{43} -4.00000i q^{44} +4.00000i q^{46} +8.00000i q^{47} -1.00000 q^{48} +9.00000 q^{49} +(-4.00000 - 1.00000i) q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000i q^{54} -12.0000i q^{56} +4.00000 q^{57} +12.0000 q^{59} +8.00000i q^{61} +4.00000 q^{62} +4.00000 q^{63} -7.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +(-4.00000 - 1.00000i) q^{68} -4.00000 q^{69} -12.0000i q^{71} -3.00000i q^{72} +8.00000i q^{74} +4.00000 q^{76} -16.0000i q^{77} +2.00000 q^{78} -4.00000i q^{79} +1.00000 q^{81} +8.00000 q^{82} +12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} -12.0000 q^{88} +10.0000 q^{89} +8.00000i q^{91} -4.00000 q^{92} +4.00000i q^{93} +8.00000 q^{94} -5.00000i q^{96} -16.0000 q^{97} -9.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} + 8 q^{7} + 2 q^{9} + 2 q^{12} - 2 q^{16} - 8 q^{17} + 8 q^{19} + 8 q^{21} - 8 q^{22} - 8 q^{23} + 4 q^{26} + 2 q^{27} + 8 q^{28} - 2 q^{34} + 2 q^{36} - 16 q^{37} - 2 q^{48} + 18 q^{49} - 8 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{62} + 8 q^{63} - 14 q^{64} - 8 q^{66} - 8 q^{68} - 8 q^{69} + 8 q^{76} + 4 q^{78} + 2 q^{81} + 16 q^{82} + 8 q^{84} - 8 q^{86} - 24 q^{88} + 20 q^{89} - 8 q^{92} + 16 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(751\) \(851\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000i 0.408248i
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 4.00000i 1.06904i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 1.00000i −0.970143 0.242536i
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000i 0.612372i
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) −1.00000 + 4.00000i −0.171499 + 0.685994i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 8.00000i 1.24939i 0.780869 + 0.624695i \(0.214777\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000i 0.603023i
\(45\) 0 0
\(46\) 4.00000i 0.589768i
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 1.00000i −0.560112 0.140028i
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) 12.0000i 1.60357i
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 12.0000i 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) −4.00000 1.00000i −0.485071 0.121268i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 8.00000i 0.929981i
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000i 1.82337i
\(78\) 2.00000 0.226455
\(79\) 4.00000i 0.450035i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) −4.00000 −0.417029
\(93\) 4.00000i 0.414781i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 5.00000i 0.510310i
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −1.00000 + 4.00000i −0.0990148 + 0.396059i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −4.00000 −0.377964
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 4.00000i 0.374634i
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) −16.0000 4.00000i −1.46672 0.366679i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 8.00000 0.724286
\(123\) 8.00000i 0.721336i
\(124\) 4.00000i 0.359211i
\(125\) 0 0
\(126\) 4.00000i 0.356348i
\(127\) 8.00000i 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 4.00000i 0.352180i
\(130\) 0 0
\(131\) 4.00000i 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 16.0000 1.38738
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −3.00000 + 12.0000i −0.257248 + 1.02899i
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) −12.0000 −1.00702
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) −8.00000 −0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 12.0000i 0.973329i
\(153\) −4.00000 1.00000i −0.323381 0.0808452i
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) 2.00000i 0.160128i
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000i 0.475831i
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000i 0.0785674i
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 8.00000i 0.624695i
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 12.0000i 0.925820i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000i 0.304997i
\(173\) −16.0000 −1.21646 −0.608229 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000i 0.301511i
\(177\) 12.0000 0.901975
\(178\) 10.0000i 0.749532i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 8.00000i 0.594635i 0.954779 + 0.297318i \(0.0960920\pi\)
−0.954779 + 0.297318i \(0.903908\pi\)
\(182\) 8.00000 0.592999
\(183\) 8.00000i 0.591377i
\(184\) 12.0000i 0.884652i
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −4.00000 + 16.0000i −0.292509 + 1.17004i
\(188\) 8.00000i 0.583460i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −7.00000 −0.505181
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 16.0000i 1.14873i
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) −4.00000 −0.284268
\(199\) 20.0000i 1.41776i −0.705328 0.708881i \(-0.749200\pi\)
0.705328 0.708881i \(-0.250800\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) −4.00000 1.00000i −0.280056 0.0700140i
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 2.00000i 0.138675i
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) 4.00000i 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 12.0000i 0.822226i
\(214\) 12.0000i 0.820303i
\(215\) 0 0
\(216\) 3.00000i 0.204124i
\(217\) 16.0000i 1.08615i
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 8.00000i 0.134535 0.538138i
\(222\) 8.00000i 0.536925i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 20.0000i 1.33631i
\(225\) 0 0
\(226\) 8.00000i 0.532152i
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 16.0000i 1.05272i
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 4.00000i 0.259828i
\(238\) −4.00000 + 16.0000i −0.259281 + 1.03713i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 16.0000i 1.03065i 0.856995 + 0.515325i \(0.172329\pi\)
−0.856995 + 0.515325i \(0.827671\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000 0.0641500
\(244\) 8.00000i 0.512148i
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 8.00000i 0.509028i
\(248\) 12.0000 0.762001
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 4.00000 0.251976
\(253\) 16.0000i 1.00591i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) −4.00000 −0.249029
\(259\) −32.0000 −1.98838
\(260\) 0 0
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 16.0000i 0.981023i
\(267\) 10.0000 0.611990
\(268\) 12.0000i 0.733017i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 4.00000 + 1.00000i 0.242536 + 0.0606339i
\(273\) 8.00000i 0.484182i
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 4.00000i 0.239474i
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000i 0.712069i
\(285\) 0 0
\(286\) 8.00000i 0.473050i
\(287\) 32.0000i 1.88890i
\(288\) 5.00000i 0.294628i
\(289\) 15.0000 + 8.00000i 0.882353 + 0.470588i
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 0 0
\(293\) 10.0000i 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 9.00000i 0.524891i
\(295\) 0 0
\(296\) 24.0000i 1.39497i
\(297\) 4.00000i 0.232104i
\(298\) 6.00000i 0.347571i
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 16.0000i 0.922225i
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −1.00000 + 4.00000i −0.0571662 + 0.228665i
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 16.0000i 0.911685i
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000i 0.680458i −0.940343 0.340229i \(-0.889495\pi\)
0.940343 0.340229i \(-0.110505\pi\)
\(312\) 6.00000 0.339683
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 4.00000i 0.225018i
\(317\) 32.0000 1.79730 0.898650 0.438667i \(-0.144549\pi\)
0.898650 + 0.438667i \(0.144549\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 16.0000i 0.891645i
\(323\) −16.0000 4.00000i −0.890264 0.222566i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000i 1.10770i
\(327\) 8.00000i 0.442401i
\(328\) 24.0000 1.32518
\(329\) 32.0000i 1.76422i
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 12.0000i 0.658586i
\(333\) −8.00000 −0.438397
\(334\) 12.0000i 0.656611i
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 4.00000i 0.216295i
\(343\) 8.00000 0.431959
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 16.0000i 0.860165i
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 2.00000i 0.106752i
\(352\) −20.0000 −1.06600
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 12.0000i 0.637793i
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −16.0000 4.00000i −0.846810 0.211702i
\(358\) 4.00000i 0.211407i
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 8.00000 0.420471
\(363\) −5.00000 −0.262432
\(364\) 8.00000i 0.419314i
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 4.00000 0.208514
\(369\) 8.00000i 0.416463i
\(370\) 0 0
\(371\) 24.0000i 1.24602i
\(372\) 4.00000i 0.207390i
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 16.0000 + 4.00000i 0.827340 + 0.206835i
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 4.00000i 0.205738i
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 8.00000i 0.409316i
\(383\) 16.0000i 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 3.00000i 0.153093i
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) −16.0000 −0.812277
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 16.0000 + 4.00000i 0.809155 + 0.202289i
\(392\) 27.0000i 1.36371i
\(393\) 4.00000i 0.201773i
\(394\) 16.0000i 0.806068i
\(395\) 0 0
\(396\) 4.00000i 0.201008i
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −20.0000 −1.00251
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 24.0000i 1.19850i 0.800561 + 0.599251i \(0.204535\pi\)
−0.800561 + 0.599251i \(0.795465\pi\)
\(402\) −12.0000 −0.598506
\(403\) −8.00000 −0.398508
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000i 1.58618i
\(408\) −3.00000 + 12.0000i −0.148522 + 0.594089i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 10.0000i 0.493264i
\(412\) 0 0
\(413\) 48.0000 2.36193
\(414\) 4.00000i 0.196589i
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 4.00000i 0.195881i
\(418\) −16.0000 −0.782586
\(419\) 4.00000i 0.195413i 0.995215 + 0.0977064i \(0.0311506\pi\)
−0.995215 + 0.0977064i \(0.968849\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −4.00000 −0.194717
\(423\) 8.00000i 0.388973i
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 32.0000i 1.54859i
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 20.0000i 0.963366i 0.876346 + 0.481683i \(0.159974\pi\)
−0.876346 + 0.481683i \(0.840026\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000i 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 8.00000i 0.383131i
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 36.0000i 1.71819i −0.511819 0.859093i \(-0.671028\pi\)
0.511819 0.859093i \(-0.328972\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −8.00000 2.00000i −0.380521 0.0951303i
\(443\) 20.0000i 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) −28.0000 −1.32288
\(449\) 8.00000i 0.377543i 0.982021 + 0.188772i \(0.0604506\pi\)
−0.982021 + 0.188772i \(0.939549\pi\)
\(450\) 0 0
\(451\) 32.0000 1.50682
\(452\) −8.00000 −0.376288
\(453\) 0 0
\(454\) 12.0000i 0.563188i
\(455\) 0 0
\(456\) 12.0000i 0.561951i
\(457\) 38.0000i 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 10.0000i 0.467269i
\(459\) −4.00000 1.00000i −0.186704 0.0466760i
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) −16.0000 −0.744387
\(463\) 40.0000i 1.85896i −0.368875 0.929479i \(-0.620257\pi\)
0.368875 0.929479i \(-0.379743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 8.00000i 0.370593i
\(467\) 4.00000i 0.185098i −0.995708 0.0925490i \(-0.970499\pi\)
0.995708 0.0925490i \(-0.0295015\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 48.0000i 2.21643i
\(470\) 0 0
\(471\) 2.00000i 0.0921551i
\(472\) 36.0000i 1.65703i
\(473\) −16.0000 −0.735681
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) −16.0000 4.00000i −0.733359 0.183340i
\(477\) 6.00000i 0.274721i
\(478\) 8.00000i 0.365911i
\(479\) 12.0000i 0.548294i 0.961688 + 0.274147i \(0.0883955\pi\)
−0.961688 + 0.274147i \(0.911605\pi\)
\(480\) 0 0
\(481\) 16.0000i 0.729537i
\(482\) 16.0000 0.728780
\(483\) −16.0000 −0.728025
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 24.0000 1.08643
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 8.00000i 0.360668i
\(493\) 0 0
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 4.00000i 0.179605i
\(497\) 48.0000i 2.15309i
\(498\) 12.0000 0.537733
\(499\) 36.0000i 1.61158i 0.592200 + 0.805791i \(0.298259\pi\)
−0.592200 + 0.805791i \(0.701741\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 12.0000i 0.535586i
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 12.0000i 0.534522i
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 9.00000 0.399704
\(508\) 8.00000i 0.354943i
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000i 0.486136i
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 4.00000i 0.176090i
\(517\) 32.0000 1.40736
\(518\) 32.0000i 1.40600i
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 24.0000i 1.05146i −0.850652 0.525730i \(-0.823792\pi\)
0.850652 0.525730i \(-0.176208\pi\)
\(522\) 0 0
\(523\) 12.0000i 0.524723i −0.964970 0.262362i \(-0.915499\pi\)
0.964970 0.262362i \(-0.0845013\pi\)
\(524\) 4.00000i 0.174741i
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 4.00000 16.0000i 0.174243 0.696971i
\(528\) 4.00000i 0.174078i
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 16.0000 0.693688
\(533\) −16.0000 −0.693037
\(534\) 10.0000i 0.432742i
\(535\) 0 0
\(536\) −36.0000 −1.55496
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 36.0000i 1.55063i
\(540\) 0 0
\(541\) 40.0000i 1.71973i −0.510518 0.859867i \(-0.670546\pi\)
0.510518 0.859867i \(-0.329454\pi\)
\(542\) 0 0
\(543\) 8.00000i 0.343313i
\(544\) −5.00000 + 20.0000i −0.214373 + 0.857493i
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 10.0000i 0.427179i
\(549\) 8.00000i 0.341432i
\(550\) 0 0
\(551\) 0 0
\(552\) 12.0000i 0.510754i
\(553\) 16.0000i 0.680389i
\(554\) 8.00000i 0.339887i
\(555\) 0 0
\(556\) 4.00000i 0.169638i
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 4.00000 0.169334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −4.00000 + 16.0000i −0.168880 + 0.675521i
\(562\) 10.0000i 0.421825i
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 8.00000i 0.336861i
\(565\) 0 0
\(566\) 4.00000i 0.168133i
\(567\) 4.00000 0.167984
\(568\) −36.0000 −1.51053
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 8.00000 0.334497
\(573\) −8.00000 −0.334205
\(574\) 32.0000 1.33565
\(575\) 0 0
\(576\) −7.00000 −0.291667
\(577\) 30.0000i 1.24892i −0.781058 0.624458i \(-0.785320\pi\)
0.781058 0.624458i \(-0.214680\pi\)
\(578\) 8.00000 15.0000i 0.332756 0.623918i
\(579\) 0 0
\(580\) 0 0
\(581\) 48.0000i 1.99138i
\(582\) 16.0000i 0.663221i
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 12.0000i 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 9.00000 0.371154
\(589\) 16.0000i 0.659269i
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 8.00000 0.328798
\(593\) 46.0000i 1.88899i −0.328521 0.944497i \(-0.606550\pi\)
0.328521 0.944497i \(-0.393450\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 20.0000i 0.818546i
\(598\) −8.00000 −0.327144
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 16.0000i 0.652654i −0.945257 0.326327i \(-0.894189\pi\)
0.945257 0.326327i \(-0.105811\pi\)
\(602\) −16.0000 −0.652111
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 6.00000i 0.243733i
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 20.0000i 0.811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −4.00000 1.00000i −0.161690 0.0404226i
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −48.0000 −1.93398
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) 36.0000i 1.44696i 0.690344 + 0.723481i \(0.257459\pi\)
−0.690344 + 0.723481i \(0.742541\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −12.0000 −0.481156
\(623\) 40.0000 1.60257
\(624\) 2.00000i 0.0800641i
\(625\) 0 0
\(626\) 16.0000i 0.639489i
\(627\) 16.0000i 0.638978i
\(628\) 2.00000i 0.0798087i
\(629\) 32.0000 + 8.00000i 1.27592 + 0.318981i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −12.0000 −0.477334
\(633\) 4.00000i 0.158986i
\(634\) 32.0000i 1.27088i
\(635\) 0 0
\(636\) 6.00000i 0.237915i
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 12.0000i 0.474713i
\(640\) 0 0
\(641\) 8.00000i 0.315981i −0.987441 0.157991i \(-0.949498\pi\)
0.987441 0.157991i \(-0.0505015\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −4.00000 + 16.0000i −0.157378 + 0.629512i
\(647\) 32.0000i 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 48.0000i 1.88416i
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 20.0000 0.783260
\(653\) −32.0000 −1.25226 −0.626128 0.779720i \(-0.715361\pi\)
−0.626128 + 0.779720i \(0.715361\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 8.00000i 0.312348i
\(657\) 0 0
\(658\) 32.0000 1.24749
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 2.00000 8.00000i 0.0776736 0.310694i
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) 8.00000i 0.309994i
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 16.0000i 0.618596i
\(670\) 0 0
\(671\) 32.0000 1.23535
\(672\) 20.0000i 0.771517i
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 16.0000i 0.616297i
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 8.00000i 0.307238i
\(679\) −64.0000 −2.45609
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 16.0000i 0.612672i
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 8.00000i 0.305441i
\(687\) −10.0000 −0.381524
\(688\) 4.00000i 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000i 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) −16.0000 −0.608229
\(693\) 16.0000i 0.607790i
\(694\) 4.00000i 0.151838i
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000 32.0000i 0.303022 1.21209i
\(698\) 2.00000i 0.0757011i
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 2.00000 0.0754851
\(703\) −32.0000 −1.20690
\(704\) 28.0000i 1.05529i
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −24.0000 −0.902613
\(708\) 12.0000 0.450988
\(709\) 40.0000i 1.50223i −0.660171 0.751116i \(-0.729516\pi\)
0.660171 0.751116i \(-0.270484\pi\)
\(710\) 0 0
\(711\) 4.00000i 0.150012i
\(712\) 30.0000i 1.12430i
\(713\) 16.0000i 0.599205i
\(714\) −4.00000 + 16.0000i −0.149696 + 0.598785i
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 8.00000 0.298765
\(718\) 32.0000i 1.19423i
\(719\) 36.0000i 1.34257i −0.741198 0.671287i \(-0.765742\pi\)
0.741198 0.671287i \(-0.234258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 16.0000i 0.595046i
\(724\) 8.00000i 0.297318i
\(725\) 0 0
\(726\) 5.00000i 0.185567i
\(727\) 8.00000i 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 24.0000 0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 + 16.0000i −0.147945 + 0.591781i
\(732\) 8.00000i 0.295689i
\(733\) 34.0000i 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 12.0000i 0.442928i
\(735\) 0 0
\(736\) 20.0000i 0.737210i
\(737\) −48.0000 −1.76810
\(738\) 8.00000 0.294484
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 24.0000 0.881068
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 12.0000i 0.439057i
\(748\) −4.00000 + 16.0000i −0.146254 + 0.585018i
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 20.0000i 0.729810i 0.931045 + 0.364905i \(0.118899\pi\)
−0.931045 + 0.364905i \(0.881101\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 20.0000 0.726433
\(759\) 16.0000i 0.580763i
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −8.00000 −0.289809
\(763\) 32.0000i 1.15848i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 24.0000i 0.866590i
\(768\) −17.0000 −0.613435
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) 6.00000i 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 48.0000i 1.72310i
\(777\) −32.0000 −1.14799
\(778\) 6.00000i 0.215110i
\(779\) 32.0000i 1.14652i
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 4.00000 16.0000i 0.143040 0.572159i
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −16.0000 −0.569976
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) −12.0000 −0.426401
\(793\) −16.0000 −0.568177
\(794\) 8.00000i 0.283909i
\(795\) 0 0
\(796\) 20.0000i 0.708881i
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 16.0000i 0.566394i
\(799\) 8.00000 32.0000i 0.283020 1.13208i
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 24.0000 0.847469
\(803\) 0 0
\(804\) 12.0000i 0.423207i
\(805\) 0 0
\(806\) 8.00000i 0.281788i
\(807\) 0 0
\(808\) 18.0000i 0.633238i
\(809\) 40.0000i 1.40633i −0.711029 0.703163i \(-0.751771\pi\)
0.711029 0.703163i \(-0.248229\pi\)
\(810\) 0 0
\(811\) 4.00000i 0.140459i −0.997531 0.0702295i \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 32.0000 1.12160
\(815\) 0 0
\(816\) 4.00000 + 1.00000i 0.140028 + 0.0350070i
\(817\) 16.0000i 0.559769i
\(818\) 10.0000i 0.349642i
\(819\) 8.00000i 0.279543i
\(820\) 0 0
\(821\) 32.0000i 1.11681i −0.829569 0.558404i \(-0.811414\pi\)
0.829569 0.558404i \(-0.188586\pi\)
\(822\) 10.0000 0.348790
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 48.0000i 1.67013i
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 14.0000i 0.485363i
\(833\) −36.0000 9.00000i −1.24733 0.311832i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 16.0000i 0.553372i
\(837\) 4.00000i 0.138260i
\(838\) 4.00000 0.138178
\(839\) 44.0000i 1.51905i 0.650479 + 0.759524i \(0.274568\pi\)
−0.650479 + 0.759524i \(0.725432\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 22.0000i 0.758170i
\(843\) 10.0000 0.344418
\(844\) 4.00000i 0.137686i
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −20.0000 −0.687208
\(848\) 6.00000i 0.206041i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 12.0000i 0.411113i
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) 36.0000i 1.23045i
\(857\) −8.00000 −0.273275 −0.136637 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 32.0000i 1.09056i
\(862\) 20.0000 0.681203
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 5.00000i 0.170103i
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 15.0000 + 8.00000i 0.509427 + 0.271694i
\(868\) 16.0000i 0.543075i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 24.0000 0.812743
\(873\) −16.0000 −0.541518
\(874\) 16.0000i 0.541208i
\(875\) 0 0
\(876\) 0 0
\(877\) 40.0000 1.35070 0.675352 0.737496i \(-0.263992\pi\)
0.675352 + 0.737496i \(0.263992\pi\)
\(878\) −36.0000 −1.21494
\(879\) 10.0000i 0.337292i
\(880\) 0 0
\(881\) 24.0000i 0.808581i −0.914631 0.404290i \(-0.867519\pi\)
0.914631 0.404290i \(-0.132481\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 20.0000i 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 2.00000 8.00000i 0.0672673 0.269069i
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 24.0000i 0.805387i
\(889\) 32.0000i 1.07325i
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 16.0000i 0.535720i
\(893\) 32.0000i 1.07084i
\(894\) 6.00000i 0.200670i
\(895\) 0 0
\(896\) 12.0000i 0.400892i
\(897\) 8.00000i 0.267112i
\(898\) 8.00000 0.266963
\(899\) 0 0
\(900\) 0 0
\(901\) 6.00000 24.0000i 0.199889 0.799556i
\(902\) 32.0000i 1.06548i
\(903\) 16.0000i 0.532447i
\(904\) 24.0000i 0.798228i
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 28.0000i 0.927681i −0.885919 0.463841i \(-0.846471\pi\)
0.885919 0.463841i \(-0.153529\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 16.0000i 0.528367i
\(918\) −1.00000 + 4.00000i −0.0330049 + 0.132020i
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 34.0000i 1.11973i
\(923\) 24.0000 0.789970
\(924\) 16.0000i 0.526361i
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) 0 0
\(929\) 24.0000i 0.787414i 0.919236 + 0.393707i \(0.128808\pi\)
−0.919236 + 0.393707i \(0.871192\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 8.00000 0.262049
\(933\) 12.0000i 0.392862i
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 26.0000i 0.849383i −0.905338 0.424691i \(-0.860383\pi\)
0.905338 0.424691i \(-0.139617\pi\)
\(938\) −48.0000 −1.56726
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 48.0000i 1.56476i 0.622804 + 0.782378i \(0.285993\pi\)
−0.622804 + 0.782378i \(0.714007\pi\)
\(942\) 2.00000 0.0651635
\(943\) 32.0000i 1.04206i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 16.0000i 0.520205i
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 4.00000i 0.129914i
\(949\) 0 0
\(950\) 0 0
\(951\) 32.0000 1.03767
\(952\) −12.0000 + 48.0000i −0.388922 + 1.55569i
\(953\) 42.0000i 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) 40.0000i 1.29167i
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) −16.0000 −0.515861
\(963\) 12.0000 0.386695
\(964\) 16.0000i 0.515325i
\(965\) 0 0
\(966\) 16.0000i 0.514792i
\(967\) 24.0000i 0.771788i −0.922543 0.385894i \(-0.873893\pi\)
0.922543 0.385894i \(-0.126107\pi\)
\(968\) 15.0000i 0.482118i
\(969\) −16.0000 4.00000i −0.513994 0.128499i
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000i 0.512936i
\(974\) 4.00000i 0.128168i
\(975\) 0 0
\(976\) 8.00000i 0.256074i
\(977\) 46.0000i 1.47167i 0.677161 + 0.735835i \(0.263210\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(978\) 20.0000i 0.639529i
\(979\) 40.0000i 1.27841i
\(980\) 0 0
\(981\) 8.00000i 0.255420i
\(982\) 20.0000i 0.638226i
\(983\) −20.0000 −0.637901 −0.318950 0.947771i \(-0.603330\pi\)
−0.318950 + 0.947771i \(0.603330\pi\)
\(984\) 24.0000 0.765092
\(985\) 0 0
\(986\) 0 0
\(987\) 32.0000i 1.01857i
\(988\) 8.00000i 0.254514i
\(989\) 16.0000i 0.508770i
\(990\) 0 0
\(991\) 12.0000i 0.381193i −0.981669 0.190596i \(-0.938958\pi\)
0.981669 0.190596i \(-0.0610421\pi\)
\(992\) 20.0000 0.635001
\(993\) −20.0000 −0.634681
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 12.0000i 0.380235i
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 36.0000 1.13956
\(999\) −8.00000 −0.253109
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 1275.2.d.d.424.1 2
5.2 odd 4 51.2.d.b.16.1 2
5.3 odd 4 1275.2.g.a.526.2 2
5.4 even 2 1275.2.d.b.424.2 2
15.2 even 4 153.2.d.a.118.2 2
17.16 even 2 1275.2.d.b.424.1 2
20.7 even 4 816.2.c.c.577.2 2
40.27 even 4 3264.2.c.d.577.1 2
40.37 odd 4 3264.2.c.e.577.2 2
60.47 odd 4 2448.2.c.j.577.1 2
85.2 odd 8 867.2.e.d.829.1 4
85.7 even 16 867.2.h.d.733.1 8
85.12 even 16 867.2.h.d.757.1 8
85.22 even 16 867.2.h.d.757.2 8
85.27 even 16 867.2.h.d.733.2 8
85.32 odd 8 867.2.e.d.829.2 4
85.33 odd 4 1275.2.g.a.526.1 2
85.37 even 16 867.2.h.d.688.2 8
85.42 odd 8 867.2.e.d.616.1 4
85.47 odd 4 867.2.a.b.1.1 1
85.57 even 16 867.2.h.d.712.1 8
85.62 even 16 867.2.h.d.712.2 8
85.67 odd 4 51.2.d.b.16.2 yes 2
85.72 odd 4 867.2.a.a.1.1 1
85.77 odd 8 867.2.e.d.616.2 4
85.82 even 16 867.2.h.d.688.1 8
85.84 even 2 inner 1275.2.d.d.424.2 2
255.47 even 4 2601.2.a.j.1.1 1
255.152 even 4 153.2.d.a.118.1 2
255.242 even 4 2601.2.a.i.1.1 1
340.67 even 4 816.2.c.c.577.1 2
680.67 even 4 3264.2.c.d.577.2 2
680.237 odd 4 3264.2.c.e.577.1 2
1020.407 odd 4 2448.2.c.j.577.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.d.b.16.1 2 5.2 odd 4
51.2.d.b.16.2 yes 2 85.67 odd 4
153.2.d.a.118.1 2 255.152 even 4
153.2.d.a.118.2 2 15.2 even 4
816.2.c.c.577.1 2 340.67 even 4
816.2.c.c.577.2 2 20.7 even 4
867.2.a.a.1.1 1 85.72 odd 4
867.2.a.b.1.1 1 85.47 odd 4
867.2.e.d.616.1 4 85.42 odd 8
867.2.e.d.616.2 4 85.77 odd 8
867.2.e.d.829.1 4 85.2 odd 8
867.2.e.d.829.2 4 85.32 odd 8
867.2.h.d.688.1 8 85.82 even 16
867.2.h.d.688.2 8 85.37 even 16
867.2.h.d.712.1 8 85.57 even 16
867.2.h.d.712.2 8 85.62 even 16
867.2.h.d.733.1 8 85.7 even 16
867.2.h.d.733.2 8 85.27 even 16
867.2.h.d.757.1 8 85.12 even 16
867.2.h.d.757.2 8 85.22 even 16
1275.2.d.b.424.1 2 17.16 even 2
1275.2.d.b.424.2 2 5.4 even 2
1275.2.d.d.424.1 2 1.1 even 1 trivial
1275.2.d.d.424.2 2 85.84 even 2 inner
1275.2.g.a.526.1 2 85.33 odd 4
1275.2.g.a.526.2 2 5.3 odd 4
2448.2.c.j.577.1 2 60.47 odd 4
2448.2.c.j.577.2 2 1020.407 odd 4
2601.2.a.i.1.1 1 255.242 even 4
2601.2.a.j.1.1 1 255.47 even 4
3264.2.c.d.577.1 2 40.27 even 4
3264.2.c.d.577.2 2 680.67 even 4
3264.2.c.e.577.1 2 680.237 odd 4
3264.2.c.e.577.2 2 40.37 odd 4