Properties

Label 3520.1.db.a.549.3
Level $3520$
Weight $1$
Character 3520.549
Analytic conductor $1.757$
Analytic rank $0$
Dimension $32$
Projective image $D_{32}$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3520,1,Mod(109,3520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3520, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 7, 8, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3520.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3520.db (of order \(16\), degree \(8\), 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: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{64})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

Embedding invariants

Embedding label 549.3
Root \(0.956940 - 0.290285i\) of defining polynomial
Character \(\chi\) \(=\) 3520.549
Dual form 3520.1.db.a.109.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0980171 + 0.995185i) q^{2} +(-0.980785 + 0.195090i) q^{4} +(0.195090 - 0.980785i) q^{5} +(-0.222174 + 0.536376i) q^{7} +(-0.290285 - 0.956940i) q^{8} +(-0.382683 - 0.923880i) q^{9} +O(q^{10})\) \(q+(0.0980171 + 0.995185i) q^{2} +(-0.980785 + 0.195090i) q^{4} +(0.195090 - 0.980785i) q^{5} +(-0.222174 + 0.536376i) q^{7} +(-0.290285 - 0.956940i) q^{8} +(-0.382683 - 0.923880i) q^{9} +(0.995185 + 0.0980171i) q^{10} +(-0.831470 + 0.555570i) q^{11} +(0.247528 + 1.24441i) q^{13} +(-0.555570 - 0.168530i) q^{14} +(0.923880 - 0.382683i) q^{16} +(-0.666656 + 0.666656i) q^{17} +(0.881921 - 0.471397i) q^{18} +1.00000i q^{20} +(-0.634393 - 0.773010i) q^{22} +(-0.923880 - 0.382683i) q^{25} +(-1.21415 + 0.368309i) q^{26} +(0.113263 - 0.569414i) q^{28} +1.66294i q^{31} +(0.471397 + 0.881921i) q^{32} +(-0.728789 - 0.598102i) q^{34} +(0.482726 + 0.322547i) q^{35} +(0.555570 + 0.831470i) q^{36} +(-0.995185 + 0.0980171i) q^{40} +(1.10579 + 1.65493i) q^{43} +(0.707107 - 0.707107i) q^{44} +(-0.980785 + 0.195090i) q^{45} +(0.468769 + 0.468769i) q^{49} +(0.290285 - 0.956940i) q^{50} +(-0.485544 - 1.17221i) q^{52} +(0.382683 + 0.923880i) q^{55} +(0.577774 + 0.0569057i) q^{56} +(-0.0761205 + 0.382683i) q^{59} +(-1.65493 + 0.162997i) q^{62} +0.580569 q^{63} +(-0.831470 + 0.555570i) q^{64} +1.26879 q^{65} +(0.523788 - 0.783904i) q^{68} +(-0.273678 + 0.512016i) q^{70} +(-0.292893 + 0.707107i) q^{71} +(-0.773010 + 0.634393i) q^{72} +(0.0750191 + 0.181112i) q^{73} +(-0.113263 - 0.569414i) q^{77} +(-0.195090 - 0.980785i) q^{80} +(-0.707107 + 0.707107i) q^{81} +(-1.24441 + 0.247528i) q^{83} +(0.523788 + 0.783904i) q^{85} +(-1.53858 + 1.26268i) q^{86} +(0.773010 + 0.634393i) q^{88} +(-1.81225 - 0.750661i) q^{89} +(-0.290285 - 0.956940i) q^{90} +(-0.722465 - 0.143707i) q^{91} +(-0.420564 + 0.512459i) q^{98} +(0.831470 + 0.555570i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{59} - 32 q^{71}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{16}\right)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(3\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(4\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(5\) 0.195090 0.980785i 0.195090 0.980785i
\(6\) 0 0
\(7\) −0.222174 + 0.536376i −0.222174 + 0.536376i −0.995185 0.0980171i \(-0.968750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(8\) −0.290285 0.956940i −0.290285 0.956940i
\(9\) −0.382683 0.923880i −0.382683 0.923880i
\(10\) 0.995185 + 0.0980171i 0.995185 + 0.0980171i
\(11\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(12\) 0 0
\(13\) 0.247528 + 1.24441i 0.247528 + 1.24441i 0.881921 + 0.471397i \(0.156250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(14\) −0.555570 0.168530i −0.555570 0.168530i
\(15\) 0 0
\(16\) 0.923880 0.382683i 0.923880 0.382683i
\(17\) −0.666656 + 0.666656i −0.666656 + 0.666656i −0.956940 0.290285i \(-0.906250\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(18\) 0.881921 0.471397i 0.881921 0.471397i
\(19\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(20\) 1.00000i 1.00000i
\(21\) 0 0
\(22\) −0.634393 0.773010i −0.634393 0.773010i
\(23\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(24\) 0 0
\(25\) −0.923880 0.382683i −0.923880 0.382683i
\(26\) −1.21415 + 0.368309i −1.21415 + 0.368309i
\(27\) 0 0
\(28\) 0.113263 0.569414i 0.113263 0.569414i
\(29\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(30\) 0 0
\(31\) 1.66294i 1.66294i 0.555570 + 0.831470i \(0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(32\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(33\) 0 0
\(34\) −0.728789 0.598102i −0.728789 0.598102i
\(35\) 0.482726 + 0.322547i 0.482726 + 0.322547i
\(36\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(37\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.995185 + 0.0980171i −0.995185 + 0.0980171i
\(41\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(42\) 0 0
\(43\) 1.10579 + 1.65493i 1.10579 + 1.65493i 0.634393 + 0.773010i \(0.281250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(44\) 0.707107 0.707107i 0.707107 0.707107i
\(45\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 0.468769 + 0.468769i 0.468769 + 0.468769i
\(50\) 0.290285 0.956940i 0.290285 0.956940i
\(51\) 0 0
\(52\) −0.485544 1.17221i −0.485544 1.17221i
\(53\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(54\) 0 0
\(55\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(56\) 0.577774 + 0.0569057i 0.577774 + 0.0569057i
\(57\) 0 0
\(58\) 0 0
\(59\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i 0.923880 + 0.382683i \(0.125000\pi\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(62\) −1.65493 + 0.162997i −1.65493 + 0.162997i
\(63\) 0.580569 0.580569
\(64\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(65\) 1.26879 1.26879
\(66\) 0 0
\(67\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(68\) 0.523788 0.783904i 0.523788 0.783904i
\(69\) 0 0
\(70\) −0.273678 + 0.512016i −0.273678 + 0.512016i
\(71\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(72\) −0.773010 + 0.634393i −0.773010 + 0.634393i
\(73\) 0.0750191 + 0.181112i 0.0750191 + 0.181112i 0.956940 0.290285i \(-0.0937500\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.113263 0.569414i −0.113263 0.569414i
\(78\) 0 0
\(79\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(80\) −0.195090 0.980785i −0.195090 0.980785i
\(81\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(82\) 0 0
\(83\) −1.24441 + 0.247528i −1.24441 + 0.247528i −0.773010 0.634393i \(-0.781250\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(84\) 0 0
\(85\) 0.523788 + 0.783904i 0.523788 + 0.783904i
\(86\) −1.53858 + 1.26268i −1.53858 + 1.26268i
\(87\) 0 0
\(88\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(89\) −1.81225 0.750661i −1.81225 0.750661i −0.980785 0.195090i \(-0.937500\pi\)
−0.831470 0.555570i \(-0.812500\pi\)
\(90\) −0.290285 0.956940i −0.290285 0.956940i
\(91\) −0.722465 0.143707i −0.722465 0.143707i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −0.420564 + 0.512459i −0.420564 + 0.512459i
\(99\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(100\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(101\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(104\) 1.11897 0.598102i 1.11897 0.598102i
\(105\) 0 0
\(106\) 0 0
\(107\) 0.979938 + 1.46658i 0.979938 + 1.46658i 0.881921 + 0.471397i \(0.156250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(108\) 0 0
\(109\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(110\) −0.881921 + 0.471397i −0.881921 + 0.471397i
\(111\) 0 0
\(112\) 0.580569i 0.580569i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.05496 0.704900i 1.05496 0.704900i
\(118\) −0.388302 0.0382444i −0.388302 0.0382444i
\(119\) −0.209464 0.505692i −0.209464 0.505692i
\(120\) 0 0
\(121\) 0.382683 0.923880i 0.382683 0.923880i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.324423 1.63099i −0.324423 1.63099i
\(125\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(126\) 0.0569057 + 0.577774i 0.0569057 + 0.577774i
\(127\) −0.196034 −0.196034 −0.0980171 0.995185i \(-0.531250\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(128\) −0.634393 0.773010i −0.634393 0.773010i
\(129\) 0 0
\(130\) 0.124363 + 1.26268i 0.124363 + 1.26268i
\(131\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.831470 + 0.444430i 0.831470 + 0.444430i
\(137\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(138\) 0 0
\(139\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(140\) −0.536376 0.222174i −0.536376 0.222174i
\(141\) 0 0
\(142\) −0.732410 0.222174i −0.732410 0.222174i
\(143\) −0.897168 0.897168i −0.897168 0.897168i
\(144\) −0.707107 0.707107i −0.707107 0.707107i
\(145\) 0 0
\(146\) −0.172887 + 0.0924099i −0.172887 + 0.0924099i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(150\) 0 0
\(151\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(152\) 0 0
\(153\) 0.871028 + 0.360791i 0.871028 + 0.360791i
\(154\) 0.555570 0.168530i 0.555570 0.168530i
\(155\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(156\) 0 0
\(157\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.956940 0.290285i 0.956940 0.290285i
\(161\) 0 0
\(162\) −0.773010 0.634393i −0.773010 0.634393i
\(163\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −0.368309 1.21415i −0.368309 1.21415i
\(167\) −1.42834 0.591637i −1.42834 0.591637i −0.471397 0.881921i \(-0.656250\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(168\) 0 0
\(169\) −0.563400 + 0.233368i −0.563400 + 0.233368i
\(170\) −0.728789 + 0.598102i −0.728789 + 0.598102i
\(171\) 0 0
\(172\) −1.40740 1.40740i −1.40740 1.40740i
\(173\) 1.72995 0.344109i 1.72995 0.344109i 0.773010 0.634393i \(-0.218750\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(174\) 0 0
\(175\) 0.410525 0.410525i 0.410525 0.410525i
\(176\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(177\) 0 0
\(178\) 0.569414 1.87711i 0.569414 1.87711i
\(179\) −0.149316 0.750661i −0.149316 0.750661i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(180\) 0.923880 0.382683i 0.923880 0.382683i
\(181\) 1.53636 1.02656i 1.53636 1.02656i 0.555570 0.831470i \(-0.312500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(182\) 0.0722012 0.733072i 0.0722012 0.733072i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.183930 0.924678i 0.183930 0.924678i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(192\) 0 0
\(193\) −1.99037 −1.99037 −0.995185 0.0980171i \(-0.968750\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.551214 0.368309i −0.551214 0.368309i
\(197\) 0.373380 1.87711i 0.373380 1.87711i −0.0980171 0.995185i \(-0.531250\pi\)
0.471397 0.881921i \(-0.343750\pi\)
\(198\) −0.471397 + 0.881921i −0.471397 + 0.881921i
\(199\) 0.707107 1.70711i 0.707107 1.70711i 1.00000i \(-0.5\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(200\) −0.0980171 + 0.995185i −0.0980171 + 0.995185i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.704900 + 1.05496i 0.704900 + 1.05496i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.36347 + 1.11897i −1.36347 + 1.11897i
\(215\) 1.83886 0.761681i 1.83886 0.761681i
\(216\) 0 0
\(217\) −0.891961 0.369462i −0.891961 0.369462i
\(218\) 0 0
\(219\) 0 0
\(220\) −0.555570 0.831470i −0.555570 0.831470i
\(221\) −0.994607 0.664575i −0.994607 0.664575i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −0.577774 + 0.0569057i −0.577774 + 0.0569057i
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) 1.28547 + 0.858923i 1.28547 + 0.858923i 0.995185 0.0980171i \(-0.0312500\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(228\) 0 0
\(229\) −1.08979 0.216773i −1.08979 0.216773i −0.382683 0.923880i \(-0.625000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.76820 + 0.732410i −1.76820 + 0.732410i −0.773010 + 0.634393i \(0.781250\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(234\) 0.804910 + 0.980785i 0.804910 + 0.980785i
\(235\) 0 0
\(236\) 0.390181i 0.390181i
\(237\) 0 0
\(238\) 0.482726 0.258022i 0.482726 0.258022i
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(242\) 0.956940 + 0.290285i 0.956940 + 0.290285i
\(243\) 0 0
\(244\) 0 0
\(245\) 0.551214 0.368309i 0.551214 0.368309i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.59133 0.482726i 1.59133 0.482726i
\(249\) 0 0
\(250\) −0.881921 0.471397i −0.881921 0.471397i
\(251\) −0.382683 + 1.92388i −0.382683 + 1.92388i 1.00000i \(0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(252\) −0.569414 + 0.113263i −0.569414 + 0.113263i
\(253\) 0 0
\(254\) −0.0192147 0.195090i −0.0192147 0.195090i
\(255\) 0 0
\(256\) 0.707107 0.707107i 0.707107 0.707107i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.24441 + 0.247528i −1.24441 + 0.247528i
\(261\) 0 0
\(262\) 0 0
\(263\) 0.761681 1.83886i 0.761681 1.83886i 0.290285 0.956940i \(-0.406250\pi\)
0.471397 0.881921i \(-0.343750\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i 1.00000 \(0\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(272\) −0.360791 + 0.871028i −0.360791 + 0.871028i
\(273\) 0 0
\(274\) 0 0
\(275\) 0.980785 0.195090i 0.980785 0.195090i
\(276\) 0 0
\(277\) 1.10579 + 1.65493i 1.10579 + 1.65493i 0.634393 + 0.773010i \(0.281250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(278\) 0 0
\(279\) 1.53636 0.636379i 1.53636 0.636379i
\(280\) 0.168530 0.555570i 0.168530 0.555570i
\(281\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(282\) 0 0
\(283\) −1.51631 0.301614i −1.51631 0.301614i −0.634393 0.773010i \(-0.718750\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(284\) 0.149316 0.750661i 0.149316 0.750661i
\(285\) 0 0
\(286\) 0.804910 0.980785i 0.804910 0.980785i
\(287\) 0 0
\(288\) 0.634393 0.773010i 0.634393 0.773010i
\(289\) 0.111140i 0.111140i
\(290\) 0 0
\(291\) 0 0
\(292\) −0.108911 0.162997i −0.108911 0.162997i
\(293\) 0.192268 + 0.0382444i 0.192268 + 0.0382444i 0.290285 0.956940i \(-0.406250\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(294\) 0 0
\(295\) 0.360480 + 0.149316i 0.360480 + 0.149316i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.13334 + 0.225436i −1.13334 + 0.225436i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −0.273678 + 0.902197i −0.273678 + 0.902197i
\(307\) 0.183930 + 0.924678i 0.183930 + 0.924678i 0.956940 + 0.290285i \(0.0937500\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(308\) 0.222174 + 0.536376i 0.222174 + 0.536376i
\(309\) 0 0
\(310\) −0.162997 + 1.65493i −0.162997 + 1.65493i
\(311\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(312\) 0 0
\(313\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) 0 0
\(315\) 0.113263 0.569414i 0.113263 0.569414i
\(316\) 0 0
\(317\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.555570 0.831470i 0.555570 0.831470i
\(325\) 0.247528 1.24441i 0.247528 1.24441i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.17588 0.785695i 1.17588 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(332\) 1.17221 0.485544i 1.17221 0.485544i
\(333\) 0 0
\(334\) 0.448786 1.47945i 0.448786 1.47945i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.35332 + 1.35332i −1.35332 + 1.35332i −0.471397 + 0.881921i \(0.656250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(338\) −0.287467 0.537813i −0.287467 0.537813i
\(339\) 0 0
\(340\) −0.666656 0.666656i −0.666656 0.666656i
\(341\) −0.923880 1.38268i −0.923880 1.38268i
\(342\) 0 0
\(343\) −0.891961 + 0.369462i −0.891961 + 0.369462i
\(344\) 1.26268 1.53858i 1.26268 1.53858i
\(345\) 0 0
\(346\) 0.512016 + 1.68789i 0.512016 + 1.68789i
\(347\) −0.192268 0.0382444i −0.192268 0.0382444i 0.0980171 0.995185i \(-0.468750\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(348\) 0 0
\(349\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(350\) 0.448786 + 0.368309i 0.448786 + 0.368309i
\(351\) 0 0
\(352\) −0.881921 0.471397i −0.881921 0.471397i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(356\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(357\) 0 0
\(358\) 0.732410 0.222174i 0.732410 0.222174i
\(359\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(360\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(361\) 0.923880 0.382683i 0.923880 0.382683i
\(362\) 1.17221 + 1.42834i 1.17221 + 1.42834i
\(363\) 0 0
\(364\) 0.736619 0.736619
\(365\) 0.192268 0.0382444i 0.192268 0.0382444i
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.59133 + 1.06330i −1.59133 + 1.06330i −0.634393 + 0.773010i \(0.718750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(374\) 0.938254 + 0.0924099i 0.938254 + 0.0924099i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.382683 1.92388i 0.382683 1.92388i 1.00000i \(-0.5\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.0382444 + 0.388302i 0.0382444 + 0.388302i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −0.580569 −0.580569
\(386\) −0.195090 1.98079i −0.195090 1.98079i
\(387\) 1.10579 1.65493i 1.10579 1.65493i
\(388\) 0 0
\(389\) 0.216773 1.08979i 0.216773 1.08979i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.312507 0.584660i 0.312507 0.584660i
\(393\) 0 0
\(394\) 1.90466 + 0.187593i 1.90466 + 0.187593i
\(395\) 0 0
\(396\) −0.923880 0.382683i −0.923880 0.382683i
\(397\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(398\) 1.76820 + 0.536376i 1.76820 + 0.536376i
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(402\) 0 0
\(403\) −2.06937 + 0.411624i −2.06937 + 0.411624i
\(404\) 0 0
\(405\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.188350 0.125852i −0.188350 0.125852i
\(414\) 0 0
\(415\) 1.26879i 1.26879i
\(416\) −0.980785 + 0.804910i −0.980785 + 0.804910i
\(417\) 0 0
\(418\) 0 0
\(419\) 1.17588 + 0.785695i 1.17588 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(420\) 0 0
\(421\) 1.81225 + 0.360480i 1.81225 + 0.360480i 0.980785 0.195090i \(-0.0625000\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.871028 0.360791i 0.871028 0.360791i
\(426\) 0 0
\(427\) 0 0
\(428\) −1.24723 1.24723i −1.24723 1.24723i
\(429\) 0 0
\(430\) 0.938254 + 1.75535i 0.938254 + 1.75535i
\(431\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0.280256 0.923880i 0.280256 0.923880i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(440\) 0.773010 0.634393i 0.773010 0.634393i
\(441\) 0.253696 0.612476i 0.253696 0.612476i
\(442\) 0.563887 1.05496i 0.563887 1.05496i
\(443\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(444\) 0 0
\(445\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(446\) 0 0
\(447\) 0 0
\(448\) −0.113263 0.569414i −0.113263 0.569414i
\(449\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(450\) −0.995185 + 0.0980171i −0.995185 + 0.0980171i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −0.728789 + 1.36347i −0.728789 + 1.36347i
\(455\) −0.281892 + 0.680547i −0.281892 + 0.680547i
\(456\) 0 0
\(457\) 0.591637 + 1.42834i 0.591637 + 1.42834i 0.881921 + 0.471397i \(0.156250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(458\) 0.108911 1.10579i 0.108911 1.10579i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.902197 1.68789i −0.902197 1.68789i
\(467\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(468\) −0.897168 + 0.897168i −0.897168 + 0.897168i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.388302 0.0382444i 0.388302 0.0382444i
\(473\) −1.83886 0.761681i −1.83886 0.761681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0.304095 + 0.455111i 0.304095 + 0.455111i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.420564 + 0.512459i 0.420564 + 0.512459i
\(491\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0.707107 0.707107i 0.707107 0.707107i
\(496\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(497\) −0.314202 0.314202i −0.314202 0.314202i
\(498\) 0 0
\(499\) 0.324423 + 1.63099i 0.324423 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(500\) 0.382683 0.923880i 0.382683 0.923880i
\(501\) 0 0
\(502\) −1.95213 0.192268i −1.95213 0.192268i
\(503\) 0.360791 + 0.871028i 0.360791 + 0.871028i 0.995185 + 0.0980171i \(0.0312500\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(504\) −0.168530 0.555570i −0.168530 0.555570i
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0.192268 0.0382444i 0.192268 0.0382444i
\(509\) 1.08979 1.63099i 1.08979 1.63099i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(510\) 0 0
\(511\) −0.113811 −0.113811
\(512\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.368309 1.21415i −0.368309 1.21415i
\(521\) −0.425215 1.02656i −0.425215 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(522\) 0 0
\(523\) −1.59133 + 1.06330i −1.59133 + 1.06330i −0.634393 + 0.773010i \(0.718750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.90466 + 0.577774i 1.90466 + 0.577774i
\(527\) −1.10861 1.10861i −1.10861 1.10861i
\(528\) 0 0
\(529\) 0.707107 0.707107i 0.707107 0.707107i
\(530\) 0 0
\(531\) 0.382683 0.0761205i 0.382683 0.0761205i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.62958 0.674993i 1.62958 0.674993i
\(536\) 0 0
\(537\) 0 0
\(538\) −0.373380 + 0.113263i −0.373380 + 0.113263i
\(539\) −0.650201 0.129333i −0.650201 0.129333i
\(540\) 0 0
\(541\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.902197 0.273678i −0.902197 0.273678i
\(545\) 0 0
\(546\) 0 0
\(547\) 1.46658 + 0.979938i 1.46658 + 0.979938i 0.995185 + 0.0980171i \(0.0312500\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.290285 + 0.956940i 0.290285 + 0.956940i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.53858 + 1.26268i −1.53858 + 1.26268i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.95213 0.388302i 1.95213 0.388302i 0.956940 0.290285i \(-0.0937500\pi\)
0.995185 0.0980171i \(-0.0312500\pi\)
\(558\) 0.783904 + 1.46658i 0.783904 + 1.46658i
\(559\) −1.78569 + 1.78569i −1.78569 + 1.78569i
\(560\) 0.569414 + 0.113263i 0.569414 + 0.113263i
\(561\) 0 0
\(562\) 0 0
\(563\) 0.373380 + 1.87711i 0.373380 + 1.87711i 0.471397 + 0.881921i \(0.343750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.151537 1.53858i 0.151537 1.53858i
\(567\) −0.222174 0.536376i −0.222174 0.536376i
\(568\) 0.761681 + 0.0750191i 0.761681 + 0.0750191i
\(569\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(570\) 0 0
\(571\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(572\) 1.05496 + 0.704900i 1.05496 + 0.704900i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.110605 + 0.0108937i −0.110605 + 0.0108937i
\(579\) 0 0
\(580\) 0 0
\(581\) 0.143707 0.722465i 0.143707 0.722465i
\(582\) 0 0
\(583\) 0 0
\(584\) 0.151537 0.124363i 0.151537 0.124363i
\(585\) −0.485544 1.17221i −0.485544 1.17221i
\(586\) −0.0192147 + 0.195090i −0.0192147 + 0.195090i
\(587\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −0.113263 + 0.373380i −0.113263 + 0.373380i
\(591\) 0 0
\(592\) 0 0
\(593\) 0.410525 0.410525i 0.410525 0.410525i −0.471397 0.881921i \(-0.656250\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(594\) 0 0
\(595\) −0.536840 + 0.106784i −0.536840 + 0.106784i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.02656 + 0.425215i −1.02656 + 0.425215i −0.831470 0.555570i \(-0.812500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(600\) 0 0
\(601\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(602\) −0.335438 1.10579i −0.335438 1.10579i
\(603\) 0 0
\(604\) 0 0
\(605\) −0.831470 0.555570i −0.831470 0.555570i
\(606\) 0 0
\(607\) 0.942793i 0.942793i −0.881921 0.471397i \(-0.843750\pi\)
0.881921 0.471397i \(-0.156250\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.924678 0.183930i −0.924678 0.183930i
\(613\) 1.72995 + 0.344109i 1.72995 + 0.344109i 0.956940 0.290285i \(-0.0937500\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(614\) −0.902197 + 0.273678i −0.902197 + 0.273678i
\(615\) 0 0
\(616\) −0.512016 + 0.273678i −0.512016 + 0.273678i
\(617\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(618\) 0 0
\(619\) −1.02656 1.53636i −1.02656 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(620\) −1.66294 −1.66294
\(621\) 0 0
\(622\) 0 0
\(623\) 0.805273 0.805273i 0.805273 0.805273i
\(624\) 0 0
\(625\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.577774 + 0.0569057i 0.577774 + 0.0569057i
\(631\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.0382444 + 0.192268i −0.0382444 + 0.192268i
\(636\) 0 0
\(637\) −0.467306 + 0.699373i −0.467306 + 0.699373i
\(638\) 0 0
\(639\) 0.765367 0.765367
\(640\) −0.881921 + 0.471397i −0.881921 + 0.471397i
\(641\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(642\) 0 0
\(643\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(648\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(649\) −0.149316 0.360480i −0.149316 0.360480i
\(650\) 1.26268 + 0.124363i 1.26268 + 0.124363i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.138617 0.138617i 0.138617 0.138617i
\(658\) 0 0
\(659\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(660\) 0 0
\(661\) 0.923880 + 1.38268i 0.923880 + 1.38268i 0.923880 + 0.382683i \(0.125000\pi\)
1.00000i \(0.5\pi\)
\(662\) 0.897168 + 1.09320i 0.897168 + 1.09320i
\(663\) 0 0
\(664\) 0.598102 + 1.11897i 0.598102 + 1.11897i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.51631 + 0.301614i 1.51631 + 0.301614i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.91388i 1.91388i −0.290285 0.956940i \(-0.593750\pi\)
0.290285 0.956940i \(-0.406250\pi\)
\(674\) −1.47945 1.21415i −1.47945 1.21415i
\(675\) 0 0
\(676\) 0.507046 0.338797i 0.507046 0.338797i
\(677\) −0.924678 0.183930i −0.924678 0.183930i −0.290285 0.956940i \(-0.593750\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.598102 0.728789i 0.598102 0.728789i
\(681\) 0 0
\(682\) 1.28547 1.05496i 1.28547 1.05496i
\(683\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.455111 0.851452i −0.455111 0.851452i
\(687\) 0 0
\(688\) 1.65493 + 1.10579i 1.65493 + 1.10579i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.216773 + 1.08979i 0.216773 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) −1.62958 + 0.674993i −1.62958 + 0.674993i
\(693\) −0.482726 + 0.322547i −0.482726 + 0.322547i
\(694\) 0.0192147 0.195090i 0.0192147 0.195090i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.322547 + 0.482726i −0.322547 + 0.482726i
\(701\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.382683 0.923880i 0.382683 0.923880i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.275899 + 1.38704i −0.275899 + 1.38704i 0.555570 + 0.831470i \(0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(710\) −0.360791 + 0.674993i −0.360791 + 0.674993i
\(711\) 0 0
\(712\) −0.192268 + 1.95213i −0.192268 + 1.95213i
\(713\) 0 0
\(714\) 0 0
\(715\) −1.05496 + 0.704900i −1.05496 + 0.704900i
\(716\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(717\) 0 0
\(718\) 0 0
\(719\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(720\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(721\) 0 0
\(722\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(723\) 0 0
\(724\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(728\) 0.0722012 + 0.733072i 0.0722012 + 0.733072i
\(729\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(730\) 0.0569057 + 0.187593i 0.0569057 + 0.187593i
\(731\) −1.84045 0.366088i −1.84045 0.366088i
\(732\) 0 0
\(733\) −0.482726 0.322547i −0.482726 0.322547i 0.290285 0.956940i \(-0.406250\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.62958 0.674993i −1.62958 0.674993i −0.634393 0.773010i \(-0.718750\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.21415 1.47945i −1.21415 1.47945i
\(747\) 0.704900 + 1.05496i 0.704900 + 1.05496i
\(748\) 0.942793i 0.942793i
\(749\) −1.00436 + 0.199779i −1.00436 + 0.199779i
\(750\) 0 0
\(751\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(758\) 1.95213 + 0.192268i 1.95213 + 0.192268i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(765\) 0.523788 0.783904i 0.523788 0.783904i
\(766\) 0 0
\(767\) −0.495056 −0.495056
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) −0.0569057 0.577774i −0.0569057 0.577774i
\(771\) 0 0
\(772\) 1.95213 0.388302i 1.95213 0.388302i
\(773\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(774\) 1.75535 + 0.938254i 1.75535 + 0.938254i
\(775\) 0.636379 1.53636i 0.636379 1.53636i
\(776\) 0 0
\(777\) 0 0
\(778\) 1.10579 + 0.108911i 1.10579 + 0.108911i
\(779\) 0 0
\(780\) 0 0
\(781\) −0.149316 0.750661i −0.149316 0.750661i
\(782\) 0 0
\(783\) 0 0
\(784\) 0.612476 + 0.253696i 0.612476 + 0.253696i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.72995 + 0.344109i −1.72995 + 0.344109i −0.956940 0.290285i \(-0.906250\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(788\) 1.91388i 1.91388i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.290285 0.956940i 0.290285 0.956940i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.360480 + 1.81225i −0.360480 + 1.81225i
\(797\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.0980171 0.995185i −0.0980171 0.995185i
\(801\) 1.96157i 1.96157i
\(802\) 0.301614 + 0.247528i 0.301614 + 0.247528i
\(803\) −0.162997 0.108911i −0.162997 0.108911i
\(804\) 0 0
\(805\) 0 0
\(806\) −0.612476 2.01906i −0.612476 2.01906i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(810\) −0.773010 + 0.634393i −0.773010 + 0.634393i
\(811\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.143707 + 0.722465i 0.143707 + 0.722465i
\(820\) 0 0
\(821\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(822\) 0 0
\(823\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0.106784 0.199779i 0.106784 0.199779i
\(827\) 0.247528 1.24441i 0.247528 1.24441i −0.634393 0.773010i \(-0.718750\pi\)
0.881921 0.471397i \(-0.156250\pi\)
\(828\) 0 0
\(829\) −0.785695 + 1.17588i −0.785695 + 1.17588i 0.195090 + 0.980785i \(0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(830\) −1.26268 + 0.124363i −1.26268 + 0.124363i
\(831\) 0 0
\(832\) −0.897168 0.897168i −0.897168 0.897168i
\(833\) −0.625015 −0.625015
\(834\) 0 0
\(835\) −0.858923 + 1.28547i −0.858923 + 1.28547i
\(836\) 0 0
\(837\) 0 0
\(838\) −0.666656 + 1.24723i −0.666656 + 1.24723i
\(839\) 0.292893 0.707107i 0.292893 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
1.00000 \(0\)
\(840\) 0 0
\(841\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(842\) −0.181112 + 1.83886i −0.181112 + 1.83886i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.118970 + 0.598102i 0.118970 + 0.598102i
\(846\) 0 0
\(847\) 0.410525 + 0.410525i 0.410525 + 0.410525i
\(848\) 0 0
\(849\) 0 0
\(850\) 0.444430 + 0.831470i 0.444430 + 0.831470i
\(851\) 0 0
\(852\) 0 0
\(853\) 0.108911 + 0.162997i 0.108911 + 0.162997i 0.881921 0.471397i \(-0.156250\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.11897 1.36347i 1.11897 1.36347i
\(857\) −0.181112 0.0750191i −0.181112 0.0750191i 0.290285 0.956940i \(-0.406250\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(858\) 0 0
\(859\) −1.38704 0.275899i −1.38704 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(860\) −1.65493 + 1.10579i −1.65493 + 1.10579i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 1.76384i 1.76384i
\(866\) 0 0
\(867\) 0 0
\(868\) 0.946901 + 0.188350i 0.946901 + 0.188350i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.322547 0.482726i −0.322547 0.482726i
\(876\) 0 0
\(877\) 0.924678 0.183930i 0.924678 0.183930i 0.290285 0.956940i \(-0.406250\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(881\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) 0.634393 + 0.192441i 0.634393 + 0.192441i
\(883\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(884\) 1.10515 + 0.457767i 1.10515 + 0.457767i
\(885\) 0 0
\(886\) 0 0
\(887\) −0.591637 1.42834i −0.591637 1.42834i −0.881921 0.471397i \(-0.843750\pi\)
0.290285 0.956940i \(-0.406250\pi\)
\(888\) 0 0
\(889\) 0.0435538 0.105148i 0.0435538 0.105148i
\(890\) −1.72995 0.924678i −1.72995 0.924678i
\(891\) 0.195090 0.980785i 0.195090 0.980785i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −0.765367 −0.765367
\(896\) 0.555570 0.168530i 0.555570 0.168530i
\(897\) 0 0
\(898\) −0.108911 1.10579i −0.108911 1.10579i
\(899\) 0 0
\(900\) −0.195090 0.980785i −0.195090 0.980785i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.707107 1.70711i −0.707107 1.70711i
\(906\) 0 0
\(907\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(908\) −1.42834 0.591637i −1.42834 0.591637i
\(909\) 0 0
\(910\) −0.704900 0.213829i −0.704900 0.213829i
\(911\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(912\) 0 0
\(913\) 0.897168 0.897168i 0.897168 0.897168i
\(914\) −1.36347 + 0.728789i −1.36347 + 0.728789i
\(915\) 0 0
\(916\) 1.11114 1.11114
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.952428 0.189450i −0.952428 0.189450i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.59133 1.06330i 1.59133 1.06330i
\(933\) 0 0
\(934\) 0 0
\(935\) −0.871028 0.360791i −0.871028 0.360791i
\(936\) −0.980785 0.804910i −0.980785 0.804910i
\(937\) 1.62958 0.674993i 1.62958 0.674993i 0.634393 0.773010i \(-0.281250\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(945\) 0 0
\(946\) 0.577774 1.90466i 0.577774 1.90466i
\(947\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(948\) 0 0
\(949\) −0.206808 + 0.138185i −0.206808 + 0.138185i
\(950\) 0 0
\(951\) 0 0
\(952\) −0.423113 + 0.347240i −0.423113 + 0.347240i
\(953\) −0.485544 + 1.17221i −0.485544 + 1.17221i 0.471397 + 0.881921i \(0.343750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(954\) 0 0
\(955\) 0.0761205 0.382683i 0.0761205 0.382683i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.76537 −1.76537
\(962\) 0 0
\(963\) 0.979938 1.46658i 0.979938 1.46658i
\(964\) 0 0
\(965\) −0.388302 + 1.95213i −0.388302 + 1.95213i
\(966\) 0 0
\(967\) 0.360791 0.871028i 0.360791 0.871028i −0.634393 0.773010i \(-0.718750\pi\)
0.995185 0.0980171i \(-0.0312500\pi\)
\(968\) −0.995185 0.0980171i −0.995185 0.0980171i
\(969\) 0 0
\(970\) 0 0
\(971\) −1.53636 + 1.02656i −1.53636 + 1.02656i −0.555570 + 0.831470i \(0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(978\) 0 0
\(979\) 1.92388 0.382683i 1.92388 0.382683i
\(980\) −0.468769 + 0.468769i −0.468769 + 0.468769i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(984\) 0 0
\(985\) −1.76820 0.732410i −1.76820 0.732410i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(991\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(992\) −1.46658 + 0.783904i −1.46658 + 0.783904i
\(993\) 0 0
\(994\) 0.281892 0.343486i 0.281892 0.343486i
\(995\) −1.53636 1.02656i −1.53636 1.02656i
\(996\) 0 0
\(997\) 0.569414 + 0.113263i 0.569414 + 0.113263i 0.471397 0.881921i \(-0.343750\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(998\) −1.59133 + 0.482726i −1.59133 + 0.482726i
\(999\) 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 3520.1.db.a.549.3 yes 32
5.4 even 2 inner 3520.1.db.a.549.2 yes 32
11.10 odd 2 inner 3520.1.db.a.549.2 yes 32
55.54 odd 2 CM 3520.1.db.a.549.3 yes 32
64.45 even 16 inner 3520.1.db.a.109.3 yes 32
320.109 even 16 inner 3520.1.db.a.109.2 32
704.109 odd 16 inner 3520.1.db.a.109.2 32
3520.109 odd 16 inner 3520.1.db.a.109.3 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3520.1.db.a.109.2 32 320.109 even 16 inner
3520.1.db.a.109.2 32 704.109 odd 16 inner
3520.1.db.a.109.3 yes 32 64.45 even 16 inner
3520.1.db.a.109.3 yes 32 3520.109 odd 16 inner
3520.1.db.a.549.2 yes 32 5.4 even 2 inner
3520.1.db.a.549.2 yes 32 11.10 odd 2 inner
3520.1.db.a.549.3 yes 32 1.1 even 1 trivial
3520.1.db.a.549.3 yes 32 55.54 odd 2 CM