Properties

Label 3525.1.bg.a.368.1
Level $3525$
Weight $1$
Character 3525.368
Analytic conductor $1.759$
Analytic rank $0$
Dimension $88$
Projective image $D_{46}$
CM discriminant -15
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3525,1,Mod(107,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(92))
 
chi = DirichletCharacter(H, H._module([46, 23, 22]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3525.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3525.bg (of order \(92\), degree \(44\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.75920416953\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(2\) over \(\Q(\zeta_{92})\)
Coefficient field: \(\Q(\zeta_{184})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{88} - x^{84} + x^{80} - x^{76} + x^{72} - x^{68} + x^{64} - x^{60} + x^{56} - x^{52} + x^{48} - x^{44} + x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{46}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{46} - \cdots)\)

Embedding invariants

Embedding label 368.1
Root \(0.366854 + 0.930278i\) of defining polynomial
Character \(\chi\) \(=\) 3525.368
Dual form 3525.1.bg.a.182.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.626895 - 1.58970i) q^{2} +(-0.753713 - 0.657204i) q^{3} +(-1.40330 + 1.31059i) q^{4} +(-0.572255 + 1.61017i) q^{6} +(1.41995 + 0.675300i) q^{8} +(0.136167 + 0.990686i) q^{9} +O(q^{10})\) \(q+(-0.626895 - 1.58970i) q^{2} +(-0.753713 - 0.657204i) q^{3} +(-1.40330 + 1.31059i) q^{4} +(-0.572255 + 1.61017i) q^{6} +(1.41995 + 0.675300i) q^{8} +(0.136167 + 0.990686i) q^{9} +(1.91901 - 0.0655554i) q^{12} +(0.0523262 - 0.764982i) q^{16} +(-0.404779 - 0.0416125i) q^{17} +(1.48953 - 0.837519i) q^{18} +(-1.24888 + 0.759461i) q^{19} +(0.741248 + 0.292310i) q^{23} +(-0.626428 - 1.44218i) q^{24} +(0.548452 - 0.836182i) q^{27} +(1.25923 + 0.0861339i) q^{31} +(0.249789 - 0.0792796i) q^{32} +(0.187602 + 0.669562i) q^{34} +(-1.48947 - 1.21177i) q^{36} +(1.99023 + 1.50924i) q^{38} -1.36161i q^{46} +(0.985460 - 0.169910i) q^{47} +(-0.542188 + 0.542188i) q^{48} +(0.398401 - 0.917211i) q^{49} +(0.277739 + 0.297386i) q^{51} +(-0.850966 - 1.78933i) q^{53} +(-1.67310 - 0.347674i) q^{54} +(1.44042 + 0.248353i) q^{57} +(1.11059 - 1.57335i) q^{61} +(-0.652480 - 2.05580i) q^{62} +(-0.766521 - 0.942181i) q^{64} +(0.622563 - 0.472104i) q^{68} +(-0.366581 - 0.707469i) q^{69} +(-0.475660 + 1.49868i) q^{72} +(0.757212 - 2.70252i) q^{76} +(1.37749 + 0.713755i) q^{79} +(-0.962917 + 0.269797i) q^{81} +(0.666248 - 0.0684924i) q^{83} +(-1.42329 + 0.561274i) q^{92} +(-0.892493 - 0.892493i) q^{93} +(-0.887885 - 1.46007i) q^{94} +(-0.240372 - 0.104408i) q^{96} +(-1.70784 - 0.0583417i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 8 q^{6} + 20 q^{16} - 12 q^{36} - 8 q^{51} + 8 q^{61} - 92 q^{76} + 4 q^{81} - 68 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{31}{46}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.626895 1.58970i −0.626895 1.58970i −0.796805 0.604236i \(-0.793478\pi\)
0.169910 0.985460i \(-0.445652\pi\)
\(3\) −0.753713 0.657204i −0.753713 0.657204i
\(4\) −1.40330 + 1.31059i −1.40330 + 1.31059i
\(5\) 0 0
\(6\) −0.572255 + 1.61017i −0.572255 + 1.61017i
\(7\) 0 0 0.836182 0.548452i \(-0.184783\pi\)
−0.836182 + 0.548452i \(0.815217\pi\)
\(8\) 1.41995 + 0.675300i 1.41995 + 0.675300i
\(9\) 0.136167 + 0.990686i 0.136167 + 0.990686i
\(10\) 0 0
\(11\) 0 0 0.631088 0.775711i \(-0.282609\pi\)
−0.631088 + 0.775711i \(0.717391\pi\)
\(12\) 1.91901 0.0655554i 1.91901 0.0655554i
\(13\) 0 0 −0.871660 0.490110i \(-0.836957\pi\)
0.871660 + 0.490110i \(0.163043\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.0523262 0.764982i 0.0523262 0.764982i
\(17\) −0.404779 0.0416125i −0.404779 0.0416125i −0.102264 0.994757i \(-0.532609\pi\)
−0.302515 + 0.953145i \(0.597826\pi\)
\(18\) 1.48953 0.837519i 1.48953 0.837519i
\(19\) −1.24888 + 0.759461i −1.24888 + 0.759461i −0.979084 0.203456i \(-0.934783\pi\)
−0.269797 + 0.962917i \(0.586957\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.741248 + 0.292310i 0.741248 + 0.292310i 0.707107 0.707107i \(-0.250000\pi\)
0.0341411 + 0.999417i \(0.489130\pi\)
\(24\) −0.626428 1.44218i −0.626428 1.44218i
\(25\) 0 0
\(26\) 0 0
\(27\) 0.548452 0.836182i 0.548452 0.836182i
\(28\) 0 0
\(29\) 0 0 −0.962917 0.269797i \(-0.913043\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(30\) 0 0
\(31\) 1.25923 + 0.0861339i 1.25923 + 0.0861339i 0.682553 0.730836i \(-0.260870\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(32\) 0.249789 0.0792796i 0.249789 0.0792796i
\(33\) 0 0
\(34\) 0.187602 + 0.669562i 0.187602 + 0.669562i
\(35\) 0 0
\(36\) −1.48947 1.21177i −1.48947 1.21177i
\(37\) 0 0 0.169910 0.985460i \(-0.445652\pi\)
−0.169910 + 0.985460i \(0.554348\pi\)
\(38\) 1.99023 + 1.50924i 1.99023 + 1.50924i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.942261 0.334880i \(-0.891304\pi\)
0.942261 + 0.334880i \(0.108696\pi\)
\(42\) 0 0
\(43\) 0 0 0.0341411 0.999417i \(-0.489130\pi\)
−0.0341411 + 0.999417i \(0.510870\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.36161i 1.36161i
\(47\) 0.985460 0.169910i 0.985460 0.169910i
\(48\) −0.542188 + 0.542188i −0.542188 + 0.542188i
\(49\) 0.398401 0.917211i 0.398401 0.917211i
\(50\) 0 0
\(51\) 0.277739 + 0.297386i 0.277739 + 0.297386i
\(52\) 0 0
\(53\) −0.850966 1.78933i −0.850966 1.78933i −0.548452 0.836182i \(-0.684783\pi\)
−0.302515 0.953145i \(-0.597826\pi\)
\(54\) −1.67310 0.347674i −1.67310 0.347674i
\(55\) 0 0
\(56\) 0 0
\(57\) 1.44042 + 0.248353i 1.44042 + 0.248353i
\(58\) 0 0
\(59\) 0 0 −0.730836 0.682553i \(-0.760870\pi\)
0.730836 + 0.682553i \(0.239130\pi\)
\(60\) 0 0
\(61\) 1.11059 1.57335i 1.11059 1.57335i 0.334880 0.942261i \(-0.391304\pi\)
0.775711 0.631088i \(-0.217391\pi\)
\(62\) −0.652480 2.05580i −0.652480 2.05580i
\(63\) 0 0
\(64\) −0.766521 0.942181i −0.766521 0.942181i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.836182 0.548452i \(-0.815217\pi\)
0.836182 + 0.548452i \(0.184783\pi\)
\(68\) 0.622563 0.472104i 0.622563 0.472104i
\(69\) −0.366581 0.707469i −0.366581 0.707469i
\(70\) 0 0
\(71\) 0 0 0.917211 0.398401i \(-0.130435\pi\)
−0.917211 + 0.398401i \(0.869565\pi\)
\(72\) −0.475660 + 1.49868i −0.475660 + 1.49868i
\(73\) 0 0 −0.604236 0.796805i \(-0.706522\pi\)
0.604236 + 0.796805i \(0.293478\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.757212 2.70252i 0.757212 2.70252i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.37749 + 0.713755i 1.37749 + 0.713755i 0.979084 0.203456i \(-0.0652174\pi\)
0.398401 + 0.917211i \(0.369565\pi\)
\(80\) 0 0
\(81\) −0.962917 + 0.269797i −0.962917 + 0.269797i
\(82\) 0 0
\(83\) 0.666248 0.0684924i 0.666248 0.0684924i 0.236764 0.971567i \(-0.423913\pi\)
0.429483 + 0.903075i \(0.358696\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.519584 0.854419i \(-0.326087\pi\)
−0.519584 + 0.854419i \(0.673913\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.42329 + 0.561274i −1.42329 + 0.561274i
\(93\) −0.892493 0.892493i −0.892493 0.892493i
\(94\) −0.887885 1.46007i −0.887885 1.46007i
\(95\) 0 0
\(96\) −0.240372 0.104408i −0.240372 0.104408i
\(97\) 0 0 0.657204 0.753713i \(-0.271739\pi\)
−0.657204 + 0.753713i \(0.728261\pi\)
\(98\) −1.70784 0.0583417i −1.70784 0.0583417i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.203456 0.979084i \(-0.434783\pi\)
−0.203456 + 0.979084i \(0.565217\pi\)
\(102\) 0.298640 0.627950i 0.298640 0.627950i
\(103\) 0 0 0.604236 0.796805i \(-0.293478\pi\)
−0.604236 + 0.796805i \(0.706522\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.31102 + 2.47450i −2.31102 + 2.47450i
\(107\) −0.977935 + 1.73926i −0.977935 + 1.73926i −0.429483 + 0.903075i \(0.641304\pi\)
−0.548452 + 0.836182i \(0.684783\pi\)
\(108\) 0.326250 + 1.89221i 0.326250 + 1.89221i
\(109\) 0.867003 1.67324i 0.867003 1.67324i 0.136167 0.990686i \(-0.456522\pi\)
0.730836 0.682553i \(-0.239130\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.58748 0.386859i −1.58748 0.386859i −0.657204 0.753713i \(-0.728261\pi\)
−0.930278 + 0.366854i \(0.880435\pi\)
\(114\) −0.508185 2.44552i −0.508185 2.44552i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.203456 0.979084i −0.203456 0.979084i
\(122\) −3.19737 0.779178i −3.19737 0.779178i
\(123\) 0 0
\(124\) −1.87997 + 1.52947i −1.87997 + 1.52947i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.169910 0.985460i \(-0.554348\pi\)
0.169910 + 0.985460i \(0.445652\pi\)
\(128\) −0.888811 + 1.58075i −0.888811 + 1.58075i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.576680 0.816970i \(-0.695652\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.546666 0.332435i −0.546666 0.332435i
\(137\) 1.88342 + 0.0643397i 1.88342 + 0.0643397i 0.953145 0.302515i \(-0.0978261\pi\)
0.930278 + 0.366854i \(0.119565\pi\)
\(138\) −0.894853 + 1.02626i −0.894853 + 1.02626i
\(139\) 1.62876 + 0.707469i 1.62876 + 0.707469i 0.997669 0.0682424i \(-0.0217391\pi\)
0.631088 + 0.775711i \(0.282609\pi\)
\(140\) 0 0
\(141\) −0.854419 0.519584i −0.854419 0.519584i
\(142\) 0 0
\(143\) 0 0
\(144\) 0.764982 0.0523262i 0.764982 0.0523262i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.903075 + 0.429483i −0.903075 + 0.429483i
\(148\) 0 0
\(149\) 0 0 0.942261 0.334880i \(-0.108696\pi\)
−0.942261 + 0.334880i \(0.891304\pi\)
\(150\) 0 0
\(151\) −0.440832 + 0.311173i −0.440832 + 0.311173i −0.775711 0.631088i \(-0.782609\pi\)
0.334880 + 0.942261i \(0.391304\pi\)
\(152\) −2.28622 + 0.235030i −2.28622 + 0.235030i
\(153\) −0.0138924 0.406675i −0.0138924 0.406675i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.753713 0.657204i \(-0.228261\pi\)
−0.753713 + 0.657204i \(0.771739\pi\)
\(158\) 0.271116 2.63723i 0.271116 2.63723i
\(159\) −0.534568 + 1.90790i −0.534568 + 1.90790i
\(160\) 0 0
\(161\) 0 0
\(162\) 1.03254 + 1.36161i 1.03254 + 1.36161i
\(163\) 0 0 0.302515 0.953145i \(-0.402174\pi\)
−0.302515 + 0.953145i \(0.597826\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −0.526549 1.01619i −0.526549 1.01619i
\(167\) 0.216997 0.164554i 0.216997 0.164554i −0.490110 0.871660i \(-0.663043\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 0.519584 + 0.854419i 0.519584 + 0.854419i
\(170\) 0 0
\(171\) −0.922444 1.13384i −0.922444 1.13384i
\(172\) 0 0
\(173\) 0.348908 + 1.09932i 0.348908 + 1.09932i 0.953145 + 0.302515i \(0.0978261\pi\)
−0.604236 + 0.796805i \(0.706522\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.990686 0.136167i \(-0.0434783\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(180\) 0 0
\(181\) 1.59976 + 0.332435i 1.59976 + 0.332435i 0.917211 0.398401i \(-0.130435\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(182\) 0 0
\(183\) −1.87108 + 0.455969i −1.87108 + 0.455969i
\(184\) 0.855140 + 0.915632i 0.855140 + 0.915632i
\(185\) 0 0
\(186\) −0.859293 + 1.97829i −0.859293 + 1.97829i
\(187\) 0 0
\(188\) −1.16021 + 1.52997i −1.16021 + 1.52997i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.0682424 0.997669i \(-0.521739\pi\)
0.0682424 + 0.997669i \(0.478261\pi\)
\(192\) −0.0414679 + 1.21389i −0.0414679 + 1.21389i
\(193\) 0 0 −0.236764 0.971567i \(-0.576087\pi\)
0.236764 + 0.971567i \(0.423913\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.643012 + 1.80926i 0.643012 + 1.80926i
\(197\) −1.53451 1.16366i −1.53451 1.16366i −0.930278 0.366854i \(-0.880435\pi\)
−0.604236 0.796805i \(-0.706522\pi\)
\(198\) 0 0
\(199\) −0.211252 0.171866i −0.211252 0.171866i 0.519584 0.854419i \(-0.326087\pi\)
−0.730836 + 0.682553i \(0.760870\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) −0.779503 0.0533194i −0.779503 0.0533194i
\(205\) 0 0
\(206\) 0 0
\(207\) −0.188654 + 0.774147i −0.188654 + 0.774147i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.794945 1.83015i −0.794945 1.83015i −0.460065 0.887885i \(-0.652174\pi\)
−0.334880 0.942261i \(-0.608696\pi\)
\(212\) 3.53924 + 1.39569i 3.53924 + 1.39569i
\(213\) 0 0
\(214\) 3.37795 + 0.464289i 3.37795 + 0.464289i
\(215\) 0 0
\(216\) 1.34345 0.816970i 1.34345 0.816970i
\(217\) 0 0
\(218\) −3.20346 0.329326i −3.20346 0.329326i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.999417 0.0341411i \(-0.0108696\pi\)
−0.999417 + 0.0341411i \(0.989130\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.380196 + 2.76613i 0.380196 + 2.76613i
\(227\) 0.487293 + 0.231746i 0.487293 + 0.231746i 0.657204 0.753713i \(-0.271739\pi\)
−0.169910 + 0.985460i \(0.554348\pi\)
\(228\) −2.34683 + 1.53929i −2.34683 + 1.53929i
\(229\) 0.180699 0.508438i 0.180699 0.508438i −0.816970 0.576680i \(-0.804348\pi\)
0.997669 + 0.0682424i \(0.0217391\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.0999066 0.253346i −0.0999066 0.253346i 0.871660 0.490110i \(-0.163043\pi\)
−0.971567 + 0.236764i \(0.923913\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.569146 1.44325i −0.569146 1.44325i
\(238\) 0 0
\(239\) 0 0 0.730836 0.682553i \(-0.239130\pi\)
−0.730836 + 0.682553i \(0.760870\pi\)
\(240\) 0 0
\(241\) −0.663521 + 1.86697i −0.663521 + 1.86697i −0.203456 + 0.979084i \(0.565217\pi\)
−0.460065 + 0.887885i \(0.652174\pi\)
\(242\) −1.42890 + 0.937216i −1.42890 + 0.937216i
\(243\) 0.903075 + 0.429483i 0.903075 + 0.429483i
\(244\) 0.503524 + 3.66341i 0.503524 + 3.66341i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 1.72989 + 0.972667i 1.72989 + 0.972667i
\(249\) −0.547173 0.386237i −0.547173 0.386237i
\(250\) 0 0
\(251\) 0 0 0.0682424 0.997669i \(-0.478261\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.86681 + 0.256587i 1.86681 + 0.256587i
\(257\) 1.86642 + 0.592374i 1.86642 + 0.592374i 0.994757 + 0.102264i \(0.0326087\pi\)
0.871660 + 0.490110i \(0.163043\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.434326 1.78226i 0.434326 1.78226i −0.169910 0.985460i \(-0.554348\pi\)
0.604236 0.796805i \(-0.293478\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.269797 0.962917i \(-0.586957\pi\)
0.269797 + 0.962917i \(0.413043\pi\)
\(270\) 0 0
\(271\) 1.05893 + 0.861502i 1.05893 + 0.861502i 0.990686 0.136167i \(-0.0434783\pi\)
0.0682424 + 0.997669i \(0.478261\pi\)
\(272\) −0.0530133 + 0.307471i −0.0530133 + 0.307471i
\(273\) 0 0
\(274\) −1.07843 3.03440i −1.07843 3.03440i
\(275\) 0 0
\(276\) 1.44163 + 0.512354i 1.44163 + 0.512354i
\(277\) 0 0 −0.236764 0.971567i \(-0.576087\pi\)
0.236764 + 0.971567i \(0.423913\pi\)
\(278\) 0.103601 3.03274i 0.103601 3.03274i
\(279\) 0.0861339 + 1.25923i 0.0861339 + 1.25923i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.290349 + 1.68399i −0.290349 + 1.68399i
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.112554 + 0.236668i 0.112554 + 0.236668i
\(289\) −0.816970 0.169768i −0.816970 0.169768i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.106270 1.03372i −0.106270 1.03372i −0.903075 0.429483i \(-0.858696\pi\)
0.796805 0.604236i \(-0.206522\pi\)
\(294\) 1.24888 + 1.16637i 1.24888 + 1.16637i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0.771025 + 0.505716i 0.771025 + 0.505716i
\(303\) 0 0
\(304\) 0.515625 + 0.995111i 0.515625 + 0.995111i
\(305\) 0 0
\(306\) −0.637780 + 0.277027i −0.637780 + 0.277027i
\(307\) 0 0 0.302515 0.953145i \(-0.402174\pi\)
−0.302515 + 0.953145i \(0.597826\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.269797 0.962917i \(-0.413043\pi\)
−0.269797 + 0.962917i \(0.586957\pi\)
\(312\) 0 0
\(313\) 0 0 0.753713 0.657204i \(-0.228261\pi\)
−0.753713 + 0.657204i \(0.771739\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.86847 + 0.803707i −2.86847 + 0.803707i
\(317\) 0.0681230 + 1.99417i 0.0681230 + 1.99417i 0.102264 + 0.994757i \(0.467391\pi\)
−0.0341411 + 0.999417i \(0.510870\pi\)
\(318\) 3.36809 0.346251i 3.36809 0.346251i
\(319\) 0 0
\(320\) 0 0
\(321\) 1.88013 0.668198i 1.88013 0.668198i
\(322\) 0 0
\(323\) 0.537123 0.255445i 0.537123 0.255445i
\(324\) 0.997669 1.64060i 0.997669 1.64060i
\(325\) 0 0
\(326\) 0 0
\(327\) −1.75313 + 0.691345i −1.75313 + 0.691345i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.614311 0.266833i −0.614311 0.266833i 0.0682424 0.997669i \(-0.478261\pi\)
−0.682553 + 0.730836i \(0.739130\pi\)
\(332\) −0.845180 + 0.969294i −0.845180 + 0.969294i
\(333\) 0 0
\(334\) −0.397624 0.241801i −0.397624 0.241801i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.429483 0.903075i \(-0.358696\pi\)
−0.429483 + 0.903075i \(0.641304\pi\)
\(338\) 1.03254 1.36161i 1.03254 1.36161i
\(339\) 0.942261 + 1.33488i 0.942261 + 1.33488i
\(340\) 0 0
\(341\) 0 0
\(342\) −1.22418 + 2.17720i −1.22418 + 2.17720i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 1.52886 1.24382i 1.52886 1.24382i
\(347\) −0.565274 1.00534i −0.565274 1.00534i −0.994757 0.102264i \(-0.967391\pi\)
0.429483 0.903075i \(-0.358696\pi\)
\(348\) 0 0
\(349\) −0.361291 1.73863i −0.361291 1.73863i −0.631088 0.775711i \(-0.717391\pi\)
0.269797 0.962917i \(-0.413043\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.500795 1.26993i 0.500795 1.26993i −0.429483 0.903075i \(-0.641304\pi\)
0.930278 0.366854i \(-0.119565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.775711 0.631088i \(-0.217391\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(360\) 0 0
\(361\) 0.522857 1.00907i 0.522857 1.00907i
\(362\) −0.474414 2.75154i −0.474414 2.75154i
\(363\) −0.490110 + 0.871660i −0.490110 + 0.871660i
\(364\) 0 0
\(365\) 0 0
\(366\) 1.89782 + 2.68860i 1.89782 + 2.68860i
\(367\) 0 0 0.604236 0.796805i \(-0.293478\pi\)
−0.604236 + 0.796805i \(0.706522\pi\)
\(368\) 0.262399 0.551746i 0.262399 0.551746i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.42213 + 0.0827424i 2.42213 + 0.0827424i
\(373\) 0 0 0.657204 0.753713i \(-0.271739\pi\)
−0.657204 + 0.753713i \(0.728261\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.51405 + 0.424216i 1.51405 + 0.424216i
\(377\) 0 0
\(378\) 0 0
\(379\) −0.136167 + 0.00931405i −0.136167 + 0.00931405i −0.136167 0.990686i \(-0.543478\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.632563 + 0.964420i 0.632563 + 0.964420i 0.999417 + 0.0341411i \(0.0108696\pi\)
−0.366854 + 0.930278i \(0.619565\pi\)
\(384\) 1.70878 0.607301i 1.70878 0.607301i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.962917 0.269797i \(-0.0869565\pi\)
−0.962917 + 0.269797i \(0.913043\pi\)
\(390\) 0 0
\(391\) −0.287878 0.149166i −0.287878 0.149166i
\(392\) 1.18510 1.03336i 1.18510 1.03336i
\(393\) 0 0
\(394\) −0.887885 + 3.16890i −0.887885 + 3.16890i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.604236 0.796805i \(-0.706522\pi\)
0.604236 + 0.796805i \(0.293478\pi\)
\(398\) −0.140782 + 0.443569i −0.140782 + 0.443569i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.460065 0.887885i \(-0.652174\pi\)
0.460065 + 0.887885i \(0.347826\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.193552 + 0.609831i 0.193552 + 0.609831i
\(409\) 1.02405 1.45075i 1.02405 1.45075i 0.136167 0.990686i \(-0.456522\pi\)
0.887885 0.460065i \(-0.152174\pi\)
\(410\) 0 0
\(411\) −1.37728 1.28629i −1.37728 1.28629i
\(412\) 0 0
\(413\) 0 0
\(414\) 1.34892 0.185405i 1.34892 0.185405i
\(415\) 0 0
\(416\) 0 0
\(417\) −0.762664 1.60365i −0.762664 1.60365i
\(418\) 0 0
\(419\) 0 0 −0.682553 0.730836i \(-0.739130\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(420\) 0 0
\(421\) −0.794945 + 1.83015i −0.794945 + 1.83015i −0.334880 + 0.942261i \(0.608696\pi\)
−0.460065 + 0.887885i \(0.652174\pi\)
\(422\) −2.41103 + 2.41103i −2.41103 + 2.41103i
\(423\) 0.302515 + 0.953145i 0.302515 + 0.953145i
\(424\) 3.11542i 3.11542i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.907117 3.72237i −0.907117 3.72237i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.334880 0.942261i \(-0.608696\pi\)
0.334880 + 0.942261i \(0.391304\pi\)
\(432\) −0.610966 0.463310i −0.610966 0.463310i
\(433\) 0 0 0.169910 0.985460i \(-0.445652\pi\)
−0.169910 + 0.985460i \(0.554348\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.976267 + 3.48434i 0.976267 + 3.48434i
\(437\) −1.14773 + 0.197888i −1.14773 + 0.197888i
\(438\) 0 0
\(439\) −0.668198 0.0457060i −0.668198 0.0457060i −0.269797 0.962917i \(-0.586957\pi\)
−0.398401 + 0.917211i \(0.630435\pi\)
\(440\) 0 0
\(441\) 0.962917 + 0.269797i 0.962917 + 0.269797i
\(442\) 0 0
\(443\) −1.03357 + 1.57580i −1.03357 + 1.57580i −0.236764 + 0.971567i \(0.576087\pi\)
−0.796805 + 0.604236i \(0.793478\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.990686 0.136167i \(-0.956522\pi\)
0.990686 + 0.136167i \(0.0434783\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.73473 1.53766i 2.73473 1.53766i
\(453\) 0.536765 + 0.0551811i 0.536765 + 0.0551811i
\(454\) 0.0629249 0.919929i 0.0629249 0.919929i
\(455\) 0 0
\(456\) 1.87761 + 1.32536i 1.87761 + 1.32536i
\(457\) 0 0 −0.871660 0.490110i \(-0.836957\pi\)
0.871660 + 0.490110i \(0.163043\pi\)
\(458\) −0.921541 + 0.0314808i −0.921541 + 0.0314808i
\(459\) −0.256797 + 0.315646i −0.256797 + 0.315646i
\(460\) 0 0
\(461\) 0 0 −0.136167 0.990686i \(-0.543478\pi\)
0.136167 + 0.990686i \(0.456522\pi\)
\(462\) 0 0
\(463\) 0 0 0.836182 0.548452i \(-0.184783\pi\)
−0.836182 + 0.548452i \(0.815217\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.340112 + 0.317642i −0.340112 + 0.317642i
\(467\) 1.33842 + 1.16704i 1.33842 + 1.16704i 0.971567 + 0.236764i \(0.0760870\pi\)
0.366854 + 0.930278i \(0.380435\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −1.93754 + 1.80954i −1.93754 + 1.80954i
\(475\) 0 0
\(476\) 0 0
\(477\) 1.65679 1.08669i 1.65679 1.08669i
\(478\) 0 0
\(479\) 0 0 −0.136167 0.990686i \(-0.543478\pi\)
0.136167 + 0.990686i \(0.456522\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.38387 0.115597i 3.38387 0.115597i
\(483\) 0 0
\(484\) 1.56869 + 1.10730i 1.56869 + 1.10730i
\(485\) 0 0
\(486\) 0.116615 1.70486i 0.116615 1.70486i
\(487\) 0 0 −0.994757 0.102264i \(-0.967391\pi\)
0.994757 + 0.102264i \(0.0326087\pi\)
\(488\) 2.63947 1.48410i 2.63947 1.48410i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.990686 0.136167i \(-0.956522\pi\)
0.990686 + 0.136167i \(0.0434783\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.131782 0.958784i 0.131782 0.958784i
\(497\) 0 0
\(498\) −0.270979 + 1.11197i −0.270979 + 1.11197i
\(499\) −0.767255 0.214975i −0.767255 0.214975i −0.136167 0.990686i \(-0.543478\pi\)
−0.631088 + 0.775711i \(0.717391\pi\)
\(500\) 0 0
\(501\) −0.271698 0.0185847i −0.271698 0.0185847i
\(502\) 0 0
\(503\) 1.02406 0.176565i 1.02406 0.176565i 0.366854 0.930278i \(-0.380435\pi\)
0.657204 + 0.753713i \(0.271739\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.169910 0.985460i 0.169910 0.985460i
\(508\) 0 0
\(509\) 0 0 −0.334880 0.942261i \(-0.608696\pi\)
0.334880 + 0.942261i \(0.391304\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.333028 1.36659i −0.333028 1.36659i
\(513\) −0.0499031 + 1.46082i −0.0499031 + 1.46082i
\(514\) −0.228352 3.33839i −0.228352 3.33839i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0.459500 1.05788i 0.459500 1.05788i
\(520\) 0 0
\(521\) 0 0 −0.682553 0.730836i \(-0.739130\pi\)
0.682553 + 0.730836i \(0.260870\pi\)
\(522\) 0 0
\(523\) 0 0 −0.429483 0.903075i \(-0.641304\pi\)
0.429483 + 0.903075i \(0.358696\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −3.10554 + 0.426846i −3.10554 + 0.426846i
\(527\) −0.506127 0.0872650i −0.506127 0.0872650i
\(528\) 0 0
\(529\) −0.266833 0.249204i −0.266833 0.249204i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.42298 0.618088i 1.42298 0.618088i 0.460065 0.887885i \(-0.347826\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(542\) 0.705690 2.22345i 0.705690 2.22345i
\(543\) −0.987286 1.30193i −0.987286 1.30193i
\(544\) −0.104408 + 0.0216963i −0.104408 + 0.0216963i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.102264 0.994757i \(-0.467391\pi\)
−0.102264 + 0.994757i \(0.532609\pi\)
\(548\) −2.72733 + 2.37811i −2.72733 + 2.37811i
\(549\) 1.70992 + 0.886009i 1.70992 + 0.886009i
\(550\) 0 0
\(551\) 0 0
\(552\) −0.0427739 1.25212i −0.0427739 1.25212i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −3.21284 + 1.14184i −3.21284 + 1.14184i
\(557\) −1.07396 1.63739i −1.07396 1.63739i −0.707107 0.707107i \(-0.750000\pi\)
−0.366854 0.930278i \(-0.619565\pi\)
\(558\) 1.94780 0.926334i 1.94780 0.926334i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.38463 + 1.38463i 1.38463 + 1.38463i 0.836182 + 0.548452i \(0.184783\pi\)
0.548452 + 0.836182i \(0.315217\pi\)
\(564\) 1.87997 0.390662i 1.87997 0.390662i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.854419 0.519584i \(-0.826087\pi\)
0.854419 + 0.519584i \(0.173913\pi\)
\(570\) 0 0
\(571\) 0.277739 1.33655i 0.277739 1.33655i −0.576680 0.816970i \(-0.695652\pi\)
0.854419 0.519584i \(-0.173913\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.829031 0.887675i 0.829031 0.887675i
\(577\) 0 0 0.490110 0.871660i \(-0.336957\pi\)
−0.490110 + 0.871660i \(0.663043\pi\)
\(578\) 0.242274 + 1.40516i 0.242274 + 1.40516i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.57668 + 0.816970i −1.57668 + 0.816970i
\(587\) 0.536221 1.35976i 0.536221 1.35976i −0.366854 0.930278i \(-0.619565\pi\)
0.903075 0.429483i \(-0.141304\pi\)
\(588\) 0.704408 1.78626i 0.704408 1.78626i
\(589\) −1.63805 + 0.848768i −1.63805 + 0.848768i
\(590\) 0 0
\(591\) 0.391823 + 1.88555i 0.391823 + 1.88555i
\(592\) 0 0
\(593\) 0.509307 + 0.905802i 0.509307 + 0.905802i 0.999417 + 0.0341411i \(0.0108696\pi\)
−0.490110 + 0.871660i \(0.663043\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.0462723 + 0.268373i 0.0462723 + 0.268373i
\(598\) 0 0
\(599\) 0 0 0.682553 0.730836i \(-0.260870\pi\)
−0.682553 + 0.730836i \(0.739130\pi\)
\(600\) 0 0
\(601\) −0.665120 0.942261i −0.665120 0.942261i 0.334880 0.942261i \(-0.391304\pi\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.210799 1.01442i 0.210799 1.01442i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.999417 0.0341411i \(-0.989130\pi\)
0.999417 + 0.0341411i \(0.0108696\pi\)
\(608\) −0.251747 + 0.288716i −0.251747 + 0.288716i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.552480 + 0.552480i 0.552480 + 0.552480i
\(613\) 0 0 0.930278 0.366854i \(-0.119565\pi\)
−0.930278 + 0.366854i \(0.880435\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.40105 0.666310i 1.40105 0.666310i 0.429483 0.903075i \(-0.358696\pi\)
0.971567 + 0.236764i \(0.0760870\pi\)
\(618\) 0 0
\(619\) 1.81464 0.644923i 1.81464 0.644923i 0.816970 0.576680i \(-0.195652\pi\)
0.997669 0.0682424i \(-0.0217391\pi\)
\(620\) 0 0
\(621\) 0.650963 0.459500i 0.650963 0.459500i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.780136 0.162114i 0.780136 0.162114i 0.203456 0.979084i \(-0.434783\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(632\) 1.47397 + 1.94371i 1.47397 + 1.94371i
\(633\) −0.603619 + 1.90185i −0.603619 + 1.90185i
\(634\) 3.12742 1.35843i 3.12742 1.35843i
\(635\) 0 0
\(636\) −1.75031 3.37795i −1.75031 3.37795i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.631088 0.775711i \(-0.717391\pi\)
0.631088 + 0.775711i \(0.282609\pi\)
\(642\) −2.24087 2.56994i −2.24087 2.56994i
\(643\) 0 0 −0.302515 0.953145i \(-0.597826\pi\)
0.302515 + 0.953145i \(0.402174\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.742799 0.693726i −0.742799 0.693726i
\(647\) −0.202623 1.97098i −0.202623 1.97098i −0.236764 0.971567i \(-0.576087\pi\)
0.0341411 0.999417i \(-0.489130\pi\)
\(648\) −1.54949 0.267159i −1.54949 0.267159i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.32629 0.323209i 1.32629 0.323209i 0.490110 0.871660i \(-0.336957\pi\)
0.836182 + 0.548452i \(0.184783\pi\)
\(654\) 2.19806 + 2.35354i 2.19806 + 2.35354i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0.0277687 + 0.405963i 0.0277687 + 0.405963i 0.990686 + 0.136167i \(0.0434783\pi\)
−0.962917 + 0.269797i \(0.913043\pi\)
\(662\) −0.0390749 + 1.14384i −0.0390749 + 1.14384i
\(663\) 0 0
\(664\) 0.992294 + 0.352661i 0.992294 + 0.352661i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.0888487 + 0.515312i −0.0888487 + 0.515312i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.953145 0.302515i \(-0.0978261\pi\)
−0.953145 + 0.302515i \(0.902174\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.84893 0.518045i −1.84893 0.518045i
\(677\) −0.420439 + 1.72528i −0.420439 + 1.72528i 0.236764 + 0.971567i \(0.423913\pi\)
−0.657204 + 0.753713i \(0.728261\pi\)
\(678\) 1.53135 2.33474i 1.53135 2.33474i
\(679\) 0 0
\(680\) 0 0
\(681\) −0.214975 0.494921i −0.214975 0.494921i
\(682\) 0 0
\(683\) 1.74847 + 0.554940i 1.74847 + 0.554940i 0.994757 0.102264i \(-0.0326087\pi\)
0.753713 + 0.657204i \(0.228261\pi\)
\(684\) 2.78046 + 0.382165i 2.78046 + 0.382165i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.470342 + 0.264460i −0.470342 + 0.264460i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.59976 1.12924i −1.59976 1.12924i −0.917211 0.398401i \(-0.869565\pi\)
−0.682553 0.730836i \(-0.739130\pi\)
\(692\) −1.93038 1.08540i −1.93038 1.08540i
\(693\) 0 0
\(694\) −1.24382 + 1.52886i −1.24382 + 1.52886i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −2.53740 + 1.66428i −2.53740 + 1.66428i
\(699\) −0.0911989 + 0.256609i −0.0911989 + 0.256609i
\(700\) 0 0
\(701\) 0 0 0.730836 0.682553i \(-0.239130\pi\)
−0.730836 + 0.682553i \(0.760870\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −2.33275 −2.33275
\(707\) 0 0
\(708\) 0 0
\(709\) 1.34066 1.25209i 1.34066 1.25209i 0.398401 0.917211i \(-0.369565\pi\)
0.942261 0.334880i \(-0.108696\pi\)
\(710\) 0 0
\(711\) −0.519540 + 1.46184i −0.519540 + 1.46184i
\(712\) 0 0
\(713\) 0.908226 + 0.431933i 0.908226 + 0.431933i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.816970 0.576680i \(-0.804348\pi\)
0.816970 + 0.576680i \(0.195652\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.93189 0.198604i −1.93189 0.198604i
\(723\) 1.72708 0.971091i 1.72708 0.971091i
\(724\) −2.68064 + 1.63013i −2.68064 + 1.63013i
\(725\) 0 0
\(726\) 1.69292 + 0.232687i 1.69292 + 0.232687i
\(727\) 0 0 −0.953145 0.302515i \(-0.902174\pi\)
0.953145 + 0.302515i \(0.0978261\pi\)
\(728\) 0 0
\(729\) −0.398401 0.917211i −0.398401 0.917211i
\(730\) 0 0
\(731\) 0 0
\(732\) 2.02809 3.09208i 2.02809 3.09208i
\(733\) 0 0 0.236764 0.971567i \(-0.423913\pi\)
−0.236764 + 0.971567i \(0.576087\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.208330 + 0.0142502i 0.208330 + 0.0142502i
\(737\) 0 0
\(738\) 0 0
\(739\) 0.311173 + 1.11059i 0.311173 + 1.11059i 0.942261 + 0.334880i \(0.108696\pi\)
−0.631088 + 0.775711i \(0.717391\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.16467 + 0.883195i 1.16467 + 0.883195i 0.994757 0.102264i \(-0.0326087\pi\)
0.169910 + 0.985460i \(0.445652\pi\)
\(744\) −0.664598 1.87000i −0.664598 1.87000i
\(745\) 0 0
\(746\) 0 0
\(747\) 0.158575 + 0.650716i 0.158575 + 0.650716i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.99534i 1.99534i 0.0682424 + 0.997669i \(0.478261\pi\)
−0.0682424 + 0.997669i \(0.521739\pi\)
\(752\) −0.0784130 0.762749i −0.0784130 0.762749i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.971567 0.236764i \(-0.0760870\pi\)
−0.971567 + 0.236764i \(0.923913\pi\)
\(758\) 0.100169 + 0.210625i 0.100169 + 0.210625i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.990686 0.136167i \(-0.0434783\pi\)
−0.990686 + 0.136167i \(0.956522\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.13658 1.61017i 1.13658 1.61017i
\(767\) 0 0
\(768\) −1.23841 1.42027i −1.23841 1.42027i
\(769\) −0.256797 0.315646i −0.256797 0.315646i 0.631088 0.775711i \(-0.282609\pi\)
−0.887885 + 0.460065i \(0.847826\pi\)
\(770\) 0 0
\(771\) −1.01743 1.67310i −1.01743 1.67310i
\(772\) 0 0
\(773\) 0.733164 0.555976i 0.733164 0.555976i −0.169910 0.985460i \(-0.554348\pi\)
0.903075 + 0.429483i \(0.141304\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −0.0566599 + 0.551149i −0.0566599 + 0.551149i
\(783\) 0 0
\(784\) −0.680803 0.352764i −0.680803 0.352764i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.0341411 0.999417i \(-0.510870\pi\)
0.0341411 + 0.999417i \(0.489130\pi\)
\(788\) 3.67847 0.378158i 3.67847 0.378158i
\(789\) −1.49867 + 1.05788i −1.49867 + 1.05788i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.521696 0.0356850i 0.521696 0.0356850i
\(797\) −1.65196 + 0.651449i −1.65196 + 0.651449i −0.994757 0.102264i \(-0.967391\pi\)
−0.657204 + 0.753713i \(0.728261\pi\)
\(798\) 0 0
\(799\) −0.405963 + 0.0277687i −0.405963 + 0.0277687i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.576680 0.816970i \(-0.695652\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(810\) 0 0
\(811\) −0.787230 + 0.842917i −0.787230 + 0.842917i −0.990686 0.136167i \(-0.956522\pi\)
0.203456 + 0.979084i \(0.434783\pi\)
\(812\) 0 0
\(813\) −0.231946 1.34526i −0.231946 1.34526i
\(814\) 0 0
\(815\) 0 0
\(816\) 0.242028 0.196904i 0.242028 0.196904i
\(817\) 0 0
\(818\) −2.94823 0.718463i −2.94823 0.718463i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.887885 0.460065i \(-0.152174\pi\)
−0.887885 + 0.460065i \(0.847826\pi\)
\(822\) −1.18140 + 2.99582i −1.18140 + 2.99582i
\(823\) 0 0 0.366854 0.930278i \(-0.380435\pi\)
−0.366854 + 0.930278i \(0.619565\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −0.893968 0.217854i −0.893968 0.217854i −0.236764 0.971567i \(-0.576087\pi\)
−0.657204 + 0.753713i \(0.728261\pi\)
\(828\) −0.749851 1.33361i −0.749851 1.33361i
\(829\) −0.806094 + 0.655806i −0.806094 + 0.655806i −0.942261 0.334880i \(-0.891304\pi\)
0.136167 + 0.990686i \(0.456522\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.199432 + 0.354689i −0.199432 + 0.354689i
\(834\) −2.07121 + 2.21773i −2.07121 + 2.21773i
\(835\) 0 0
\(836\) 0 0
\(837\) 0.762653 1.00571i 0.762653 1.00571i
\(838\) 0 0
\(839\) 0 0 0.203456 0.979084i \(-0.434783\pi\)
−0.203456 + 0.979084i \(0.565217\pi\)
\(840\) 0 0
\(841\) 0.854419 + 0.519584i 0.854419 + 0.519584i
\(842\) 3.40772 + 0.116411i 3.40772 + 0.116411i
\(843\) 0 0
\(844\) 3.51412 + 1.52640i 3.51412 + 1.52640i
\(845\) 0 0
\(846\) 1.32557 1.07843i 1.32557 1.07843i
\(847\) 0 0
\(848\) −1.41333 + 0.557345i −1.41333 + 0.557345i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.548452 0.836182i \(-0.684783\pi\)
0.548452 + 0.836182i \(0.315217\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.56314 + 1.80926i −2.56314 + 1.80926i
\(857\) −1.98488 + 0.204051i −1.98488 + 0.204051i −0.985460 + 0.169910i \(0.945652\pi\)
−0.999417 + 0.0341411i \(0.989130\pi\)
\(858\) 0 0
\(859\) −1.00063 + 0.280364i −1.00063 + 0.280364i −0.730836 0.682553i \(-0.760870\pi\)
−0.269797 + 0.962917i \(0.586957\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.158655 + 1.54329i −0.158655 + 1.54329i 0.548452 + 0.836182i \(0.315217\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0.0707053 0.252350i 0.0707053 0.252350i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.504189 + 0.664872i 0.504189 + 0.664872i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 2.36104 1.79043i 2.36104 1.79043i
\(873\) 0 0
\(874\) 1.03409 + 1.70048i 1.03409 + 1.70048i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.657204 0.753713i \(-0.728261\pi\)
0.657204 + 0.753713i \(0.271739\pi\)
\(878\) 0.346231 + 1.09088i 0.346231 + 1.09088i
\(879\) −0.599268 + 0.848969i −0.599268 + 0.848969i
\(880\) 0 0
\(881\) 0 0 −0.730836 0.682553i \(-0.760870\pi\)
0.730836 + 0.682553i \(0.239130\pi\)
\(882\) −0.174753 1.69988i −0.174753 1.69988i
\(883\) 0 0 −0.985460 0.169910i \(-0.945652\pi\)
0.985460 + 0.169910i \(0.0543478\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.15299 + 0.655198i 3.15299 + 0.655198i
\(887\) −0.627764 1.32000i −0.627764 1.32000i −0.930278 0.366854i \(-0.880435\pi\)
0.302515 0.953145i \(-0.402174\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.10168 + 0.960616i −1.10168 + 0.960616i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0.269995 + 0.759692i 0.269995 + 0.759692i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.99290 1.62135i −1.99290 1.62135i
\(905\) 0 0
\(906\) −0.248774 0.887885i −0.248774 0.887885i
\(907\) 0 0 0.985460 0.169910i \(-0.0543478\pi\)
−0.985460 + 0.169910i \(0.945652\pi\)
\(908\) −0.987544 + 0.313432i −0.987544 + 0.313432i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.962917 0.269797i \(-0.913043\pi\)
0.962917 + 0.269797i \(0.0869565\pi\)
\(912\) 0.265357 1.08890i 0.265357 1.08890i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.412779 + 0.950313i 0.412779 + 0.950313i
\(917\) 0 0
\(918\) 0.662766 + 0.210353i 0.662766 + 0.210353i
\(919\) 0.789381 + 0.108498i 0.789381 + 0.108498i 0.519584 0.854419i \(-0.326087\pi\)
0.269797 + 0.962917i \(0.413043\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 0.631088 0.775711i \(-0.282609\pi\)
−0.631088 + 0.775711i \(0.717391\pi\)
\(930\) 0 0
\(931\) 0.199031 + 1.44806i 0.199031 + 1.44806i
\(932\) 0.472232 + 0.224583i 0.472232 + 0.224583i
\(933\) 0 0
\(934\) 1.01619 2.85930i 1.01619 2.85930i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.753713 0.657204i \(-0.771739\pi\)
0.753713 + 0.657204i \(0.228261\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.769396 + 0.504647i −0.769396 + 0.504647i −0.871660 0.490110i \(-0.836957\pi\)
0.102264 + 0.994757i \(0.467391\pi\)
\(948\) 2.69020 + 1.27940i 2.69020 + 1.27940i
\(949\) 0 0
\(950\) 0 0
\(951\) 1.25923 1.54781i 1.25923 1.54781i
\(952\) 0 0
\(953\) −0.905802 0.509307i −0.905802 0.509307i −0.0341411 0.999417i \(-0.510870\pi\)
−0.871660 + 0.490110i \(0.836957\pi\)
\(954\) −2.76613 1.95255i −2.76613 1.95255i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.587564 + 0.0807588i 0.587564 + 0.0807588i
\(962\) 0 0
\(963\) −1.85622 0.731998i −1.85622 0.731998i
\(964\) −1.51571 3.48952i −1.51571 3.48952i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.548452 0.836182i \(-0.315217\pi\)
−0.548452 + 0.836182i \(0.684783\pi\)
\(968\) 0.372277 1.52765i 0.372277 1.52765i
\(969\) −0.572716 0.160468i −0.572716 0.160468i
\(970\) 0 0
\(971\) 0 0 −0.997669 0.0682424i \(-0.978261\pi\)
0.997669 + 0.0682424i \(0.0217391\pi\)
\(972\) −1.83016 + 0.580867i −1.83016 + 0.580867i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.14547 0.931909i −1.14547 0.931909i
\(977\) −0.195968 + 1.13659i −0.195968 + 1.13659i 0.707107 + 0.707107i \(0.250000\pi\)
−0.903075 + 0.429483i \(0.858696\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.77571 + 0.631088i 1.77571 + 0.631088i
\(982\) 0 0
\(983\) −0.0184223 + 0.539279i −0.0184223 + 0.539279i 0.953145 + 0.302515i \(0.0978261\pi\)
−0.971567 + 0.236764i \(0.923913\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.0931581 0.0997480i −0.0931581 0.0997480i 0.682553 0.730836i \(-0.260870\pi\)
−0.775711 + 0.631088i \(0.782609\pi\)
\(992\) 0.321372 0.0783162i 0.321372 0.0783162i
\(993\) 0.287650 + 0.604843i 0.287650 + 0.604843i
\(994\) 0 0
\(995\) 0 0
\(996\) 1.27405 0.175114i 1.27405 0.175114i
\(997\) 0 0 −0.985460 0.169910i \(-0.945652\pi\)
0.985460 + 0.169910i \(0.0543478\pi\)
\(998\) 0.139244 + 1.35447i 0.139244 + 1.35447i
\(999\) 0 0
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 3525.1.bg.a.368.1 yes 88
3.2 odd 2 inner 3525.1.bg.a.368.2 yes 88
5.2 odd 4 inner 3525.1.bg.a.932.2 yes 88
5.3 odd 4 inner 3525.1.bg.a.932.1 yes 88
5.4 even 2 inner 3525.1.bg.a.368.2 yes 88
15.2 even 4 inner 3525.1.bg.a.932.1 yes 88
15.8 even 4 inner 3525.1.bg.a.932.2 yes 88
15.14 odd 2 CM 3525.1.bg.a.368.1 yes 88
47.41 odd 46 inner 3525.1.bg.a.3143.1 yes 88
141.41 even 46 inner 3525.1.bg.a.3143.2 yes 88
235.88 even 92 inner 3525.1.bg.a.182.1 88
235.182 even 92 inner 3525.1.bg.a.182.2 yes 88
235.229 odd 46 inner 3525.1.bg.a.3143.2 yes 88
705.182 odd 92 inner 3525.1.bg.a.182.1 88
705.323 odd 92 inner 3525.1.bg.a.182.2 yes 88
705.464 even 46 inner 3525.1.bg.a.3143.1 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.1.bg.a.182.1 88 235.88 even 92 inner
3525.1.bg.a.182.1 88 705.182 odd 92 inner
3525.1.bg.a.182.2 yes 88 235.182 even 92 inner
3525.1.bg.a.182.2 yes 88 705.323 odd 92 inner
3525.1.bg.a.368.1 yes 88 1.1 even 1 trivial
3525.1.bg.a.368.1 yes 88 15.14 odd 2 CM
3525.1.bg.a.368.2 yes 88 3.2 odd 2 inner
3525.1.bg.a.368.2 yes 88 5.4 even 2 inner
3525.1.bg.a.932.1 yes 88 5.3 odd 4 inner
3525.1.bg.a.932.1 yes 88 15.2 even 4 inner
3525.1.bg.a.932.2 yes 88 5.2 odd 4 inner
3525.1.bg.a.932.2 yes 88 15.8 even 4 inner
3525.1.bg.a.3143.1 yes 88 47.41 odd 46 inner
3525.1.bg.a.3143.1 yes 88 705.464 even 46 inner
3525.1.bg.a.3143.2 yes 88 141.41 even 46 inner
3525.1.bg.a.3143.2 yes 88 235.229 odd 46 inner