Properties

Label 1600.2.j.c.1007.6
Level $1600$
Weight $2$
Character 1600.1007
Analytic conductor $12.776$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(143,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1007.6
Root \(-0.859408 - 1.12313i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1007
Dual form 1600.2.j.c.143.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.54564i q^{3} +(-1.17442 - 1.17442i) q^{7} +0.611000 q^{9} +O(q^{10})\) \(q+1.54564i q^{3} +(-1.17442 - 1.17442i) q^{7} +0.611000 q^{9} +(-2.04567 - 2.04567i) q^{11} +2.14367 q^{13} +(-2.07308 - 2.07308i) q^{17} +(-4.47190 - 4.47190i) q^{19} +(1.81523 - 1.81523i) q^{21} +(-4.86373 + 4.86373i) q^{23} +5.58130i q^{27} +(-5.51757 + 5.51757i) q^{29} -5.72181i q^{31} +(3.16187 - 3.16187i) q^{33} -11.0214 q^{37} +3.31334i q^{39} -11.4241i q^{41} -0.251676 q^{43} +(0.119541 - 0.119541i) q^{47} -4.24147i q^{49} +(3.20423 - 3.20423i) q^{51} +2.69520i q^{53} +(6.91195 - 6.91195i) q^{57} +(-1.24990 + 1.24990i) q^{59} +(-2.48034 - 2.48034i) q^{61} +(-0.717571 - 0.717571i) q^{63} +9.23670 q^{67} +(-7.51757 - 7.51757i) q^{69} +8.85246 q^{71} +(-7.85956 - 7.85956i) q^{73} +4.80496i q^{77} -4.86934 q^{79} -6.79368 q^{81} -4.94936i q^{83} +(-8.52818 - 8.52818i) q^{87} -3.63047 q^{89} +(-2.51757 - 2.51757i) q^{91} +8.84385 q^{93} +(9.89595 + 9.89595i) q^{97} +(-1.24990 - 1.24990i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 8 q^{11} + 8 q^{19} + 16 q^{29} + 48 q^{51} + 8 q^{59} - 16 q^{61} - 16 q^{69} + 32 q^{71} - 80 q^{79} + 16 q^{81} + 64 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 1.54564i 0.892375i 0.894939 + 0.446188i \(0.147219\pi\)
−0.894939 + 0.446188i \(0.852781\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.17442 1.17442i −0.443890 0.443890i 0.449427 0.893317i \(-0.351628\pi\)
−0.893317 + 0.449427i \(0.851628\pi\)
\(8\) 0 0
\(9\) 0.611000 0.203667
\(10\) 0 0
\(11\) −2.04567 2.04567i −0.616793 0.616793i 0.327915 0.944707i \(-0.393654\pi\)
−0.944707 + 0.327915i \(0.893654\pi\)
\(12\) 0 0
\(13\) 2.14367 0.594547 0.297273 0.954792i \(-0.403923\pi\)
0.297273 + 0.954792i \(0.403923\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.07308 2.07308i −0.502796 0.502796i 0.409510 0.912306i \(-0.365700\pi\)
−0.912306 + 0.409510i \(0.865700\pi\)
\(18\) 0 0
\(19\) −4.47190 4.47190i −1.02592 1.02592i −0.999655 0.0262700i \(-0.991637\pi\)
−0.0262700 0.999655i \(-0.508363\pi\)
\(20\) 0 0
\(21\) 1.81523 1.81523i 0.396116 0.396116i
\(22\) 0 0
\(23\) −4.86373 + 4.86373i −1.01416 + 1.01416i −0.0142597 + 0.999898i \(0.504539\pi\)
−0.999898 + 0.0142597i \(0.995461\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.58130i 1.07412i
\(28\) 0 0
\(29\) −5.51757 + 5.51757i −1.02459 + 1.02459i −0.0248976 + 0.999690i \(0.507926\pi\)
−0.999690 + 0.0248976i \(0.992074\pi\)
\(30\) 0 0
\(31\) 5.72181i 1.02767i −0.857890 0.513833i \(-0.828225\pi\)
0.857890 0.513833i \(-0.171775\pi\)
\(32\) 0 0
\(33\) 3.16187 3.16187i 0.550411 0.550411i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.0214 −1.81191 −0.905956 0.423373i \(-0.860846\pi\)
−0.905956 + 0.423373i \(0.860846\pi\)
\(38\) 0 0
\(39\) 3.31334i 0.530559i
\(40\) 0 0
\(41\) 11.4241i 1.78415i −0.451885 0.892076i \(-0.649248\pi\)
0.451885 0.892076i \(-0.350752\pi\)
\(42\) 0 0
\(43\) −0.251676 −0.0383802 −0.0191901 0.999816i \(-0.506109\pi\)
−0.0191901 + 0.999816i \(0.506109\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.119541 0.119541i 0.0174368 0.0174368i −0.698335 0.715771i \(-0.746075\pi\)
0.715771 + 0.698335i \(0.246075\pi\)
\(48\) 0 0
\(49\) 4.24147i 0.605924i
\(50\) 0 0
\(51\) 3.20423 3.20423i 0.448682 0.448682i
\(52\) 0 0
\(53\) 2.69520i 0.370214i 0.982718 + 0.185107i \(0.0592632\pi\)
−0.982718 + 0.185107i \(0.940737\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.91195 6.91195i 0.915510 0.915510i
\(58\) 0 0
\(59\) −1.24990 + 1.24990i −0.162724 + 0.162724i −0.783772 0.621049i \(-0.786707\pi\)
0.621049 + 0.783772i \(0.286707\pi\)
\(60\) 0 0
\(61\) −2.48034 2.48034i −0.317575 0.317575i 0.530260 0.847835i \(-0.322094\pi\)
−0.847835 + 0.530260i \(0.822094\pi\)
\(62\) 0 0
\(63\) −0.717571 0.717571i −0.0904055 0.0904055i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.23670 1.12844 0.564221 0.825623i \(-0.309176\pi\)
0.564221 + 0.825623i \(0.309176\pi\)
\(68\) 0 0
\(69\) −7.51757 7.51757i −0.905009 0.905009i
\(70\) 0 0
\(71\) 8.85246 1.05059 0.525297 0.850919i \(-0.323954\pi\)
0.525297 + 0.850919i \(0.323954\pi\)
\(72\) 0 0
\(73\) −7.85956 7.85956i −0.919892 0.919892i 0.0771295 0.997021i \(-0.475424\pi\)
−0.997021 + 0.0771295i \(0.975424\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.80496i 0.547576i
\(78\) 0 0
\(79\) −4.86934 −0.547844 −0.273922 0.961752i \(-0.588321\pi\)
−0.273922 + 0.961752i \(0.588321\pi\)
\(80\) 0 0
\(81\) −6.79368 −0.754853
\(82\) 0 0
\(83\) 4.94936i 0.543263i −0.962401 0.271632i \(-0.912437\pi\)
0.962401 0.271632i \(-0.0875632\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.52818 8.52818i −0.914317 0.914317i
\(88\) 0 0
\(89\) −3.63047 −0.384829 −0.192414 0.981314i \(-0.561632\pi\)
−0.192414 + 0.981314i \(0.561632\pi\)
\(90\) 0 0
\(91\) −2.51757 2.51757i −0.263913 0.263913i
\(92\) 0 0
\(93\) 8.84385 0.917064
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.89595 + 9.89595i 1.00478 + 1.00478i 0.999989 + 0.00479337i \(0.00152578\pi\)
0.00479337 + 0.999989i \(0.498474\pi\)
\(98\) 0 0
\(99\) −1.24990 1.24990i −0.125620 0.125620i
\(100\) 0 0
\(101\) 6.29557 6.29557i 0.626433 0.626433i −0.320736 0.947169i \(-0.603930\pi\)
0.947169 + 0.320736i \(0.103930\pi\)
\(102\) 0 0
\(103\) −3.06642 + 3.06642i −0.302143 + 0.302143i −0.841852 0.539709i \(-0.818534\pi\)
0.539709 + 0.841852i \(0.318534\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.1186i 1.26822i 0.773242 + 0.634111i \(0.218634\pi\)
−0.773242 + 0.634111i \(0.781366\pi\)
\(108\) 0 0
\(109\) 2.00000 2.00000i 0.191565 0.191565i −0.604807 0.796372i \(-0.706750\pi\)
0.796372 + 0.604807i \(0.206750\pi\)
\(110\) 0 0
\(111\) 17.0351i 1.61690i
\(112\) 0 0
\(113\) 0.551529 0.551529i 0.0518835 0.0518835i −0.680689 0.732572i \(-0.738319\pi\)
0.732572 + 0.680689i \(0.238319\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.30978 0.121089
\(118\) 0 0
\(119\) 4.86934i 0.446372i
\(120\) 0 0
\(121\) 2.63047i 0.239133i
\(122\) 0 0
\(123\) 17.6576 1.59213
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.39015 9.39015i 0.833241 0.833241i −0.154717 0.987959i \(-0.549447\pi\)
0.987959 + 0.154717i \(0.0494467\pi\)
\(128\) 0 0
\(129\) 0.389000i 0.0342496i
\(130\) 0 0
\(131\) 0.471903 0.471903i 0.0412303 0.0412303i −0.686191 0.727421i \(-0.740719\pi\)
0.727421 + 0.686191i \(0.240719\pi\)
\(132\) 0 0
\(133\) 10.5038i 0.910795i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.2564 16.2564i 1.38888 1.38888i 0.561186 0.827690i \(-0.310345\pi\)
0.827690 0.561186i \(-0.189655\pi\)
\(138\) 0 0
\(139\) −5.26767 + 5.26767i −0.446798 + 0.446798i −0.894289 0.447491i \(-0.852318\pi\)
0.447491 + 0.894289i \(0.352318\pi\)
\(140\) 0 0
\(141\) 0.184767 + 0.184767i 0.0155602 + 0.0155602i
\(142\) 0 0
\(143\) −4.38524 4.38524i −0.366712 0.366712i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.55577 0.540711
\(148\) 0 0
\(149\) 3.70234 + 3.70234i 0.303308 + 0.303308i 0.842306 0.538999i \(-0.181197\pi\)
−0.538999 + 0.842306i \(0.681197\pi\)
\(150\) 0 0
\(151\) −11.3133 −0.920667 −0.460333 0.887746i \(-0.652270\pi\)
−0.460333 + 0.887746i \(0.652270\pi\)
\(152\) 0 0
\(153\) −1.26665 1.26665i −0.102403 0.102403i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.7622i 1.49739i 0.662916 + 0.748694i \(0.269319\pi\)
−0.662916 + 0.748694i \(0.730681\pi\)
\(158\) 0 0
\(159\) −4.16580 −0.330370
\(160\) 0 0
\(161\) 11.4241 0.900349
\(162\) 0 0
\(163\) 17.9093i 1.40276i −0.712786 0.701382i \(-0.752567\pi\)
0.712786 0.701382i \(-0.247433\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.7939 12.7939i −0.990020 0.990020i 0.00993072 0.999951i \(-0.496839\pi\)
−0.999951 + 0.00993072i \(0.996839\pi\)
\(168\) 0 0
\(169\) −8.40468 −0.646514
\(170\) 0 0
\(171\) −2.73233 2.73233i −0.208947 0.208947i
\(172\) 0 0
\(173\) −17.2040 −1.30799 −0.653997 0.756497i \(-0.726909\pi\)
−0.653997 + 0.756497i \(0.726909\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.93190 1.93190i −0.145210 0.145210i
\(178\) 0 0
\(179\) 3.76748 + 3.76748i 0.281594 + 0.281594i 0.833745 0.552150i \(-0.186193\pi\)
−0.552150 + 0.833745i \(0.686193\pi\)
\(180\) 0 0
\(181\) −4.29557 + 4.29557i −0.319288 + 0.319288i −0.848493 0.529206i \(-0.822490\pi\)
0.529206 + 0.848493i \(0.322490\pi\)
\(182\) 0 0
\(183\) 3.83371 3.83371i 0.283396 0.283396i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.48168i 0.620242i
\(188\) 0 0
\(189\) 6.55481 6.55481i 0.476792 0.476792i
\(190\) 0 0
\(191\) 25.8876i 1.87316i 0.350451 + 0.936581i \(0.386028\pi\)
−0.350451 + 0.936581i \(0.613972\pi\)
\(192\) 0 0
\(193\) −10.1105 + 10.1105i −0.727770 + 0.727770i −0.970175 0.242405i \(-0.922064\pi\)
0.242405 + 0.970175i \(0.422064\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.6118 −1.11230 −0.556149 0.831083i \(-0.687722\pi\)
−0.556149 + 0.831083i \(0.687722\pi\)
\(198\) 0 0
\(199\) 25.2178i 1.78764i −0.448422 0.893822i \(-0.648014\pi\)
0.448422 0.893822i \(-0.351986\pi\)
\(200\) 0 0
\(201\) 14.2766i 1.00699i
\(202\) 0 0
\(203\) 12.9599 0.909608
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.97174 + 2.97174i −0.206550 + 0.206550i
\(208\) 0 0
\(209\) 18.2961i 1.26557i
\(210\) 0 0
\(211\) 6.84144 6.84144i 0.470984 0.470984i −0.431249 0.902233i \(-0.641927\pi\)
0.902233 + 0.431249i \(0.141927\pi\)
\(212\) 0 0
\(213\) 13.6827i 0.937524i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.71982 + 6.71982i −0.456171 + 0.456171i
\(218\) 0 0
\(219\) 12.1480 12.1480i 0.820888 0.820888i
\(220\) 0 0
\(221\) −4.44400 4.44400i −0.298936 0.298936i
\(222\) 0 0
\(223\) −1.31883 1.31883i −0.0883151 0.0883151i 0.661569 0.749884i \(-0.269891\pi\)
−0.749884 + 0.661569i \(0.769891\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.63692 −0.307763 −0.153882 0.988089i \(-0.549177\pi\)
−0.153882 + 0.988089i \(0.549177\pi\)
\(228\) 0 0
\(229\) 12.7396 + 12.7396i 0.841855 + 0.841855i 0.989100 0.147245i \(-0.0470407\pi\)
−0.147245 + 0.989100i \(0.547041\pi\)
\(230\) 0 0
\(231\) −7.42674 −0.488643
\(232\) 0 0
\(233\) −12.0057 12.0057i −0.786521 0.786521i 0.194401 0.980922i \(-0.437724\pi\)
−0.980922 + 0.194401i \(0.937724\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.52625i 0.488882i
\(238\) 0 0
\(239\) 2.64356 0.170997 0.0854987 0.996338i \(-0.472752\pi\)
0.0854987 + 0.996338i \(0.472752\pi\)
\(240\) 0 0
\(241\) 7.01568 0.451920 0.225960 0.974137i \(-0.427448\pi\)
0.225960 + 0.974137i \(0.427448\pi\)
\(242\) 0 0
\(243\) 6.24333i 0.400510i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.58628 9.58628i −0.609961 0.609961i
\(248\) 0 0
\(249\) 7.64993 0.484795
\(250\) 0 0
\(251\) 15.7675 + 15.7675i 0.995234 + 0.995234i 0.999989 0.00475439i \(-0.00151338\pi\)
−0.00475439 + 0.999989i \(0.501513\pi\)
\(252\) 0 0
\(253\) 19.8992 1.25105
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.27135 + 7.27135i 0.453574 + 0.453574i 0.896539 0.442965i \(-0.146073\pi\)
−0.442965 + 0.896539i \(0.646073\pi\)
\(258\) 0 0
\(259\) 12.9438 + 12.9438i 0.804289 + 0.804289i
\(260\) 0 0
\(261\) −3.37123 + 3.37123i −0.208674 + 0.208674i
\(262\) 0 0
\(263\) −5.32058 + 5.32058i −0.328081 + 0.328081i −0.851856 0.523775i \(-0.824523\pi\)
0.523775 + 0.851856i \(0.324523\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.61139i 0.343411i
\(268\) 0 0
\(269\) −2.29766 + 2.29766i −0.140091 + 0.140091i −0.773674 0.633584i \(-0.781583\pi\)
0.633584 + 0.773674i \(0.281583\pi\)
\(270\) 0 0
\(271\) 14.9396i 0.907518i 0.891124 + 0.453759i \(0.149917\pi\)
−0.891124 + 0.453759i \(0.850083\pi\)
\(272\) 0 0
\(273\) 3.89126 3.89126i 0.235510 0.235510i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.2360 −0.675104 −0.337552 0.941307i \(-0.609599\pi\)
−0.337552 + 0.941307i \(0.609599\pi\)
\(278\) 0 0
\(279\) 3.49602i 0.209301i
\(280\) 0 0
\(281\) 15.0546i 0.898083i 0.893511 + 0.449041i \(0.148234\pi\)
−0.893511 + 0.449041i \(0.851766\pi\)
\(282\) 0 0
\(283\) −18.8230 −1.11891 −0.559455 0.828861i \(-0.688990\pi\)
−0.559455 + 0.828861i \(0.688990\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.4168 + 13.4168i −0.791967 + 0.791967i
\(288\) 0 0
\(289\) 8.40468i 0.494393i
\(290\) 0 0
\(291\) −15.2956 + 15.2956i −0.896642 + 0.896642i
\(292\) 0 0
\(293\) 6.18256i 0.361189i 0.983558 + 0.180594i \(0.0578021\pi\)
−0.983558 + 0.180594i \(0.942198\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 11.4175 11.4175i 0.662511 0.662511i
\(298\) 0 0
\(299\) −10.4262 + 10.4262i −0.602965 + 0.602965i
\(300\) 0 0
\(301\) 0.295574 + 0.295574i 0.0170366 + 0.0170366i
\(302\) 0 0
\(303\) 9.73069 + 9.73069i 0.559013 + 0.559013i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.85636 −0.277167 −0.138584 0.990351i \(-0.544255\pi\)
−0.138584 + 0.990351i \(0.544255\pi\)
\(308\) 0 0
\(309\) −4.73957 4.73957i −0.269625 0.269625i
\(310\) 0 0
\(311\) −2.09551 −0.118826 −0.0594128 0.998233i \(-0.518923\pi\)
−0.0594128 + 0.998233i \(0.518923\pi\)
\(312\) 0 0
\(313\) 13.6684 + 13.6684i 0.772586 + 0.772586i 0.978558 0.205972i \(-0.0660355\pi\)
−0.205972 + 0.978558i \(0.566035\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.21646i 0.349151i 0.984644 + 0.174576i \(0.0558554\pi\)
−0.984644 + 0.174576i \(0.944145\pi\)
\(318\) 0 0
\(319\) 22.5743 1.26392
\(320\) 0 0
\(321\) −20.2766 −1.13173
\(322\) 0 0
\(323\) 18.5412i 1.03166i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.09128 + 3.09128i 0.170948 + 0.170948i
\(328\) 0 0
\(329\) −0.280783 −0.0154801
\(330\) 0 0
\(331\) −8.28922 8.28922i −0.455617 0.455617i 0.441597 0.897214i \(-0.354412\pi\)
−0.897214 + 0.441597i \(0.854412\pi\)
\(332\) 0 0
\(333\) −6.73409 −0.369026
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11.4175 11.4175i −0.621951 0.621951i 0.324079 0.946030i \(-0.394946\pi\)
−0.946030 + 0.324079i \(0.894946\pi\)
\(338\) 0 0
\(339\) 0.852465 + 0.852465i 0.0462995 + 0.0462995i
\(340\) 0 0
\(341\) −11.7049 + 11.7049i −0.633857 + 0.633857i
\(342\) 0 0
\(343\) −13.2022 + 13.2022i −0.712853 + 0.712853i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.73654i 0.0932226i 0.998913 + 0.0466113i \(0.0148422\pi\)
−0.998913 + 0.0466113i \(0.985158\pi\)
\(348\) 0 0
\(349\) 20.1087 20.1087i 1.07640 1.07640i 0.0795655 0.996830i \(-0.474647\pi\)
0.996830 0.0795655i \(-0.0253533\pi\)
\(350\) 0 0
\(351\) 11.9645i 0.638616i
\(352\) 0 0
\(353\) 14.7572 14.7572i 0.785448 0.785448i −0.195296 0.980744i \(-0.562567\pi\)
0.980744 + 0.195296i \(0.0625667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.52625 −0.398331
\(358\) 0 0
\(359\) 25.3481i 1.33782i 0.743343 + 0.668911i \(0.233239\pi\)
−0.743343 + 0.668911i \(0.766761\pi\)
\(360\) 0 0
\(361\) 20.9958i 1.10504i
\(362\) 0 0
\(363\) 4.06575 0.213397
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.4798 19.4798i 1.01684 1.01684i 0.0169808 0.999856i \(-0.494595\pi\)
0.999856 0.0169808i \(-0.00540540\pi\)
\(368\) 0 0
\(369\) 6.98015i 0.363372i
\(370\) 0 0
\(371\) 3.16530 3.16530i 0.164334 0.164334i
\(372\) 0 0
\(373\) 25.4963i 1.32015i −0.751200 0.660074i \(-0.770525\pi\)
0.751200 0.660074i \(-0.229475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11.8279 + 11.8279i −0.609165 + 0.609165i
\(378\) 0 0
\(379\) 6.82367 6.82367i 0.350508 0.350508i −0.509790 0.860299i \(-0.670277\pi\)
0.860299 + 0.509790i \(0.170277\pi\)
\(380\) 0 0
\(381\) 14.5138 + 14.5138i 0.743564 + 0.743564i
\(382\) 0 0
\(383\) −2.08490 2.08490i −0.106533 0.106533i 0.651831 0.758364i \(-0.274001\pi\)
−0.758364 + 0.651831i \(0.774001\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.153774 −0.00781677
\(388\) 0 0
\(389\) 7.92395 + 7.92395i 0.401760 + 0.401760i 0.878853 0.477093i \(-0.158309\pi\)
−0.477093 + 0.878853i \(0.658309\pi\)
\(390\) 0 0
\(391\) 20.1658 1.01983
\(392\) 0 0
\(393\) 0.729391 + 0.729391i 0.0367929 + 0.0367929i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.1865i 1.21389i −0.794745 0.606943i \(-0.792395\pi\)
0.794745 0.606943i \(-0.207605\pi\)
\(398\) 0 0
\(399\) −16.2351 −0.812771
\(400\) 0 0
\(401\) 13.9958 0.698918 0.349459 0.936952i \(-0.386365\pi\)
0.349459 + 0.936952i \(0.386365\pi\)
\(402\) 0 0
\(403\) 12.2657i 0.610996i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.5462 + 22.5462i 1.11757 + 1.11757i
\(408\) 0 0
\(409\) 11.4241 0.564888 0.282444 0.959284i \(-0.408855\pi\)
0.282444 + 0.959284i \(0.408855\pi\)
\(410\) 0 0
\(411\) 25.1265 + 25.1265i 1.23940 + 1.23940i
\(412\) 0 0
\(413\) 2.93583 0.144463
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.14192 8.14192i −0.398711 0.398711i
\(418\) 0 0
\(419\) 16.5455 + 16.5455i 0.808299 + 0.808299i 0.984376 0.176077i \(-0.0563407\pi\)
−0.176077 + 0.984376i \(0.556341\pi\)
\(420\) 0 0
\(421\) −15.5900 + 15.5900i −0.759808 + 0.759808i −0.976287 0.216479i \(-0.930543\pi\)
0.216479 + 0.976287i \(0.430543\pi\)
\(422\) 0 0
\(423\) 0.0730395 0.0730395i 0.00355130 0.00355130i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.82594i 0.281937i
\(428\) 0 0
\(429\) 6.77800 6.77800i 0.327245 0.327245i
\(430\) 0 0
\(431\) 18.4609i 0.889229i −0.895722 0.444615i \(-0.853341\pi\)
0.895722 0.444615i \(-0.146659\pi\)
\(432\) 0 0
\(433\) −13.7872 + 13.7872i −0.662571 + 0.662571i −0.955985 0.293414i \(-0.905208\pi\)
0.293414 + 0.955985i \(0.405208\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 43.5003 2.08090
\(438\) 0 0
\(439\) 16.9049i 0.806826i 0.915018 + 0.403413i \(0.132176\pi\)
−0.915018 + 0.403413i \(0.867824\pi\)
\(440\) 0 0
\(441\) 2.59153i 0.123406i
\(442\) 0 0
\(443\) −12.2979 −0.584291 −0.292145 0.956374i \(-0.594369\pi\)
−0.292145 + 0.956374i \(0.594369\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.72248 + 5.72248i −0.270664 + 0.270664i
\(448\) 0 0
\(449\) 9.34968i 0.441239i 0.975360 + 0.220619i \(0.0708079\pi\)
−0.975360 + 0.220619i \(0.929192\pi\)
\(450\) 0 0
\(451\) −23.3700 + 23.3700i −1.10045 + 1.10045i
\(452\) 0 0
\(453\) 17.4863i 0.821580i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −14.9127 + 14.9127i −0.697586 + 0.697586i −0.963889 0.266303i \(-0.914198\pi\)
0.266303 + 0.963889i \(0.414198\pi\)
\(458\) 0 0
\(459\) 11.5705 11.5705i 0.540064 0.540064i
\(460\) 0 0
\(461\) 18.0005 + 18.0005i 0.838367 + 0.838367i 0.988644 0.150277i \(-0.0480165\pi\)
−0.150277 + 0.988644i \(0.548017\pi\)
\(462\) 0 0
\(463\) −6.25392 6.25392i −0.290644 0.290644i 0.546690 0.837335i \(-0.315887\pi\)
−0.837335 + 0.546690i \(0.815887\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.9988 1.11053 0.555267 0.831672i \(-0.312616\pi\)
0.555267 + 0.831672i \(0.312616\pi\)
\(468\) 0 0
\(469\) −10.8478 10.8478i −0.500904 0.500904i
\(470\) 0 0
\(471\) −28.9996 −1.33623
\(472\) 0 0
\(473\) 0.514846 + 0.514846i 0.0236727 + 0.0236727i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.64677i 0.0754002i
\(478\) 0 0
\(479\) 23.6618 1.08114 0.540568 0.841300i \(-0.318209\pi\)
0.540568 + 0.841300i \(0.318209\pi\)
\(480\) 0 0
\(481\) −23.6263 −1.07727
\(482\) 0 0
\(483\) 17.6576i 0.803449i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.1809 + 21.1809i 0.959797 + 0.959797i 0.999223 0.0394255i \(-0.0125528\pi\)
−0.0394255 + 0.999223i \(0.512553\pi\)
\(488\) 0 0
\(489\) 27.6813 1.25179
\(490\) 0 0
\(491\) −13.2854 13.2854i −0.599563 0.599563i 0.340633 0.940196i \(-0.389359\pi\)
−0.940196 + 0.340633i \(0.889359\pi\)
\(492\) 0 0
\(493\) 22.8767 1.03032
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.3965 10.3965i −0.466348 0.466348i
\(498\) 0 0
\(499\) −24.8026 24.8026i −1.11032 1.11032i −0.993107 0.117211i \(-0.962605\pi\)
−0.117211 0.993107i \(-0.537395\pi\)
\(500\) 0 0
\(501\) 19.7747 19.7747i 0.883469 0.883469i
\(502\) 0 0
\(503\) 23.2205 23.2205i 1.03535 1.03535i 0.0359989 0.999352i \(-0.488539\pi\)
0.999352 0.0359989i \(-0.0114613\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.9906i 0.576933i
\(508\) 0 0
\(509\) −30.5917 + 30.5917i −1.35595 + 1.35595i −0.477105 + 0.878847i \(0.658314\pi\)
−0.878847 + 0.477105i \(0.841686\pi\)
\(510\) 0 0
\(511\) 18.4609i 0.816661i
\(512\) 0 0
\(513\) 24.9590 24.9590i 1.10197 1.10197i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.489083 −0.0215098
\(518\) 0 0
\(519\) 26.5911i 1.16722i
\(520\) 0 0
\(521\) 13.1096i 0.574342i −0.957879 0.287171i \(-0.907285\pi\)
0.957879 0.287171i \(-0.0927149\pi\)
\(522\) 0 0
\(523\) −31.0886 −1.35941 −0.679706 0.733485i \(-0.737893\pi\)
−0.679706 + 0.733485i \(0.737893\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.8618 + 11.8618i −0.516706 + 0.516706i
\(528\) 0 0
\(529\) 24.3118i 1.05703i
\(530\) 0 0
\(531\) −0.763691 + 0.763691i −0.0331413 + 0.0331413i
\(532\) 0 0
\(533\) 24.4896i 1.06076i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.82316 + 5.82316i −0.251288 + 0.251288i
\(538\) 0 0
\(539\) −8.67664 + 8.67664i −0.373729 + 0.373729i
\(540\) 0 0
\(541\) −0.926425 0.926425i −0.0398301 0.0398301i 0.686911 0.726741i \(-0.258966\pi\)
−0.726741 + 0.686911i \(0.758966\pi\)
\(542\) 0 0
\(543\) −6.63941 6.63941i −0.284924 0.284924i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.1171 −0.731875 −0.365938 0.930639i \(-0.619252\pi\)
−0.365938 + 0.930639i \(0.619252\pi\)
\(548\) 0 0
\(549\) −1.51549 1.51549i −0.0646794 0.0646794i
\(550\) 0 0
\(551\) 49.3481 2.10230
\(552\) 0 0
\(553\) 5.71866 + 5.71866i 0.243182 + 0.243182i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.5797i 0.533017i −0.963833 0.266509i \(-0.914130\pi\)
0.963833 0.266509i \(-0.0858701\pi\)
\(558\) 0 0
\(559\) −0.539510 −0.0228189
\(560\) 0 0
\(561\) −13.1096 −0.553488
\(562\) 0 0
\(563\) 19.4227i 0.818569i 0.912407 + 0.409284i \(0.134222\pi\)
−0.912407 + 0.409284i \(0.865778\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.97865 + 7.97865i 0.335072 + 0.335072i
\(568\) 0 0
\(569\) −5.57547 −0.233736 −0.116868 0.993147i \(-0.537285\pi\)
−0.116868 + 0.993147i \(0.537285\pi\)
\(570\) 0 0
\(571\) −5.58062 5.58062i −0.233542 0.233542i 0.580627 0.814169i \(-0.302807\pi\)
−0.814169 + 0.580627i \(0.802807\pi\)
\(572\) 0 0
\(573\) −40.0129 −1.67156
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 27.7472 + 27.7472i 1.15513 + 1.15513i 0.985509 + 0.169624i \(0.0542553\pi\)
0.169624 + 0.985509i \(0.445745\pi\)
\(578\) 0 0
\(579\) −15.6272 15.6272i −0.649444 0.649444i
\(580\) 0 0
\(581\) −5.81264 + 5.81264i −0.241149 + 0.241149i
\(582\) 0 0
\(583\) 5.51349 5.51349i 0.228345 0.228345i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.8977i 1.27529i −0.770332 0.637643i \(-0.779909\pi\)
0.770332 0.637643i \(-0.220091\pi\)
\(588\) 0 0
\(589\) −25.5874 + 25.5874i −1.05431 + 1.05431i
\(590\) 0 0
\(591\) 24.1303i 0.992587i
\(592\) 0 0
\(593\) −7.82287 + 7.82287i −0.321247 + 0.321247i −0.849245 0.527998i \(-0.822943\pi\)
0.527998 + 0.849245i \(0.322943\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 38.9777 1.59525
\(598\) 0 0
\(599\) 24.3316i 0.994163i 0.867704 + 0.497081i \(0.165595\pi\)
−0.867704 + 0.497081i \(0.834405\pi\)
\(600\) 0 0
\(601\) 30.6420i 1.24991i −0.780660 0.624956i \(-0.785117\pi\)
0.780660 0.624956i \(-0.214883\pi\)
\(602\) 0 0
\(603\) 5.64362 0.229826
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0615 10.0615i 0.408386 0.408386i −0.472790 0.881175i \(-0.656753\pi\)
0.881175 + 0.472790i \(0.156753\pi\)
\(608\) 0 0
\(609\) 20.0314i 0.811712i
\(610\) 0 0
\(611\) 0.256256 0.256256i 0.0103670 0.0103670i
\(612\) 0 0
\(613\) 9.66991i 0.390564i 0.980747 + 0.195282i \(0.0625622\pi\)
−0.980747 + 0.195282i \(0.937438\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.46520 + 4.46520i −0.179762 + 0.179762i −0.791252 0.611490i \(-0.790570\pi\)
0.611490 + 0.791252i \(0.290570\pi\)
\(618\) 0 0
\(619\) 5.32437 5.32437i 0.214004 0.214004i −0.591962 0.805966i \(-0.701646\pi\)
0.805966 + 0.591962i \(0.201646\pi\)
\(620\) 0 0
\(621\) −27.1460 27.1460i −1.08933 1.08933i
\(622\) 0 0
\(623\) 4.26370 + 4.26370i 0.170822 + 0.170822i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −28.2791 −1.12936
\(628\) 0 0
\(629\) 22.8483 + 22.8483i 0.911021 + 0.911021i
\(630\) 0 0
\(631\) −24.1303 −0.960611 −0.480306 0.877101i \(-0.659474\pi\)
−0.480306 + 0.877101i \(0.659474\pi\)
\(632\) 0 0
\(633\) 10.5744 + 10.5744i 0.420294 + 0.420294i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.09230i 0.360250i
\(638\) 0 0
\(639\) 5.40885 0.213971
\(640\) 0 0
\(641\) 40.1642 1.58639 0.793196 0.608967i \(-0.208416\pi\)
0.793196 + 0.608967i \(0.208416\pi\)
\(642\) 0 0
\(643\) 38.2226i 1.50735i −0.657245 0.753677i \(-0.728278\pi\)
0.657245 0.753677i \(-0.271722\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.07480 + 4.07480i 0.160197 + 0.160197i 0.782654 0.622457i \(-0.213866\pi\)
−0.622457 + 0.782654i \(0.713866\pi\)
\(648\) 0 0
\(649\) 5.11378 0.200733
\(650\) 0 0
\(651\) −10.3864 10.3864i −0.407076 0.407076i
\(652\) 0 0
\(653\) −18.5270 −0.725016 −0.362508 0.931981i \(-0.618079\pi\)
−0.362508 + 0.931981i \(0.618079\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.80219 4.80219i −0.187351 0.187351i
\(658\) 0 0
\(659\) −11.4021 11.4021i −0.444163 0.444163i 0.449245 0.893409i \(-0.351693\pi\)
−0.893409 + 0.449245i \(0.851693\pi\)
\(660\) 0 0
\(661\) 6.03893 6.03893i 0.234887 0.234887i −0.579842 0.814729i \(-0.696886\pi\)
0.814729 + 0.579842i \(0.196886\pi\)
\(662\) 0 0
\(663\) 6.86882 6.86882i 0.266763 0.266763i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 53.6720i 2.07819i
\(668\) 0 0
\(669\) 2.03843 2.03843i 0.0788102 0.0788102i
\(670\) 0 0
\(671\) 10.1479i 0.391756i
\(672\) 0 0
\(673\) −20.5070 + 20.5070i −0.790488 + 0.790488i −0.981573 0.191085i \(-0.938799\pi\)
0.191085 + 0.981573i \(0.438799\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.8942 0.610864 0.305432 0.952214i \(-0.401199\pi\)
0.305432 + 0.952214i \(0.401199\pi\)
\(678\) 0 0
\(679\) 23.2441i 0.892025i
\(680\) 0 0
\(681\) 7.16700i 0.274640i
\(682\) 0 0
\(683\) 7.34149 0.280914 0.140457 0.990087i \(-0.455143\pi\)
0.140457 + 0.990087i \(0.455143\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −19.6908 + 19.6908i −0.751250 + 0.751250i
\(688\) 0 0
\(689\) 5.77762i 0.220110i
\(690\) 0 0
\(691\) 28.5464 28.5464i 1.08595 1.08595i 0.0900145 0.995940i \(-0.471309\pi\)
0.995940 0.0900145i \(-0.0286913\pi\)
\(692\) 0 0
\(693\) 2.93583i 0.111523i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −23.6832 + 23.6832i −0.897064 + 0.897064i
\(698\) 0 0
\(699\) 18.5565 18.5565i 0.701871 0.701871i
\(700\) 0 0
\(701\) 17.7049 + 17.7049i 0.668706 + 0.668706i 0.957416 0.288710i \(-0.0932265\pi\)
−0.288710 + 0.957416i \(0.593227\pi\)
\(702\) 0 0
\(703\) 49.2867 + 49.2867i 1.85888 + 1.85888i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.7873 −0.556135
\(708\) 0 0
\(709\) −31.2225 31.2225i −1.17259 1.17259i −0.981591 0.190995i \(-0.938829\pi\)
−0.190995 0.981591i \(-0.561171\pi\)
\(710\) 0 0
\(711\) −2.97517 −0.111577
\(712\) 0 0
\(713\) 27.8293 + 27.8293i 1.04222 + 1.04222i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.08598i 0.152594i
\(718\) 0 0
\(719\) −30.7570 −1.14704 −0.573520 0.819191i \(-0.694423\pi\)
−0.573520 + 0.819191i \(0.694423\pi\)
\(720\) 0 0
\(721\) 7.20253 0.268236
\(722\) 0 0
\(723\) 10.8437i 0.403282i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20.0526 + 20.0526i 0.743711 + 0.743711i 0.973290 0.229579i \(-0.0737350\pi\)
−0.229579 + 0.973290i \(0.573735\pi\)
\(728\) 0 0
\(729\) −30.0310 −1.11226
\(730\) 0 0
\(731\) 0.521745 + 0.521745i 0.0192974 + 0.0192974i
\(732\) 0 0
\(733\) 4.69214 0.173308 0.0866540 0.996238i \(-0.472383\pi\)
0.0866540 + 0.996238i \(0.472383\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.8953 18.8953i −0.696016 0.696016i
\(738\) 0 0
\(739\) −5.45035 5.45035i −0.200494 0.200494i 0.599717 0.800212i \(-0.295280\pi\)
−0.800212 + 0.599717i \(0.795280\pi\)
\(740\) 0 0
\(741\) 14.8169 14.8169i 0.544314 0.544314i
\(742\) 0 0
\(743\) 28.8916 28.8916i 1.05993 1.05993i 0.0618433 0.998086i \(-0.480302\pi\)
0.998086 0.0618433i \(-0.0196979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.02406i 0.110645i
\(748\) 0 0
\(749\) 15.4068 15.4068i 0.562951 0.562951i
\(750\) 0 0
\(751\) 34.7738i 1.26892i −0.772958 0.634458i \(-0.781224\pi\)
0.772958 0.634458i \(-0.218776\pi\)
\(752\) 0 0
\(753\) −24.3708 + 24.3708i −0.888122 + 0.888122i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −13.0634 −0.474796 −0.237398 0.971412i \(-0.576295\pi\)
−0.237398 + 0.971412i \(0.576295\pi\)
\(758\) 0 0
\(759\) 30.7570i 1.11641i
\(760\) 0 0
\(761\) 7.48332i 0.271270i 0.990759 + 0.135635i \(0.0433074\pi\)
−0.990759 + 0.135635i \(0.956693\pi\)
\(762\) 0 0
\(763\) −4.69769 −0.170068
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.67938 + 2.67938i −0.0967468 + 0.0967468i
\(768\) 0 0
\(769\) 21.5518i 0.777179i −0.921411 0.388589i \(-0.872962\pi\)
0.921411 0.388589i \(-0.127038\pi\)
\(770\) 0 0
\(771\) −11.2389 + 11.2389i −0.404758 + 0.404758i
\(772\) 0 0
\(773\) 28.6752i 1.03138i 0.856777 + 0.515688i \(0.172464\pi\)
−0.856777 + 0.515688i \(0.827536\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −20.0065 + 20.0065i −0.717727 + 0.717727i
\(778\) 0 0
\(779\) −51.0877 + 51.0877i −1.83041 + 1.83041i
\(780\) 0 0
\(781\) −18.1092 18.1092i −0.647999 0.647999i
\(782\) 0 0
\(783\) −30.7952 30.7952i −1.10053 1.10053i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −35.9487 −1.28143 −0.640716 0.767778i \(-0.721363\pi\)
−0.640716 + 0.767778i \(0.721363\pi\)
\(788\) 0 0
\(789\) −8.22370 8.22370i −0.292771 0.292771i
\(790\) 0 0
\(791\) −1.29546 −0.0460611
\(792\) 0 0
\(793\) −5.31703 5.31703i −0.188813 0.188813i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.7052i 0.379196i −0.981862 0.189598i \(-0.939282\pi\)
0.981862 0.189598i \(-0.0607185\pi\)
\(798\) 0 0
\(799\) −0.495636 −0.0175343
\(800\) 0 0
\(801\) −2.21821 −0.0783767
\(802\) 0 0
\(803\) 32.1561i 1.13476i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.55135 3.55135i −0.125014 0.125014i
\(808\) 0 0
\(809\) −10.3002 −0.362137 −0.181069 0.983470i \(-0.557956\pi\)
−0.181069 + 0.983470i \(0.557956\pi\)
\(810\) 0 0
\(811\) −27.0808 27.0808i −0.950936 0.950936i 0.0479153 0.998851i \(-0.484742\pi\)
−0.998851 + 0.0479153i \(0.984742\pi\)
\(812\) 0 0
\(813\) −23.0913 −0.809847
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.12547 + 1.12547i 0.0393752 + 0.0393752i
\(818\) 0 0
\(819\) −1.53824 1.53824i −0.0537503 0.0537503i
\(820\) 0 0
\(821\) 6.00000 6.00000i 0.209401 0.209401i −0.594612 0.804013i \(-0.702694\pi\)
0.804013 + 0.594612i \(0.202694\pi\)
\(822\) 0 0
\(823\) 17.4197 17.4197i 0.607214 0.607214i −0.335003 0.942217i \(-0.608737\pi\)
0.942217 + 0.335003i \(0.108737\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.05415i 0.106203i 0.998589 + 0.0531016i \(0.0169107\pi\)
−0.998589 + 0.0531016i \(0.983089\pi\)
\(828\) 0 0
\(829\) 2.18527 2.18527i 0.0758976 0.0758976i −0.668139 0.744037i \(-0.732909\pi\)
0.744037 + 0.668139i \(0.232909\pi\)
\(830\) 0 0
\(831\) 17.3668i 0.602446i
\(832\) 0 0
\(833\) −8.79290 + 8.79290i −0.304656 + 0.304656i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 31.9351 1.10384
\(838\) 0 0
\(839\) 1.68627i 0.0582167i 0.999576 + 0.0291083i \(0.00926678\pi\)
−0.999576 + 0.0291083i \(0.990733\pi\)
\(840\) 0 0
\(841\) 31.8872i 1.09956i
\(842\) 0 0
\(843\) −23.2690 −0.801427
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.08928 + 3.08928i −0.106149 + 0.106149i
\(848\) 0 0
\(849\) 29.0935i 0.998488i
\(850\) 0 0
\(851\) 53.6052 53.6052i 1.83756 1.83756i
\(852\) 0 0
\(853\) 38.0838i 1.30396i −0.758235 0.651982i \(-0.773938\pi\)
0.758235 0.651982i \(-0.226062\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.9027 + 27.9027i −0.953138 + 0.953138i −0.998950 0.0458123i \(-0.985412\pi\)
0.0458123 + 0.998950i \(0.485412\pi\)
\(858\) 0 0
\(859\) 7.51122 7.51122i 0.256280 0.256280i −0.567260 0.823539i \(-0.691996\pi\)
0.823539 + 0.567260i \(0.191996\pi\)
\(860\) 0 0
\(861\) −20.7375 20.7375i −0.706732 0.706732i
\(862\) 0 0
\(863\) −35.6257 35.6257i −1.21271 1.21271i −0.970131 0.242580i \(-0.922006\pi\)
−0.242580 0.970131i \(-0.577994\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 12.9906 0.441184
\(868\) 0 0
\(869\) 9.96107 + 9.96107i 0.337906 + 0.337906i
\(870\) 0 0
\(871\) 19.8004 0.670912
\(872\) 0 0
\(873\) 6.04642 + 6.04642i 0.204640 + 0.204640i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.6461i 1.03484i −0.855730 0.517422i \(-0.826892\pi\)
0.855730 0.517422i \(-0.173108\pi\)
\(878\) 0 0
\(879\) −9.55600 −0.322316
\(880\) 0 0
\(881\) −20.5286 −0.691625 −0.345813 0.938304i \(-0.612397\pi\)
−0.345813 + 0.938304i \(0.612397\pi\)
\(882\) 0 0
\(883\) 20.5650i 0.692068i −0.938222 0.346034i \(-0.887528\pi\)
0.938222 0.346034i \(-0.112472\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.6473 23.6473i −0.793997 0.793997i 0.188144 0.982141i \(-0.439753\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(888\) 0 0
\(889\) −22.0560 −0.739735
\(890\) 0 0
\(891\) 13.8976 + 13.8976i 0.465588 + 0.465588i
\(892\) 0 0
\(893\) −1.06915 −0.0357778
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −16.1152 16.1152i −0.538071 0.538071i
\(898\) 0 0
\(899\) 31.5705 + 31.5705i 1.05293 + 1.05293i
\(900\) 0 0
\(901\) 5.58736 5.58736i 0.186142 0.186142i
\(902\) 0 0
\(903\) −0.456851 + 0.456851i −0.0152030 + 0.0152030i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.63021i 0.120539i −0.998182 0.0602696i \(-0.980804\pi\)
0.998182 0.0602696i \(-0.0191961\pi\)
\(908\) 0 0
\(909\) 3.84659 3.84659i 0.127583 0.127583i
\(910\) 0 0
\(911\) 44.3485i 1.46933i −0.678430 0.734665i \(-0.737339\pi\)
0.678430 0.734665i \(-0.262661\pi\)
\(912\) 0 0
\(913\) −10.1248 + 10.1248i −0.335081 + 0.335081i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.10843 −0.0366035
\(918\) 0 0
\(919\) 13.3312i 0.439756i −0.975527 0.219878i \(-0.929434\pi\)
0.975527 0.219878i \(-0.0705660\pi\)
\(920\) 0 0
\(921\) 7.50618i 0.247337i
\(922\) 0 0
\(923\) 18.9768 0.624628
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.87358 + 1.87358i −0.0615364 + 0.0615364i
\(928\) 0 0
\(929\) 13.5713i 0.445260i 0.974903 + 0.222630i \(0.0714641\pi\)
−0.974903 + 0.222630i \(0.928536\pi\)
\(930\) 0 0
\(931\) −18.9674 + 18.9674i −0.621632 + 0.621632i
\(932\) 0 0
\(933\) 3.23891i 0.106037i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.5066 10.5066i 0.343235 0.343235i −0.514347 0.857582i \(-0.671966\pi\)
0.857582 + 0.514347i \(0.171966\pi\)
\(938\) 0 0
\(939\) −21.1265 + 21.1265i −0.689437 + 0.689437i
\(940\) 0 0
\(941\) −4.25664 4.25664i −0.138763 0.138763i 0.634313 0.773076i \(-0.281283\pi\)
−0.773076 + 0.634313i \(0.781283\pi\)
\(942\) 0 0
\(943\) 55.5640 + 55.5640i 1.80941 + 1.80941i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27.2068 −0.884101 −0.442050 0.896990i \(-0.645749\pi\)
−0.442050 + 0.896990i \(0.645749\pi\)
\(948\) 0 0
\(949\) −16.8483 16.8483i −0.546919 0.546919i
\(950\) 0 0
\(951\) −9.60841 −0.311574
\(952\) 0 0
\(953\) −0.537259 0.537259i −0.0174035 0.0174035i 0.698351 0.715755i \(-0.253917\pi\)
−0.715755 + 0.698351i \(0.753917\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 34.8917i 1.12789i
\(958\) 0 0
\(959\) −38.1837 −1.23302
\(960\) 0 0
\(961\) −1.73907 −0.0560990
\(962\) 0 0
\(963\) 8.01545i 0.258294i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9.89350 + 9.89350i 0.318154 + 0.318154i 0.848058 0.529904i \(-0.177772\pi\)
−0.529904 + 0.848058i \(0.677772\pi\)
\(968\) 0 0
\(969\) −28.6580 −0.920629
\(970\) 0 0
\(971\) −14.8635 14.8635i −0.476992 0.476992i 0.427176 0.904168i \(-0.359508\pi\)
−0.904168 + 0.427176i \(0.859508\pi\)
\(972\) 0 0
\(973\) 12.3729 0.396658
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.4685 + 11.4685i 0.366908 + 0.366908i 0.866348 0.499440i \(-0.166461\pi\)
−0.499440 + 0.866348i \(0.666461\pi\)
\(978\) 0 0
\(979\) 7.42674 + 7.42674i 0.237360 + 0.237360i
\(980\) 0 0
\(981\) 1.22200 1.22200i 0.0390154 0.0390154i
\(982\) 0 0
\(983\) −34.9827 + 34.9827i −1.11577 + 1.11577i −0.123420 + 0.992355i \(0.539386\pi\)
−0.992355 + 0.123420i \(0.960614\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.433989i 0.0138140i
\(988\) 0 0
\(989\) 1.22408 1.22408i 0.0389236 0.0389236i
\(990\) 0 0
\(991\) 42.0365i 1.33533i 0.744460 + 0.667667i \(0.232707\pi\)
−0.744460 + 0.667667i \(0.767293\pi\)
\(992\) 0 0
\(993\) 12.8121 12.8121i 0.406581 0.406581i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 31.7128 1.00435 0.502177 0.864765i \(-0.332533\pi\)
0.502177 + 0.864765i \(0.332533\pi\)
\(998\) 0 0
\(999\) 61.5139i 1.94621i
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 1600.2.j.c.1007.6 16
4.3 odd 2 400.2.j.c.307.6 yes 16
5.2 odd 4 1600.2.s.c.943.6 16
5.3 odd 4 1600.2.s.c.943.3 16
5.4 even 2 inner 1600.2.j.c.1007.3 16
16.5 even 4 400.2.s.c.107.7 yes 16
16.11 odd 4 1600.2.s.c.207.3 16
20.3 even 4 400.2.s.c.243.7 yes 16
20.7 even 4 400.2.s.c.243.2 yes 16
20.19 odd 2 400.2.j.c.307.3 yes 16
80.27 even 4 inner 1600.2.j.c.143.6 16
80.37 odd 4 400.2.j.c.43.3 16
80.43 even 4 inner 1600.2.j.c.143.3 16
80.53 odd 4 400.2.j.c.43.6 yes 16
80.59 odd 4 1600.2.s.c.207.6 16
80.69 even 4 400.2.s.c.107.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.c.43.3 16 80.37 odd 4
400.2.j.c.43.6 yes 16 80.53 odd 4
400.2.j.c.307.3 yes 16 20.19 odd 2
400.2.j.c.307.6 yes 16 4.3 odd 2
400.2.s.c.107.2 yes 16 80.69 even 4
400.2.s.c.107.7 yes 16 16.5 even 4
400.2.s.c.243.2 yes 16 20.7 even 4
400.2.s.c.243.7 yes 16 20.3 even 4
1600.2.j.c.143.3 16 80.43 even 4 inner
1600.2.j.c.143.6 16 80.27 even 4 inner
1600.2.j.c.1007.3 16 5.4 even 2 inner
1600.2.j.c.1007.6 16 1.1 even 1 trivial
1600.2.s.c.207.3 16 16.11 odd 4
1600.2.s.c.207.6 16 80.59 odd 4
1600.2.s.c.943.3 16 5.3 odd 4
1600.2.s.c.943.6 16 5.2 odd 4