Properties

Label 3520.1.db.a.109.4
Level $3520$
Weight $1$
Character 3520.109
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 109.4
Root \(0.290285 - 0.956940i\) of defining polynomial
Character \(\chi\) \(=\) 3520.109
Dual form 3520.1.db.a.549.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.995185 + 0.0980171i) q^{2} +(0.980785 + 0.195090i) q^{4} +(-0.195090 - 0.980785i) q^{5} +(0.732410 + 1.76820i) q^{7} +(0.956940 + 0.290285i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\) \(q+(0.995185 + 0.0980171i) q^{2} +(0.980785 + 0.195090i) q^{4} +(-0.195090 - 0.980785i) q^{5} +(0.732410 + 1.76820i) q^{7} +(0.956940 + 0.290285i) q^{8} +(-0.382683 + 0.923880i) q^{9} +(-0.0980171 - 0.995185i) q^{10} +(0.831470 + 0.555570i) q^{11} +(0.301614 - 1.51631i) q^{13} +(0.555570 + 1.83147i) q^{14} +(0.923880 + 0.382683i) q^{16} +(-1.24723 - 1.24723i) q^{17} +(-0.471397 + 0.881921i) q^{18} -1.00000i q^{20} +(0.773010 + 0.634393i) q^{22} +(-0.923880 + 0.382683i) q^{25} +(0.448786 - 1.47945i) q^{26} +(0.373380 + 1.87711i) q^{28} +1.66294i q^{31} +(0.881921 + 0.471397i) q^{32} +(-1.11897 - 1.36347i) q^{34} +(1.59133 - 1.06330i) q^{35} +(-0.555570 + 0.831470i) q^{36} +(0.0980171 - 0.995185i) q^{40} +(0.108911 - 0.162997i) q^{43} +(0.707107 + 0.707107i) q^{44} +(0.980785 + 0.195090i) q^{45} +(-1.88298 + 1.88298i) q^{49} +(-0.956940 + 0.290285i) q^{50} +(0.591637 - 1.42834i) q^{52} +(0.382683 - 0.923880i) q^{55} +(0.187593 + 1.90466i) q^{56} +(-0.0761205 - 0.382683i) q^{59} +(-0.162997 + 1.65493i) q^{62} -1.91388 q^{63} +(0.831470 + 0.555570i) q^{64} -1.54602 q^{65} +(-0.979938 - 1.46658i) q^{68} +(1.68789 - 0.902197i) q^{70} +(-0.292893 - 0.707107i) q^{71} +(-0.634393 + 0.773010i) q^{72} +(0.761681 - 1.83886i) q^{73} +(-0.373380 + 1.87711i) q^{77} +(0.195090 - 0.980785i) q^{80} +(-0.707107 - 0.707107i) q^{81} +(-1.51631 - 0.301614i) q^{83} +(-0.979938 + 1.46658i) q^{85} +(0.124363 - 0.151537i) q^{86} +(0.634393 + 0.773010i) q^{88} +(1.81225 - 0.750661i) q^{89} +(0.956940 + 0.290285i) q^{90} +(2.90205 - 0.577253i) q^{91} +(-2.05848 + 1.68935i) q^{98} +(-0.831470 + 0.555570i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) 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 \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{16}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.995185 + 0.0980171i 0.995185 + 0.0980171i
\(3\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(4\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(5\) −0.195090 0.980785i −0.195090 0.980785i
\(6\) 0 0
\(7\) 0.732410 + 1.76820i 0.732410 + 1.76820i 0.634393 + 0.773010i \(0.281250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(8\) 0.956940 + 0.290285i 0.956940 + 0.290285i
\(9\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(10\) −0.0980171 0.995185i −0.0980171 0.995185i
\(11\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(12\) 0 0
\(13\) 0.301614 1.51631i 0.301614 1.51631i −0.471397 0.881921i \(-0.656250\pi\)
0.773010 0.634393i \(-0.218750\pi\)
\(14\) 0.555570 + 1.83147i 0.555570 + 1.83147i
\(15\) 0 0
\(16\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(17\) −1.24723 1.24723i −1.24723 1.24723i −0.956940 0.290285i \(-0.906250\pi\)
−0.290285 0.956940i \(-0.593750\pi\)
\(18\) −0.471397 + 0.881921i −0.471397 + 0.881921i
\(19\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(20\) 1.00000i 1.00000i
\(21\) 0 0
\(22\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(23\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(24\) 0 0
\(25\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(26\) 0.448786 1.47945i 0.448786 1.47945i
\(27\) 0 0
\(28\) 0.373380 + 1.87711i 0.373380 + 1.87711i
\(29\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(30\) 0 0
\(31\) 1.66294i 1.66294i 0.555570 + 0.831470i \(0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(32\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(33\) 0 0
\(34\) −1.11897 1.36347i −1.11897 1.36347i
\(35\) 1.59133 1.06330i 1.59133 1.06330i
\(36\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(37\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.0980171 0.995185i 0.0980171 0.995185i
\(41\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(42\) 0 0
\(43\) 0.108911 0.162997i 0.108911 0.162997i −0.773010 0.634393i \(-0.781250\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(44\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) −1.88298 + 1.88298i −1.88298 + 1.88298i
\(50\) −0.956940 + 0.290285i −0.956940 + 0.290285i
\(51\) 0 0
\(52\) 0.591637 1.42834i 0.591637 1.42834i
\(53\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(54\) 0 0
\(55\) 0.382683 0.923880i 0.382683 0.923880i
\(56\) 0.187593 + 1.90466i 0.187593 + 1.90466i
\(57\) 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 \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(62\) −0.162997 + 1.65493i −0.162997 + 1.65493i
\(63\) −1.91388 −1.91388
\(64\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(65\) −1.54602 −1.54602
\(66\) 0 0
\(67\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(68\) −0.979938 1.46658i −0.979938 1.46658i
\(69\) 0 0
\(70\) 1.68789 0.902197i 1.68789 0.902197i
\(71\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(72\) −0.634393 + 0.773010i −0.634393 + 0.773010i
\(73\) 0.761681 1.83886i 0.761681 1.83886i 0.290285 0.956940i \(-0.406250\pi\)
0.471397 0.881921i \(-0.343750\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.373380 + 1.87711i −0.373380 + 1.87711i
\(78\) 0 0
\(79\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) 0.195090 0.980785i 0.195090 0.980785i
\(81\) −0.707107 0.707107i −0.707107 0.707107i
\(82\) 0 0
\(83\) −1.51631 0.301614i −1.51631 0.301614i −0.634393 0.773010i \(-0.718750\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(84\) 0 0
\(85\) −0.979938 + 1.46658i −0.979938 + 1.46658i
\(86\) 0.124363 0.151537i 0.124363 0.151537i
\(87\) 0 0
\(88\) 0.634393 + 0.773010i 0.634393 + 0.773010i
\(89\) 1.81225 0.750661i 1.81225 0.750661i 0.831470 0.555570i \(-0.187500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(90\) 0.956940 + 0.290285i 0.956940 + 0.290285i
\(91\) 2.90205 0.577253i 2.90205 0.577253i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −2.05848 + 1.68935i −2.05848 + 1.68935i
\(99\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(100\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(101\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(102\) 0 0
\(103\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(104\) 0.728789 1.36347i 0.728789 1.36347i
\(105\) 0 0
\(106\) 0 0
\(107\) 0.523788 0.783904i 0.523788 0.783904i −0.471397 0.881921i \(-0.656250\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(108\) 0 0
\(109\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(110\) 0.471397 0.881921i 0.471397 0.881921i
\(111\) 0 0
\(112\) 1.91388i 1.91388i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.28547 + 0.858923i 1.28547 + 0.858923i
\(118\) −0.0382444 0.388302i −0.0382444 0.388302i
\(119\) 1.29186 3.11882i 1.29186 3.11882i
\(120\) 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
\(126\) −1.90466 0.187593i −1.90466 0.187593i
\(127\) −1.99037 −1.99037 −0.995185 0.0980171i \(-0.968750\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(128\) 0.773010 + 0.634393i 0.773010 + 0.634393i
\(129\) 0 0
\(130\) −1.53858 0.151537i −1.53858 0.151537i
\(131\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.831470 1.55557i −0.831470 1.55557i
\(137\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(140\) 1.76820 0.732410i 1.76820 0.732410i
\(141\) 0 0
\(142\) −0.222174 0.732410i −0.222174 0.732410i
\(143\) 1.09320 1.09320i 1.09320 1.09320i
\(144\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(145\) 0 0
\(146\) 0.938254 1.75535i 0.938254 1.75535i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(150\) 0 0
\(151\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(152\) 0 0
\(153\) 1.62958 0.674993i 1.62958 0.674993i
\(154\) −0.555570 + 1.83147i −0.555570 + 1.83147i
\(155\) 1.63099 0.324423i 1.63099 0.324423i
\(156\) 0 0
\(157\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.290285 0.956940i 0.290285 0.956940i
\(161\) 0 0
\(162\) −0.634393 0.773010i −0.634393 0.773010i
\(163\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −1.47945 0.448786i −1.47945 0.448786i
\(167\) −1.17221 + 0.485544i −1.17221 + 0.485544i −0.881921 0.471397i \(-0.843750\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(168\) 0 0
\(169\) −1.28436 0.531999i −1.28436 0.531999i
\(170\) −1.11897 + 1.36347i −1.11897 + 1.36347i
\(171\) 0 0
\(172\) 0.138617 0.138617i 0.138617 0.138617i
\(173\) 0.924678 + 0.183930i 0.924678 + 0.183930i 0.634393 0.773010i \(-0.281250\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(174\) 0 0
\(175\) −1.35332 1.35332i −1.35332 1.35332i
\(176\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(177\) 0 0
\(178\) 1.87711 0.569414i 1.87711 0.569414i
\(179\) 0.149316 0.750661i 0.149316 0.750661i −0.831470 0.555570i \(-0.812500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(180\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(181\) −1.53636 1.02656i −1.53636 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(182\) 2.94465 0.290023i 2.94465 0.290023i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.344109 1.72995i −0.344109 1.72995i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(192\) 0 0
\(193\) 0.196034 0.196034 0.0980171 0.995185i \(-0.468750\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.21415 + 1.47945i −2.21415 + 1.47945i
\(197\) −0.113263 0.569414i −0.113263 0.569414i −0.995185 0.0980171i \(-0.968750\pi\)
0.881921 0.471397i \(-0.156250\pi\)
\(198\) −0.881921 + 0.471397i −0.881921 + 0.471397i
\(199\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.995185 + 0.0980171i −0.995185 + 0.0980171i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.858923 1.28547i 0.858923 1.28547i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.598102 0.728789i 0.598102 0.728789i
\(215\) −0.181112 0.0750191i −0.181112 0.0750191i
\(216\) 0 0
\(217\) −2.94040 + 1.21795i −2.94040 + 1.21795i
\(218\) 0 0
\(219\) 0 0
\(220\) 0.555570 0.831470i 0.555570 0.831470i
\(221\) −2.26737 + 1.51501i −2.26737 + 1.51501i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −0.187593 + 1.90466i −0.187593 + 1.90466i
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) −1.05496 + 0.704900i −1.05496 + 0.704900i −0.956940 0.290285i \(-0.906250\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(228\) 0 0
\(229\) −1.08979 + 0.216773i −1.08979 + 0.216773i −0.707107 0.707107i \(-0.750000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.536376 0.222174i −0.536376 0.222174i 0.0980171 0.995185i \(-0.468750\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(234\) 1.19509 + 0.980785i 1.19509 + 0.980785i
\(235\) 0 0
\(236\) 0.390181i 0.390181i
\(237\) 0 0
\(238\) 1.59133 2.97718i 1.59133 2.97718i
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(242\) 0.290285 + 0.956940i 0.290285 + 0.956940i
\(243\) 0 0
\(244\) 0 0
\(245\) 2.21415 + 1.47945i 2.21415 + 1.47945i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.482726 + 1.59133i −0.482726 + 1.59133i
\(249\) 0 0
\(250\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(251\) −0.382683 1.92388i −0.382683 1.92388i −0.382683 0.923880i \(-0.625000\pi\)
1.00000i \(-0.5\pi\)
\(252\) −1.87711 0.373380i −1.87711 0.373380i
\(253\) 0 0
\(254\) −1.98079 0.195090i −1.98079 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
\(260\) −1.51631 0.301614i −1.51631 0.301614i
\(261\) 0 0
\(262\) 0 0
\(263\) −0.0750191 0.181112i −0.0750191 0.181112i 0.881921 0.471397i \(-0.156250\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.0761205 0.382683i 0.0761205 0.382683i −0.923880 0.382683i \(-0.875000\pi\)
1.00000 \(0\)
\(270\) 0 0
\(271\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(272\) −0.674993 1.62958i −0.674993 1.62958i
\(273\) 0 0
\(274\) 0 0
\(275\) −0.980785 0.195090i −0.980785 0.195090i
\(276\) 0 0
\(277\) 0.108911 0.162997i 0.108911 0.162997i −0.773010 0.634393i \(-0.781250\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(278\) 0 0
\(279\) −1.53636 0.636379i −1.53636 0.636379i
\(280\) 1.83147 0.555570i 1.83147 0.555570i
\(281\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(282\) 0 0
\(283\) 1.24441 0.247528i 1.24441 0.247528i 0.471397 0.881921i \(-0.343750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(284\) −0.149316 0.750661i −0.149316 0.750661i
\(285\) 0 0
\(286\) 1.19509 0.980785i 1.19509 0.980785i
\(287\) 0 0
\(288\) −0.773010 + 0.634393i −0.773010 + 0.634393i
\(289\) 2.11114i 2.11114i
\(290\) 0 0
\(291\) 0 0
\(292\) 1.10579 1.65493i 1.10579 1.65493i
\(293\) −1.95213 + 0.388302i −1.95213 + 0.388302i −0.956940 + 0.290285i \(0.906250\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(294\) 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
\(301\) 0.367977 + 0.0731952i 0.367977 + 0.0731952i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 1.68789 0.512016i 1.68789 0.512016i
\(307\) −0.344109 + 1.72995i −0.344109 + 1.72995i 0.290285 + 0.956940i \(0.406250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(308\) −0.732410 + 1.76820i −0.732410 + 1.76820i
\(309\) 0 0
\(310\) 1.65493 0.162997i 1.65493 0.162997i
\(311\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(314\) 0 0
\(315\) 0.373380 + 1.87711i 0.373380 + 1.87711i
\(316\) 0 0
\(317\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(318\) 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
\(325\) 0.301614 + 1.51631i 0.301614 + 1.51631i
\(326\) 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 \(-0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(332\) −1.42834 0.591637i −1.42834 0.591637i
\(333\) 0 0
\(334\) −1.21415 + 0.368309i −1.21415 + 0.368309i
\(335\) 0 0
\(336\) 0 0
\(337\) −0.410525 0.410525i −0.410525 0.410525i 0.471397 0.881921i \(-0.343750\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(338\) −1.22603 0.655327i −1.22603 0.655327i
\(339\) 0 0
\(340\) −1.24723 + 1.24723i −1.24723 + 1.24723i
\(341\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(342\) 0 0
\(343\) −2.94040 1.21795i −2.94040 1.21795i
\(344\) 0.151537 0.124363i 0.151537 0.124363i
\(345\) 0 0
\(346\) 0.902197 + 0.273678i 0.902197 + 0.273678i
\(347\) 1.95213 0.388302i 1.95213 0.388302i 0.956940 0.290285i \(-0.0937500\pi\)
0.995185 0.0980171i \(-0.0312500\pi\)
\(348\) 0 0
\(349\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(350\) −1.21415 1.47945i −1.21415 1.47945i
\(351\) 0 0
\(352\) 0.471397 + 0.881921i 0.471397 + 0.881921i
\(353\) 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
\(358\) 0.222174 0.732410i 0.222174 0.732410i
\(359\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(360\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(361\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(362\) −1.42834 1.17221i −1.42834 1.17221i
\(363\) 0 0
\(364\) 2.95890 2.95890
\(365\) −1.95213 0.388302i −1.95213 0.388302i
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.482726 + 0.322547i 0.482726 + 0.322547i 0.773010 0.634393i \(-0.218750\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(374\) −0.172887 1.75535i −0.172887 1.75535i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.382683 + 1.92388i 0.382683 + 1.92388i 0.382683 + 0.923880i \(0.375000\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.388302 0.0382444i −0.388302 0.0382444i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 1.91388 1.91388
\(386\) 0.195090 + 0.0192147i 0.195090 + 0.0192147i
\(387\) 0.108911 + 0.162997i 0.108911 + 0.162997i
\(388\) 0 0
\(389\) 0.216773 + 1.08979i 0.216773 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.34850 + 1.25530i −2.34850 + 1.25530i
\(393\) 0 0
\(394\) −0.0569057 0.577774i −0.0569057 0.577774i
\(395\) 0 0
\(396\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(397\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(398\) 0.536376 + 1.76820i 0.536376 + 1.76820i
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(402\) 0 0
\(403\) 2.52154 + 0.501565i 2.52154 + 0.501565i
\(404\) 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 \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.620908 0.414877i 0.620908 0.414877i
\(414\) 0 0
\(415\) 1.54602i 1.54602i
\(416\) 0.980785 1.19509i 0.980785 1.19509i
\(417\) 0 0
\(418\) 0 0
\(419\) −1.17588 + 0.785695i −1.17588 + 0.785695i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(420\) 0 0
\(421\) −1.81225 + 0.360480i −1.81225 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.62958 + 0.674993i 1.62958 + 0.674993i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.666656 0.666656i 0.666656 0.666656i
\(429\) 0 0
\(430\) −0.172887 0.0924099i −0.172887 0.0924099i
\(431\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) −3.04562 + 0.923880i −3.04562 + 0.923880i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(440\) 0.634393 0.773010i 0.634393 0.773010i
\(441\) −1.01906 2.46024i −1.01906 2.46024i
\(442\) −2.40494 + 1.28547i −2.40494 + 1.28547i
\(443\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(444\) 0 0
\(445\) −1.08979 1.63099i −1.08979 1.63099i
\(446\) 0 0
\(447\) 0 0
\(448\) −0.373380 + 1.87711i −0.373380 + 1.87711i
\(449\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(450\) 0.0980171 0.995185i 0.0980171 0.995185i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −1.11897 + 0.598102i −1.11897 + 0.598102i
\(455\) −1.13232 2.73367i −1.13232 2.73367i
\(456\) 0 0
\(457\) 0.485544 1.17221i 0.485544 1.17221i −0.471397 0.881921i \(-0.656250\pi\)
0.956940 0.290285i \(-0.0937500\pi\)
\(458\) −1.10579 + 0.108911i −1.10579 + 0.108911i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.512016 0.273678i −0.512016 0.273678i
\(467\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(468\) 1.09320 + 1.09320i 1.09320 + 1.09320i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.0382444 0.388302i 0.0382444 0.388302i
\(473\) 0.181112 0.0750191i 0.181112 0.0750191i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.87549 2.80686i 1.87549 2.80686i
\(477\) 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 \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 2.05848 + 1.68935i 2.05848 + 1.68935i
\(491\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(492\) 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
\(497\) 1.03578 1.03578i 1.03578 1.03578i
\(498\) 0 0
\(499\) 0.324423 1.63099i 0.324423 1.63099i −0.382683 0.923880i \(-0.625000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(500\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(501\) 0 0
\(502\) −0.192268 1.95213i −0.192268 1.95213i
\(503\) 0.674993 1.62958i 0.674993 1.62958i −0.0980171 0.995185i \(-0.531250\pi\)
0.773010 0.634393i \(-0.218750\pi\)
\(504\) −1.83147 0.555570i −1.83147 0.555570i
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −1.95213 0.388302i −1.95213 0.388302i
\(509\) 1.08979 + 1.63099i 1.08979 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(510\) 0 0
\(511\) 3.80933 3.80933
\(512\) 0.634393 + 0.773010i 0.634393 + 0.773010i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −1.47945 0.448786i −1.47945 0.448786i
\(521\) 0.425215 1.02656i 0.425215 1.02656i −0.555570 0.831470i \(-0.687500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(522\) 0 0
\(523\) 0.482726 + 0.322547i 0.482726 + 0.322547i 0.773010 0.634393i \(-0.218750\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.0569057 0.187593i −0.0569057 0.187593i
\(527\) 2.07406 2.07406i 2.07406 2.07406i
\(528\) 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
\(535\) −0.871028 0.360791i −0.871028 0.360791i
\(536\) 0 0
\(537\) 0 0
\(538\) 0.113263 0.373380i 0.113263 0.373380i
\(539\) −2.61177 + 0.519514i −2.61177 + 0.519514i
\(540\) 0 0
\(541\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.512016 1.68789i −0.512016 1.68789i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.783904 0.523788i 0.783904 0.523788i −0.0980171 0.995185i \(-0.531250\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −0.956940 0.290285i −0.956940 0.290285i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.124363 0.151537i 0.124363 0.151537i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.192268 + 0.0382444i 0.192268 + 0.0382444i 0.290285 0.956940i \(-0.406250\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(558\) −1.46658 0.783904i −1.46658 0.783904i
\(559\) −0.214305 0.214305i −0.214305 0.214305i
\(560\) 1.87711 0.373380i 1.87711 0.373380i
\(561\) 0 0
\(562\) 0 0
\(563\) −0.113263 + 0.569414i −0.113263 + 0.569414i 0.881921 + 0.471397i \(0.156250\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.26268 0.124363i 1.26268 0.124363i
\(567\) 0.732410 1.76820i 0.732410 1.76820i
\(568\) −0.0750191 0.761681i −0.0750191 0.761681i
\(569\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(572\) 1.28547 0.858923i 1.28547 0.858923i
\(573\) 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 \(\pi\)
\(578\) −0.206928 + 2.10097i −0.206928 + 2.10097i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.577253 2.90205i −0.577253 2.90205i
\(582\) 0 0
\(583\) 0 0
\(584\) 1.26268 1.53858i 1.26268 1.53858i
\(585\) 0.591637 1.42834i 0.591637 1.42834i
\(586\) −1.98079 + 0.195090i −1.98079 + 0.195090i
\(587\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −0.373380 + 0.113263i −0.373380 + 0.113263i
\(591\) 0 0
\(592\) 0 0
\(593\) −1.35332 1.35332i −1.35332 1.35332i −0.881921 0.471397i \(-0.843750\pi\)
−0.471397 0.881921i \(-0.656250\pi\)
\(594\) 0 0
\(595\) −3.31092 0.658583i −3.31092 0.658583i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.02656 + 0.425215i 1.02656 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(600\) 0 0
\(601\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(602\) 0.359031 + 0.108911i 0.359031 + 0.108911i
\(603\) 0 0
\(604\) 0 0
\(605\) 0.831470 0.555570i 0.831470 0.555570i
\(606\) 0 0
\(607\) 1.76384i 1.76384i 0.471397 + 0.881921i \(0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.72995 0.344109i 1.72995 0.344109i
\(613\) 0.924678 0.183930i 0.924678 0.183930i 0.290285 0.956940i \(-0.406250\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(614\) −0.512016 + 1.68789i −0.512016 + 1.68789i
\(615\) 0 0
\(616\) −0.902197 + 1.68789i −0.902197 + 1.68789i
\(617\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(618\) 0 0
\(619\) 1.02656 1.53636i 1.02656 1.53636i 0.195090 0.980785i \(-0.437500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(620\) 1.66294 1.66294
\(621\) 0 0
\(622\) 0 0
\(623\) 2.65463 + 2.65463i 2.65463 + 2.65463i
\(624\) 0 0
\(625\) 0.707107 0.707107i 0.707107 0.707107i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.187593 + 1.90466i 0.187593 + 1.90466i
\(631\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.388302 + 1.95213i 0.388302 + 1.95213i
\(636\) 0 0
\(637\) 2.28726 + 3.42313i 2.28726 + 3.42313i
\(638\) 0 0
\(639\) 0.765367 0.765367
\(640\) 0.471397 0.881921i 0.471397 0.881921i
\(641\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(642\) 0 0
\(643\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(648\) −0.471397 0.881921i −0.471397 0.881921i
\(649\) 0.149316 0.360480i 0.149316 0.360480i
\(650\) 0.151537 + 1.53858i 0.151537 + 1.53858i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.40740 + 1.40740i 1.40740 + 1.40740i
\(658\) 0 0
\(659\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(660\) 0 0
\(661\) 0.923880 1.38268i 0.923880 1.38268i 1.00000i \(-0.5\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(662\) −1.09320 0.897168i −1.09320 0.897168i
\(663\) 0 0
\(664\) −1.36347 0.728789i −1.36347 0.728789i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.24441 + 0.247528i −1.24441 + 0.247528i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.580569i 0.580569i 0.956940 + 0.290285i \(0.0937500\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(674\) −0.368309 0.448786i −0.368309 0.448786i
\(675\) 0 0
\(676\) −1.15589 0.772343i −1.15589 0.772343i
\(677\) 1.72995 0.344109i 1.72995 0.344109i 0.773010 0.634393i \(-0.218750\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.36347 + 1.11897i −1.36347 + 1.11897i
\(681\) 0 0
\(682\) −1.05496 + 1.28547i −1.05496 + 1.28547i
\(683\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.80686 1.50030i −2.80686 1.50030i
\(687\) 0 0
\(688\) 0.162997 0.108911i 0.162997 0.108911i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.216773 1.08979i 0.216773 1.08979i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(692\) 0.871028 + 0.360791i 0.871028 + 0.360791i
\(693\) −1.59133 1.06330i −1.59133 1.06330i
\(694\) 1.98079 0.195090i 1.98079 0.195090i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.06330 1.59133i −1.06330 1.59133i
\(701\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(702\) 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 0.831470 + 0.555570i \(0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(710\) −0.674993 + 0.360791i −0.674993 + 0.360791i
\(711\) 0 0
\(712\) 1.95213 0.192268i 1.95213 0.192268i
\(713\) 0 0
\(714\) 0 0
\(715\) −1.28547 0.858923i −1.28547 0.858923i
\(716\) 0.292893 0.707107i 0.292893 0.707107i
\(717\) 0 0
\(718\) 0 0
\(719\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(720\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(721\) 0 0
\(722\) 0.881921 + 0.471397i 0.881921 + 0.471397i
\(723\) 0 0
\(724\) −1.30656 1.30656i −1.30656 1.30656i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(728\) 2.94465 + 0.290023i 2.94465 + 0.290023i
\(729\) 0.923880 0.382683i 0.923880 0.382683i
\(730\) −1.90466 0.577774i −1.90466 0.577774i
\(731\) −0.339130 + 0.0674571i −0.339130 + 0.0674571i
\(732\) 0 0
\(733\) −1.59133 + 1.06330i −1.59133 + 1.06330i −0.634393 + 0.773010i \(0.718750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.871028 0.360791i 0.871028 0.360791i 0.0980171 0.995185i \(-0.468750\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.448786 + 0.368309i 0.448786 + 0.368309i
\(747\) 0.858923 1.28547i 0.858923 1.28547i
\(748\) 1.76384i 1.76384i
\(749\) 1.76972 + 0.352020i 1.76972 + 0.352020i
\(750\) 0 0
\(751\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(758\) 0.192268 + 1.95213i 0.192268 + 1.95213i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −0.382683 0.0761205i −0.382683 0.0761205i
\(765\) −0.979938 1.46658i −0.979938 1.46658i
\(766\) 0 0
\(767\) −0.603227 −0.603227
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 1.90466 + 0.187593i 1.90466 + 0.187593i
\(771\) 0 0
\(772\) 0.192268 + 0.0382444i 0.192268 + 0.0382444i
\(773\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(774\) 0.0924099 + 0.172887i 0.0924099 + 0.172887i
\(775\) −0.636379 1.53636i −0.636379 1.53636i
\(776\) 0 0
\(777\) 0 0
\(778\) 0.108911 + 1.10579i 0.108911 + 1.10579i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.149316 0.750661i 0.149316 0.750661i
\(782\) 0 0
\(783\) 0 0
\(784\) −2.46024 + 1.01906i −2.46024 + 1.01906i
\(785\) 0 0
\(786\) 0 0
\(787\) −0.924678 0.183930i −0.924678 0.183930i −0.290285 0.956940i \(-0.593750\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(788\) 0.580569i 0.580569i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.956940 + 0.290285i −0.956940 + 0.290285i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(797\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.995185 0.0980171i −0.995185 0.0980171i
\(801\) 1.96157i 1.96157i
\(802\) −0.247528 0.301614i −0.247528 0.301614i
\(803\) 1.65493 1.10579i 1.65493 1.10579i
\(804\) 0 0
\(805\) 0 0
\(806\) 2.46024 + 0.746304i 2.46024 + 0.746304i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(810\) −0.634393 + 0.773010i −0.634393 + 0.773010i
\(811\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −0.577253 + 2.90205i −0.577253 + 2.90205i
\(820\) 0 0
\(821\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(822\) 0 0
\(823\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0.658583 0.352020i 0.658583 0.352020i
\(827\) 0.301614 + 1.51631i 0.301614 + 1.51631i 0.773010 + 0.634393i \(0.218750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(828\) 0 0
\(829\) 0.785695 + 1.17588i 0.785695 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(830\) −0.151537 + 1.53858i −0.151537 + 1.53858i
\(831\) 0 0
\(832\) 1.09320 1.09320i 1.09320 1.09320i
\(833\) 4.69701 4.69701
\(834\) 0 0
\(835\) 0.704900 + 1.05496i 0.704900 + 1.05496i
\(836\) 0 0
\(837\) 0 0
\(838\) −1.24723 + 0.666656i −1.24723 + 0.666656i
\(839\) 0.292893 + 0.707107i 0.292893 + 0.707107i 1.00000 \(0\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(840\) 0 0
\(841\) 0.382683 0.923880i 0.382683 0.923880i
\(842\) −1.83886 + 0.181112i −1.83886 + 0.181112i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.271211 + 1.36347i −0.271211 + 1.36347i
\(846\) 0 0
\(847\) −1.35332 + 1.35332i −1.35332 + 1.35332i
\(848\) 0 0
\(849\) 0 0
\(850\) 1.55557 + 0.831470i 1.55557 + 0.831470i
\(851\) 0 0
\(852\) 0 0
\(853\) −1.10579 + 1.65493i −1.10579 + 1.65493i −0.471397 + 0.881921i \(0.656250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.728789 0.598102i 0.728789 0.598102i
\(857\) −1.83886 + 0.761681i −1.83886 + 0.761681i −0.881921 + 0.471397i \(0.843750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(858\) 0 0
\(859\) 1.38704 0.275899i 1.38704 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(860\) −0.162997 0.108911i −0.162997 0.108911i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0.942793i 0.942793i
\(866\) 0 0
\(867\) 0 0
\(868\) −3.12151 + 0.620908i −3.12151 + 0.620908i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.06330 + 1.59133i −1.06330 + 1.59133i
\(876\) 0 0
\(877\) −1.72995 0.344109i −1.72995 0.344109i −0.773010 0.634393i \(-0.781250\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0.707107 0.707107i 0.707107 0.707107i
\(881\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(882\) −0.773010 2.54827i −0.773010 2.54827i
\(883\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(884\) −2.51936 + 1.04355i −2.51936 + 1.04355i
\(885\) 0 0
\(886\) 0 0
\(887\) −0.485544 + 1.17221i −0.485544 + 1.17221i 0.471397 + 0.881921i \(0.343750\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(888\) 0 0
\(889\) −1.45777 3.51936i −1.45777 3.51936i
\(890\) −0.924678 1.72995i −0.924678 1.72995i
\(891\) −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 + 1.83147i −0.555570 + 1.83147i
\(897\) 0 0
\(898\) 1.10579 + 0.108911i 1.10579 + 0.108911i
\(899\) 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 \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(908\) −1.17221 + 0.485544i −1.17221 + 0.485544i
\(909\) 0 0
\(910\) −0.858923 2.83149i −0.858923 2.83149i
\(911\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(912\) 0 0
\(913\) −1.09320 1.09320i −1.09320 1.09320i
\(914\) 0.598102 1.11897i 0.598102 1.11897i
\(915\) 0 0
\(916\) −1.11114 −1.11114
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.16054 + 0.230845i −1.16054 + 0.230845i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.482726 0.322547i −0.482726 0.322547i
\(933\) 0 0
\(934\) 0 0
\(935\) −1.62958 + 0.674993i −1.62958 + 0.674993i
\(936\) 0.980785 + 1.19509i 0.980785 + 1.19509i
\(937\) −0.871028 0.360791i −0.871028 0.360791i −0.0980171 0.995185i \(-0.531250\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.0761205 0.382683i 0.0761205 0.382683i
\(945\) 0 0
\(946\) 0.187593 0.0569057i 0.187593 0.0569057i
\(947\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(948\) 0 0
\(949\) −2.55856 1.70957i −2.55856 1.70957i
\(950\) 0 0
\(951\) 0 0
\(952\) 2.14157 2.60952i 2.14157 2.60952i
\(953\) 0.591637 + 1.42834i 0.591637 + 1.42834i 0.881921 + 0.471397i \(0.156250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(954\) 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
\(963\) 0.523788 + 0.783904i 0.523788 + 0.783904i
\(964\) 0 0
\(965\) −0.0382444 0.192268i −0.0382444 0.192268i
\(966\) 0 0
\(967\) 0.674993 + 1.62958i 0.674993 + 1.62958i 0.773010 + 0.634393i \(0.218750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(968\) 0.0980171 + 0.995185i 0.0980171 + 0.995185i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.53636 + 1.02656i 1.53636 + 1.02656i 0.980785 + 0.195090i \(0.0625000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(980\) 1.88298 + 1.88298i 1.88298 + 1.88298i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(984\) 0 0
\(985\) −0.536376 + 0.222174i −0.536376 + 0.222174i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.634393 + 0.773010i 0.634393 + 0.773010i
\(991\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(992\) −0.783904 + 1.46658i −0.783904 + 1.46658i
\(993\) 0 0
\(994\) 1.13232 0.929273i 1.13232 0.929273i
\(995\) 1.53636 1.02656i 1.53636 1.02656i
\(996\) 0 0
\(997\) 1.87711 0.373380i 1.87711 0.373380i 0.881921 0.471397i \(-0.156250\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(998\) 0.482726 1.59133i 0.482726 1.59133i
\(999\) 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.109.4 yes 32
5.4 even 2 inner 3520.1.db.a.109.1 32
11.10 odd 2 inner 3520.1.db.a.109.1 32
55.54 odd 2 CM 3520.1.db.a.109.4 yes 32
64.37 even 16 inner 3520.1.db.a.549.4 yes 32
320.229 even 16 inner 3520.1.db.a.549.1 yes 32
704.549 odd 16 inner 3520.1.db.a.549.1 yes 32
3520.549 odd 16 inner 3520.1.db.a.549.4 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3520.1.db.a.109.1 32 5.4 even 2 inner
3520.1.db.a.109.1 32 11.10 odd 2 inner
3520.1.db.a.109.4 yes 32 1.1 even 1 trivial
3520.1.db.a.109.4 yes 32 55.54 odd 2 CM
3520.1.db.a.549.1 yes 32 320.229 even 16 inner
3520.1.db.a.549.1 yes 32 704.549 odd 16 inner
3520.1.db.a.549.4 yes 32 64.37 even 16 inner
3520.1.db.a.549.4 yes 32 3520.549 odd 16 inner