Properties

Label 88.2.k.a.51.2
Level $88$
Weight $2$
Character 88.51
Analytic conductor $0.703$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [88,2,Mod(19,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 88.k (of order \(10\), degree \(4\), 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: \(0.702683537787\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label 51.2
Root \(0.831254 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 88.51
Dual form 88.2.k.a.19.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.831254 - 1.14412i) q^{2} +(0.0137431 - 0.0422971i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(-0.0369690 - 0.0508834i) q^{6} +(-2.68999 - 0.874032i) q^{8} +(2.42545 + 1.76219i) q^{9} +O(q^{10})\) \(q+(0.831254 - 1.14412i) q^{2} +(0.0137431 - 0.0422971i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(-0.0369690 - 0.0508834i) q^{6} +(-2.68999 - 0.874032i) q^{8} +(2.42545 + 1.76219i) q^{9} +(-2.27205 + 2.41615i) q^{11} -0.0889475 q^{12} +(-3.23607 + 2.35114i) q^{16} +(2.32662 + 3.20231i) q^{17} +(4.03233 - 1.31018i) q^{18} +(-4.30278 - 1.39806i) q^{19} +(0.875728 + 4.60794i) q^{22} +(-0.0739380 + 0.101767i) q^{24} +(1.54508 - 4.75528i) q^{25} +(0.215809 - 0.156794i) q^{27} +5.65685i q^{32} +(0.0709711 + 0.129307i) q^{33} +5.59785 q^{34} +(1.85288 - 5.70258i) q^{36} +(-5.17625 + 3.76077i) q^{38} +(-12.0591 - 3.91824i) q^{41} -12.7426i q^{43} +(6.00000 + 2.82843i) q^{44} +(0.0549726 + 0.169188i) q^{48} +(5.66312 - 4.11450i) q^{49} +(-4.15627 - 5.72061i) q^{50} +(0.167423 - 0.0543992i) q^{51} -0.377248i q^{54} +(-0.118267 + 0.162781i) q^{57} +(3.58332 + 11.0283i) q^{59} +(6.47214 + 4.70228i) q^{64} +(0.206938 + 0.0262869i) q^{66} +12.3962 q^{67} +(4.65323 - 6.40462i) q^{68} +(-4.98423 - 6.86021i) q^{72} +(-11.9395 + 3.87937i) q^{73} +(-0.179900 - 0.130705i) q^{75} +9.04842i q^{76} +(2.77565 + 8.54258i) q^{81} +(-14.5071 + 10.5400i) q^{82} +(7.56384 + 10.4107i) q^{83} +(-14.5791 - 10.5923i) q^{86} +(8.22359 - 4.51360i) q^{88} -17.8873 q^{89} +(0.239268 + 0.0777430i) q^{96} +(14.6151 + 10.6185i) q^{97} -9.89949i q^{98} +(-9.76847 + 1.85647i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{4} - 20 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{4} - 20 q^{6} - 2 q^{9} + 6 q^{11} + 8 q^{12} - 8 q^{16} + 40 q^{18} - 10 q^{19} - 4 q^{22} - 40 q^{24} - 10 q^{25} + 38 q^{27} - 38 q^{33} + 16 q^{34} + 24 q^{36} - 24 q^{38} + 48 q^{44} - 16 q^{48} + 14 q^{49} - 70 q^{51} + 70 q^{57} - 18 q^{59} + 16 q^{64} - 8 q^{66} - 28 q^{67} + 30 q^{75} - 8 q^{81} - 48 q^{82} + 90 q^{83} - 36 q^{86} + 8 q^{88} - 36 q^{89} + 30 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{7}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.831254 1.14412i 0.587785 0.809017i
\(3\) 0.0137431 0.0422971i 0.00793461 0.0244202i −0.947011 0.321202i \(-0.895913\pi\)
0.954945 + 0.296781i \(0.0959133\pi\)
\(4\) −0.618034 1.90211i −0.309017 0.951057i
\(5\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) −0.0369690 0.0508834i −0.0150925 0.0207731i
\(7\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) −2.68999 0.874032i −0.951057 0.309017i
\(9\) 2.42545 + 1.76219i 0.808484 + 0.587398i
\(10\) 0 0
\(11\) −2.27205 + 2.41615i −0.685048 + 0.728498i
\(12\) −0.0889475 −0.0256769
\(13\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.23607 + 2.35114i −0.809017 + 0.587785i
\(17\) 2.32662 + 3.20231i 0.564287 + 0.776675i 0.991864 0.127304i \(-0.0406325\pi\)
−0.427576 + 0.903979i \(0.640633\pi\)
\(18\) 4.03233 1.31018i 0.950429 0.308813i
\(19\) −4.30278 1.39806i −0.987125 0.320736i −0.229416 0.973329i \(-0.573682\pi\)
−0.757709 + 0.652592i \(0.773682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.875728 + 4.60794i 0.186706 + 0.982416i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −0.0739380 + 0.101767i −0.0150925 + 0.0207731i
\(25\) 1.54508 4.75528i 0.309017 0.951057i
\(26\) 0 0
\(27\) 0.215809 0.156794i 0.0415325 0.0301751i
\(28\) 0 0
\(29\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0.0709711 + 0.129307i 0.0123545 + 0.0225094i
\(34\) 5.59785 0.960023
\(35\) 0 0
\(36\) 1.85288 5.70258i 0.308813 0.950429i
\(37\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) −5.17625 + 3.76077i −0.839699 + 0.610077i
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0591 3.91824i −1.88332 0.611926i −0.984994 0.172588i \(-0.944787\pi\)
−0.898322 0.439338i \(-0.855213\pi\)
\(42\) 0 0
\(43\) 12.7426i 1.94323i −0.236575 0.971613i \(-0.576025\pi\)
0.236575 0.971613i \(-0.423975\pi\)
\(44\) 6.00000 + 2.82843i 0.904534 + 0.426401i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(48\) 0.0549726 + 0.169188i 0.00793461 + 0.0244202i
\(49\) 5.66312 4.11450i 0.809017 0.587785i
\(50\) −4.15627 5.72061i −0.587785 0.809017i
\(51\) 0.167423 0.0543992i 0.0234440 0.00761741i
\(52\) 0 0
\(53\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(54\) 0.377248i 0.0513369i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.118267 + 0.162781i −0.0156649 + 0.0215609i
\(58\) 0 0
\(59\) 3.58332 + 11.0283i 0.466509 + 1.43577i 0.857075 + 0.515191i \(0.172279\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.47214 + 4.70228i 0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0.206938 + 0.0262869i 0.0254722 + 0.00323569i
\(67\) 12.3962 1.51444 0.757220 0.653160i \(-0.226557\pi\)
0.757220 + 0.653160i \(0.226557\pi\)
\(68\) 4.65323 6.40462i 0.564287 0.776675i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) −4.98423 6.86021i −0.587398 0.808484i
\(73\) −11.9395 + 3.87937i −1.39741 + 0.454046i −0.908352 0.418206i \(-0.862659\pi\)
−0.489057 + 0.872252i \(0.662659\pi\)
\(74\) 0 0
\(75\) −0.179900 0.130705i −0.0207731 0.0150925i
\(76\) 9.04842i 1.03792i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(80\) 0 0
\(81\) 2.77565 + 8.54258i 0.308406 + 0.949176i
\(82\) −14.5071 + 10.5400i −1.60204 + 1.16395i
\(83\) 7.56384 + 10.4107i 0.830240 + 1.14273i 0.987878 + 0.155230i \(0.0496119\pi\)
−0.157639 + 0.987497i \(0.550388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −14.5791 10.5923i −1.57210 1.14220i
\(87\) 0 0
\(88\) 8.22359 4.51360i 0.876638 0.481151i
\(89\) −17.8873 −1.89605 −0.948026 0.318192i \(-0.896924\pi\)
−0.948026 + 0.318192i \(0.896924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.239268 + 0.0777430i 0.0244202 + 0.00793461i
\(97\) 14.6151 + 10.6185i 1.48394 + 1.07814i 0.976262 + 0.216592i \(0.0694942\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.89949i 1.00000i
\(99\) −9.76847 + 1.85647i −0.981768 + 0.186583i
\(100\) −10.0000 −1.00000
\(101\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(102\) 0.0769320 0.236772i 0.00761741 0.0234440i
\(103\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.8797 + 3.85993i 1.14845 + 0.373154i 0.820559 0.571562i \(-0.193662\pi\)
0.327891 + 0.944716i \(0.393662\pi\)
\(108\) −0.431618 0.313589i −0.0415325 0.0301751i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.60617 4.94328i 0.151096 0.465024i −0.846649 0.532152i \(-0.821383\pi\)
0.997744 + 0.0671276i \(0.0213835\pi\)
\(114\) 0.0879314 + 0.270625i 0.00823553 + 0.0253464i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 15.5964 + 5.06758i 1.43577 + 0.466509i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.675596 10.9792i −0.0614178 0.998112i
\(122\) 0 0
\(123\) −0.331460 + 0.456216i −0.0298868 + 0.0411356i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(128\) 10.7600 3.49613i 0.951057 0.309017i
\(129\) −0.538974 0.175123i −0.0474540 0.0154187i
\(130\) 0 0
\(131\) 21.4892i 1.87752i −0.344574 0.938759i \(-0.611977\pi\)
0.344574 0.938759i \(-0.388023\pi\)
\(132\) 0.202093 0.214911i 0.0175899 0.0187056i
\(133\) 0 0
\(134\) 10.3044 14.1828i 0.890165 1.22521i
\(135\) 0 0
\(136\) −3.45966 10.6477i −0.296663 0.913036i
\(137\) 14.6870 10.6707i 1.25480 0.911664i 0.256307 0.966595i \(-0.417494\pi\)
0.998490 + 0.0549317i \(0.0174941\pi\)
\(138\) 0 0
\(139\) −8.06998 + 2.62210i −0.684487 + 0.222403i −0.630559 0.776142i \(-0.717174\pi\)
−0.0539282 + 0.998545i \(0.517174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −11.9921 −0.999341
\(145\) 0 0
\(146\) −5.48626 + 16.8850i −0.454046 + 1.39741i
\(147\) −0.0962020 0.296079i −0.00793461 0.0244202i
\(148\) 0 0
\(149\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(150\) −0.299085 + 0.0971787i −0.0244202 + 0.00793461i
\(151\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(152\) 10.3525 + 7.52153i 0.839699 + 0.610077i
\(153\) 11.8670i 0.959390i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 12.0810 + 3.92537i 0.949176 + 0.308406i
\(163\) −19.0863 13.8670i −1.49495 1.08615i −0.972339 0.233575i \(-0.924958\pi\)
−0.522612 0.852570i \(-0.675042\pi\)
\(164\) 25.3594i 1.98024i
\(165\) 0 0
\(166\) 18.1986 1.41249
\(167\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(168\) 0 0
\(169\) −4.01722 12.3637i −0.309017 0.951057i
\(170\) 0 0
\(171\) −7.97253 10.9732i −0.609675 0.839145i
\(172\) −24.2378 + 7.87535i −1.84812 + 0.600490i
\(173\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.67178 13.1607i 0.126015 0.992028i
\(177\) 0.515712 0.0387633
\(178\) −14.8689 + 20.4653i −1.11447 + 1.53394i
\(179\) −7.53762 + 23.1984i −0.563388 + 1.73393i 0.109303 + 0.994008i \(0.465138\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −13.0235 1.65435i −0.952370 0.120978i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(192\) 0.287840 0.209128i 0.0207731 0.0150925i
\(193\) 9.97505 + 13.7295i 0.718020 + 0.988269i 0.999587 + 0.0287278i \(0.00914559\pi\)
−0.281568 + 0.959541i \(0.590854\pi\)
\(194\) 24.2977 7.89479i 1.74447 0.566813i
\(195\) 0 0
\(196\) −11.3262 8.22899i −0.809017 0.587785i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −5.99604 + 12.7195i −0.426120 + 0.903938i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −8.31254 + 11.4412i −0.587785 + 0.809017i
\(201\) 0.170363 0.524324i 0.0120165 0.0369829i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.206947 0.284838i −0.0144892 0.0199426i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.1540 7.21972i 0.909884 0.499398i
\(210\) 0 0
\(211\) −7.26809 + 10.0037i −0.500356 + 0.688681i −0.982256 0.187545i \(-0.939947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 14.2912 10.3832i 0.976930 0.709781i
\(215\) 0 0
\(216\) −0.717568 + 0.233152i −0.0488243 + 0.0158640i
\(217\) 0 0
\(218\) 0 0
\(219\) 0.558319i 0.0377277i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(224\) 0 0
\(225\) 12.1273 8.81097i 0.808484 0.587398i
\(226\) −4.32058 5.94677i −0.287401 0.395573i
\(227\) 27.9665 9.08687i 1.85620 0.603117i 0.860617 0.509252i \(-0.170078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) 0.382722 + 0.124354i 0.0253464 + 0.00823553i
\(229\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.67475 + 10.5634i −0.502790 + 0.692031i −0.982683 0.185296i \(-0.940675\pi\)
0.479893 + 0.877327i \(0.340675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.7625 13.6318i 1.22134 0.887352i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(240\) 0 0
\(241\) 29.9717i 1.93064i 0.261061 + 0.965322i \(0.415928\pi\)
−0.261061 + 0.965322i \(0.584072\pi\)
\(242\) −13.1232 8.35357i −0.843590 0.536988i
\(243\) 1.19974 0.0769631
\(244\) 0 0
\(245\) 0 0
\(246\) 0.246439 + 0.758462i 0.0157124 + 0.0483578i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.544295 0.176852i 0.0344933 0.0112075i
\(250\) 0 0
\(251\) −4.85410 3.52671i −0.306388 0.222604i 0.423957 0.905682i \(-0.360641\pi\)
−0.730345 + 0.683078i \(0.760641\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.94427 15.2169i 0.309017 0.951057i
\(257\) −5.44503 16.7581i −0.339651 1.04534i −0.964385 0.264502i \(-0.914792\pi\)
0.624734 0.780838i \(-0.285208\pi\)
\(258\) −0.648387 + 0.471081i −0.0403668 + 0.0293282i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −24.5863 17.8630i −1.51894 1.10358i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −0.0778938 0.409865i −0.00479403 0.0252254i
\(265\) 0 0
\(266\) 0 0
\(267\) −0.245828 + 0.756581i −0.0150444 + 0.0463020i
\(268\) −7.66129 23.5790i −0.467987 1.44032i
\(269\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(270\) 0 0
\(271\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(272\) −15.0582 4.89270i −0.913036 0.296663i
\(273\) 0 0
\(274\) 25.6739i 1.55101i
\(275\) 7.97899 + 14.5374i 0.481151 + 0.876638i
\(276\) 0 0
\(277\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(278\) −3.70820 + 11.4127i −0.222403 + 0.684487i
\(279\) 0 0
\(280\) 0 0
\(281\) −7.23112 9.95279i −0.431373 0.593734i 0.536895 0.843649i \(-0.319597\pi\)
−0.968268 + 0.249916i \(0.919597\pi\)
\(282\) 0 0
\(283\) 24.2099 + 7.86629i 1.43913 + 0.467602i 0.921627 0.388078i \(-0.126861\pi\)
0.517505 + 0.855680i \(0.326861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −9.96847 + 13.7204i −0.587398 + 0.808484i
\(289\) 0.411629 1.26686i 0.0242135 0.0745214i
\(290\) 0 0
\(291\) 0.649987 0.472243i 0.0381029 0.0276834i
\(292\) 14.7580 + 20.3126i 0.863647 + 1.18871i
\(293\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(294\) −0.418720 0.136050i −0.0244202 0.00793461i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.111489 + 0.877672i −0.00646925 + 0.0509277i
\(298\) 0 0
\(299\) 0 0
\(300\) −0.137431 + 0.422971i −0.00793461 + 0.0244202i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 17.2111 5.59223i 0.987125 0.320736i
\(305\) 0 0
\(306\) 13.5773 + 9.86449i 0.776163 + 0.563915i
\(307\) 13.1200i 0.748796i 0.927268 + 0.374398i \(0.122151\pi\)
−0.927268 + 0.374398i \(0.877849\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(312\) 0 0
\(313\) −23.6150 + 17.1573i −1.33480 + 0.969788i −0.335181 + 0.942154i \(0.608798\pi\)
−0.999618 + 0.0276348i \(0.991202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.326528 0.449427i 0.0182250 0.0250846i
\(322\) 0 0
\(323\) −5.53390 17.0316i −0.307914 0.947663i
\(324\) 14.5335 10.5592i 0.807417 0.586623i
\(325\) 0 0
\(326\) −31.7311 + 10.3100i −1.75742 + 0.571021i
\(327\) 0 0
\(328\) 29.0143 + 21.0801i 1.60204 + 1.16395i
\(329\) 0 0
\(330\) 0 0
\(331\) 35.9970 1.97857 0.989287 0.145981i \(-0.0466339\pi\)
0.989287 + 0.145981i \(0.0466339\pi\)
\(332\) 15.1277 20.8215i 0.830240 1.14273i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.68807 + 0.873408i −0.146429 + 0.0475775i −0.381314 0.924445i \(-0.624528\pi\)
0.234886 + 0.972023i \(0.424528\pi\)
\(338\) −17.4850 5.68121i −0.951057 0.309017i
\(339\) −0.187012 0.135872i −0.0101571 0.00737958i
\(340\) 0 0
\(341\) 0 0
\(342\) −19.1819 −1.03724
\(343\) 0 0
\(344\) −11.1374 + 34.2775i −0.600490 + 1.84812i
\(345\) 0 0
\(346\) 0 0
\(347\) −3.32456 4.57586i −0.178472 0.245645i 0.710404 0.703795i \(-0.248512\pi\)
−0.888875 + 0.458149i \(0.848512\pi\)
\(348\) 0 0
\(349\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −13.6678 12.8526i −0.728498 0.685048i
\(353\) −30.7905 −1.63881 −0.819405 0.573214i \(-0.805696\pi\)
−0.819405 + 0.573214i \(0.805696\pi\)
\(354\) 0.428688 0.590038i 0.0227845 0.0313602i
\(355\) 0 0
\(356\) 11.0550 + 34.0237i 0.585912 + 1.80325i
\(357\) 0 0
\(358\) 20.2762 + 27.9077i 1.07163 + 1.47497i
\(359\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(360\) 0 0
\(361\) 1.18802 + 0.863144i 0.0625272 + 0.0454286i
\(362\) 0 0
\(363\) −0.473674 0.122314i −0.0248614 0.00641980i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) 0 0
\(369\) −22.3441 30.7540i −1.16319 1.60099i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −12.7186 + 13.5253i −0.657662 + 0.699374i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −28.9063 + 21.0017i −1.48482 + 1.07878i −0.508853 + 0.860853i \(0.669930\pi\)
−0.975964 + 0.217930i \(0.930070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(384\) 0.503163i 0.0256769i
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 22.4549 30.9065i 1.14145 1.57107i
\(388\) 11.1649 34.3621i 0.566813 1.74447i
\(389\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −18.8300 + 6.11822i −0.951057 + 0.309017i
\(393\) −0.908929 0.295329i −0.0458494 0.0148974i
\(394\) 0 0
\(395\) 0 0
\(396\) 9.56847 + 17.4334i 0.480834 + 0.876060i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.18034 + 19.0211i 0.309017 + 0.951057i
\(401\) −14.9029 + 10.8276i −0.744216 + 0.540704i −0.894029 0.448010i \(-0.852133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −0.458276 0.630762i −0.0228567 0.0314596i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.497915 −0.0246504
\(409\) 19.9501 27.4589i 0.986469 1.35776i 0.0531978 0.998584i \(-0.483059\pi\)
0.933271 0.359174i \(-0.116941\pi\)
\(410\) 0 0
\(411\) −0.249495 0.767868i −0.0123067 0.0378761i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.377372i 0.0184800i
\(418\) 2.67410 21.0513i 0.130795 1.02965i
\(419\) −36.1749 −1.76726 −0.883630 0.468186i \(-0.844908\pi\)
−0.883630 + 0.468186i \(0.844908\pi\)
\(420\) 0 0
\(421\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(422\) 5.40380 + 16.6312i 0.263053 + 0.809593i
\(423\) 0 0
\(424\) 0 0
\(425\) 18.8227 6.11587i 0.913036 0.296663i
\(426\) 0 0
\(427\) 0 0
\(428\) 24.9820i 1.20755i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(432\) −0.329727 + 1.01479i −0.0158640 + 0.0488243i
\(433\) −1.35885 4.18210i −0.0653020 0.200979i 0.913082 0.407777i \(-0.133696\pi\)
−0.978384 + 0.206798i \(0.933696\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.638786 + 0.464105i 0.0305224 + 0.0221758i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 20.9862 0.999341
\(442\) 0 0
\(443\) 11.0137 33.8968i 0.523279 1.61049i −0.244416 0.969670i \(-0.578596\pi\)
0.767695 0.640816i \(-0.221404\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.14750 4.46642i −0.290118 0.210783i 0.433200 0.901298i \(-0.357384\pi\)
−0.723319 + 0.690514i \(0.757384\pi\)
\(450\) 21.1992i 0.999341i
\(451\) 36.8659 20.2342i 1.73595 0.952792i
\(452\) −10.3953 −0.488956
\(453\) 0 0
\(454\) 12.8508 39.5506i 0.603117 1.85620i
\(455\) 0 0
\(456\) 0.460415 0.334511i 0.0215609 0.0156649i
\(457\) 7.15719 + 9.85102i 0.334799 + 0.460811i 0.942913 0.333038i \(-0.108074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) 1.00421 + 0.326287i 0.0468725 + 0.0152298i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 5.70615 + 17.5617i 0.264332 + 0.813531i
\(467\) −24.2705 + 17.6336i −1.12311 + 0.815984i −0.984677 0.174389i \(-0.944205\pi\)
−0.138428 + 0.990372i \(0.544205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 32.7981i 1.50965i
\(473\) 30.7881 + 28.9518i 1.41564 + 1.33120i
\(474\) 0 0
\(475\) −13.2963 + 18.3008i −0.610077 + 0.839699i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 34.2913 + 24.9141i 1.56192 + 1.13480i
\(483\) 0 0
\(484\) −20.4662 + 8.07060i −0.930282 + 0.366845i
\(485\) 0 0
\(486\) 0.997285 1.37265i 0.0452378 0.0622645i
\(487\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(488\) 0 0
\(489\) −0.848838 + 0.616717i −0.0383858 + 0.0278889i
\(490\) 0 0
\(491\) −12.5975 + 4.09316i −0.568515 + 0.184722i −0.579149 0.815222i \(-0.696615\pi\)
0.0106338 + 0.999943i \(0.496615\pi\)
\(492\) 1.07263 + 0.348518i 0.0483578 + 0.0157124i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0.250107 0.769749i 0.0112075 0.0344933i
\(499\) 11.2061 + 34.4890i 0.501656 + 1.54394i 0.806321 + 0.591479i \(0.201456\pi\)
−0.304664 + 0.952460i \(0.598544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −8.06998 + 2.62210i −0.360181 + 0.117030i
\(503\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.578159 −0.0256769
\(508\) 0 0
\(509\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −13.3001 18.3060i −0.587785 0.809017i
\(513\) −1.14779 + 0.372938i −0.0506760 + 0.0164656i
\(514\) −23.6995 7.70043i −1.04534 0.339651i
\(515\) 0 0
\(516\) 1.13342i 0.0498961i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.7271 + 39.1700i 0.557585 + 1.71607i 0.689017 + 0.724745i \(0.258042\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −16.6190 22.8740i −0.726696 1.00021i −0.999275 0.0380781i \(-0.987876\pi\)
0.272578 0.962134i \(-0.412124\pi\)
\(524\) −40.8748 + 13.2810i −1.78563 + 0.580185i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.533685 0.251582i −0.0232257 0.0109487i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −10.7429 + 33.0632i −0.466201 + 1.43482i
\(532\) 0 0
\(533\) 0 0
\(534\) 0.661276 + 0.910169i 0.0286162 + 0.0393868i
\(535\) 0 0
\(536\) −33.3458 10.8347i −1.44032 0.467987i
\(537\) 0.877634 + 0.637639i 0.0378727 + 0.0275161i
\(538\) 0 0
\(539\) −2.92562 + 23.0313i −0.126015 + 0.992028i
\(540\) 0 0
\(541\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −18.1150 + 13.1613i −0.776675 + 0.564287i
\(545\) 0 0
\(546\) 0 0
\(547\) −44.1012 14.3293i −1.88563 0.612678i −0.983409 0.181402i \(-0.941936\pi\)
−0.902220 0.431276i \(-0.858064\pi\)
\(548\) −29.3741 21.3415i −1.25480 0.911664i
\(549\) 0 0
\(550\) 23.2651 + 2.95532i 0.992028 + 0.126015i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 9.97505 + 13.7295i 0.423036 + 0.582259i
\(557\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.248957 + 0.528118i −0.0105110 + 0.0222972i
\(562\) −17.3981 −0.733895
\(563\) 7.12022 9.80014i 0.300081 0.413027i −0.632175 0.774826i \(-0.717837\pi\)
0.932256 + 0.361799i \(0.117837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 29.1246 21.1603i 1.22420 0.889432i
\(567\) 0 0
\(568\) 0 0
\(569\) 44.6394 + 14.5042i 1.87138 + 0.608048i 0.991015 + 0.133753i \(0.0427029\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 42.4264i 1.77549i 0.460336 + 0.887745i \(0.347729\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 7.41152 + 22.8103i 0.308813 + 0.950429i
\(577\) −34.6150 + 25.1493i −1.44104 + 1.04698i −0.453218 + 0.891400i \(0.649724\pi\)
−0.987824 + 0.155579i \(0.950276\pi\)
\(578\) −1.10728 1.52404i −0.0460567 0.0633917i
\(579\) 0.717805 0.233229i 0.0298310 0.00969266i
\(580\) 0 0
\(581\) 0 0
\(582\) 1.13622i 0.0470978i
\(583\) 0 0
\(584\) 35.5078 1.46932
\(585\) 0 0
\(586\) 0 0
\(587\) −14.7043 45.2551i −0.606910 1.86788i −0.483087 0.875573i \(-0.660484\pi\)
−0.123823 0.992304i \(-0.539516\pi\)
\(588\) −0.503720 + 0.365974i −0.0207731 + 0.0150925i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.1035i 1.27727i −0.769510 0.638634i \(-0.779500\pi\)
0.769510 0.638634i \(-0.220500\pi\)
\(594\) 0.911489 + 0.857126i 0.0373988 + 0.0351683i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(600\) 0.369690 + 0.508834i 0.0150925 + 0.0207731i
\(601\) 12.6573 4.11260i 0.516301 0.167756i −0.0392649 0.999229i \(-0.512502\pi\)
0.555566 + 0.831472i \(0.312502\pi\)
\(602\) 0 0
\(603\) 30.0664 + 21.8445i 1.22440 + 0.889578i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(608\) 7.90861 24.3402i 0.320736 0.987125i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 22.5724 7.33421i 0.912434 0.296468i
\(613\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(614\) 15.0108 + 10.9060i 0.605789 + 0.440131i
\(615\) 0 0
\(616\) 0 0
\(617\) 28.3894 1.14291 0.571457 0.820632i \(-0.306378\pi\)
0.571457 + 0.820632i \(0.306378\pi\)
\(618\) 0 0
\(619\) 14.9516 46.0163i 0.600955 1.84955i 0.0784409 0.996919i \(-0.475006\pi\)
0.522514 0.852631i \(-0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.2254 14.6946i −0.809017 0.587785i
\(626\) 41.2806i 1.64990i
\(627\) −0.124595 0.655599i −0.00497584 0.0261821i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(632\) 0 0
\(633\) 0.323239 + 0.444901i 0.0128476 + 0.0176832i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.36257 16.5043i 0.211809 0.651880i −0.787556 0.616243i \(-0.788654\pi\)
0.999365 0.0356372i \(-0.0113461\pi\)
\(642\) −0.242772 0.747176i −0.00958145 0.0294887i
\(643\) 28.6904 20.8448i 1.13144 0.822039i 0.145537 0.989353i \(-0.453509\pi\)
0.985904 + 0.167313i \(0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0863 7.82611i −0.947663 0.307914i
\(647\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) 25.4055i 0.998023i
\(649\) −34.7876 16.3990i −1.36553 0.643718i
\(650\) 0 0
\(651\) 0 0
\(652\) −14.5806 + 44.8745i −0.571021 + 1.75742i
\(653\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 48.2364 15.6730i 1.88332 0.611926i
\(657\) −35.7948 11.6304i −1.39649 0.453746i
\(658\) 0 0
\(659\) 2.26047i 0.0880555i −0.999030 0.0440278i \(-0.985981\pi\)
0.999030 0.0440278i \(-0.0140190\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 29.9227 41.1850i 1.16298 1.60070i
\(663\) 0 0
\(664\) −11.2474 34.6159i −0.436483 1.34336i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 3.65721 5.03371i 0.140975 0.194035i −0.732691 0.680561i \(-0.761736\pi\)
0.873666 + 0.486526i \(0.161736\pi\)
\(674\) −1.23518 + 3.80151i −0.0475775 + 0.146429i
\(675\) −0.412159 1.26849i −0.0158640 0.0488243i
\(676\) −21.0344 + 15.2824i −0.809017 + 0.587785i
\(677\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(678\) −0.310909 + 0.101021i −0.0119404 + 0.00387967i
\(679\) 0 0
\(680\) 0 0
\(681\) 1.30778i 0.0501144i
\(682\) 0 0
\(683\) −42.0000 −1.60709 −0.803543 0.595247i \(-0.797054\pi\)
−0.803543 + 0.595247i \(0.797054\pi\)
\(684\) −15.9451 + 21.9465i −0.609675 + 0.839145i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 29.9596 + 41.2359i 1.14220 + 1.57210i
\(689\) 0 0
\(690\) 0 0
\(691\) −8.08626 5.87501i −0.307616 0.223496i 0.423257 0.906010i \(-0.360887\pi\)
−0.730873 + 0.682514i \(0.760887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −7.99890 −0.303634
\(695\) 0 0
\(696\) 0 0
\(697\) −15.5095 47.7333i −0.587463 1.80803i
\(698\) 0 0
\(699\) 0.341325 + 0.469794i 0.0129101 + 0.0177692i
\(700\) 0 0
\(701\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −26.0664 + 4.95386i −0.982416 + 0.186706i
\(705\) 0 0
\(706\) −25.5947 + 35.2281i −0.963269 + 1.32583i
\(707\) 0 0
\(708\) −0.318727 0.980942i −0.0119785 0.0368661i
\(709\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 48.1168 + 15.6341i 1.80325 + 0.585912i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 48.7845 1.82316
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.97509 0.641744i 0.0735051 0.0238833i
\(723\) 1.26771 + 0.411905i 0.0471468 + 0.0153189i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.533685 + 0.440268i −0.0198069 + 0.0163399i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −8.31047 + 25.5770i −0.307795 + 0.947296i
\(730\) 0 0
\(731\) 40.8057 29.6471i 1.50925 1.09654i
\(732\) 0 0
\(733\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −28.1648 + 29.9512i −1.03746 + 1.10327i
\(738\) −53.7599 −1.97893
\(739\) 11.3004 15.5537i 0.415693 0.572152i −0.548902 0.835886i \(-0.684954\pi\)
0.964595 + 0.263734i \(0.0849541\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 38.5797i 1.41156i
\(748\) 4.90219 + 25.7945i 0.179242 + 0.943142i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(752\) 0 0
\(753\) −0.215880 + 0.156846i −0.00786711 + 0.00571579i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(758\) 50.5301i 1.83534i
\(759\) 0 0
\(760\) 0 0
\(761\) −32.2419 + 44.3772i −1.16877 + 1.60867i −0.497475 + 0.867479i \(0.665739\pi\)
−0.671293 + 0.741192i \(0.734261\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.575680 0.418256i −0.0207731 0.0150925i
\(769\) 50.9117i 1.83592i −0.396670 0.917961i \(-0.629834\pi\)
0.396670 0.917961i \(-0.370166\pi\)
\(770\) 0 0
\(771\) −0.783649 −0.0282224
\(772\) 19.9501 27.4589i 0.718020 0.988269i
\(773\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(774\) −16.6951 51.3823i −0.600094 1.84690i
\(775\) 0 0
\(776\) −30.0336 41.3377i −1.07814 1.48394i
\(777\) 0 0
\(778\) 0 0
\(779\) 46.4097 + 33.7186i 1.66280 + 1.20810i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −8.65248 + 26.6296i −0.309017 + 0.951057i
\(785\) 0 0
\(786\) −1.09344 + 0.794433i −0.0390018 + 0.0283365i
\(787\) 32.5745 + 44.8350i 1.16116 + 1.59820i 0.706877 + 0.707336i \(0.250103\pi\)
0.454280 + 0.890859i \(0.349897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 27.8997 + 3.54405i 0.991374 + 0.125932i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 26.8999 + 8.74032i 0.951057 + 0.309017i
\(801\) −43.3848 31.5209i −1.53293 1.11374i
\(802\) 26.0512i 0.919901i
\(803\) 17.7539 37.6617i 0.626522 1.32905i
\(804\) −1.10261 −0.0388862
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13.6290 + 18.7587i 0.479169 + 0.659519i 0.978345 0.206981i \(-0.0663639\pi\)
−0.499176 + 0.866501i \(0.666364\pi\)
\(810\) 0 0
\(811\) −11.4011 3.70445i −0.400347 0.130081i 0.101921 0.994792i \(-0.467501\pi\)
−0.502268 + 0.864712i \(0.667501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.413893 + 0.569675i −0.0144892 + 0.0199426i
\(817\) −17.8149 + 54.8285i −0.623264 + 1.91821i
\(818\) −14.8328 45.6507i −0.518617 1.59614i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(822\) −1.08593 0.352840i −0.0378761 0.0123067i
\(823\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(824\) 0 0
\(825\) 0.724545 0.137698i 0.0252254 0.00479403i
\(826\) 0 0
\(827\) −26.5907 + 36.5990i −0.924650 + 1.27267i 0.0372604 + 0.999306i \(0.488137\pi\)
−0.961910 + 0.273366i \(0.911863\pi\)
\(828\) 0 0
\(829\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.3518 + 8.56222i 0.913036 + 0.296663i
\(834\) 0.431760 + 0.313692i 0.0149506 + 0.0108623i
\(835\) 0 0
\(836\) −21.8624 20.5584i −0.756126 0.711029i
\(837\) 0 0
\(838\) −30.0705 + 41.3885i −1.03877 + 1.42974i
\(839\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(840\) 0 0
\(841\) 23.4615 17.0458i 0.809017 0.587785i
\(842\) 0 0
\(843\) −0.520352 + 0.169073i −0.0179219 + 0.00582317i
\(844\) 23.5200 + 7.64212i 0.809593 + 0.263053i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.665442 0.915902i 0.0228379 0.0314337i
\(850\) 8.64915 26.6193i 0.296663 0.913036i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −28.5825 20.7664i −0.976930 0.709781i
\(857\) 13.4344i 0.458912i −0.973319 0.229456i \(-0.926305\pi\)
0.973319 0.229456i \(-0.0736947\pi\)
\(858\) 0 0
\(859\) 25.9930 0.886869 0.443434 0.896307i \(-0.353760\pi\)
0.443434 + 0.896307i \(0.353760\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(864\) 0.886963 + 1.22080i 0.0301751 + 0.0415325i
\(865\) 0 0
\(866\) −5.91438 1.92170i −0.200979 0.0653020i
\(867\) −0.0479275 0.0348214i −0.00162770 0.00118260i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 16.7363 + 51.5091i 0.566439 + 1.74332i
\(874\) 0 0
\(875\) 0 0
\(876\) 1.06199 0.345060i 0.0358812 0.0116585i
\(877\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.6878 0.629610 0.314805 0.949156i \(-0.398061\pi\)
0.314805 + 0.949156i \(0.398061\pi\)
\(882\) 17.4448 24.0107i 0.587398 0.808484i
\(883\) −11.2886 + 34.7428i −0.379892 + 1.16919i 0.560227 + 0.828339i \(0.310714\pi\)
−0.940119 + 0.340848i \(0.889286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −29.6269 40.7779i −0.995335 1.36996i
\(887\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −26.9466 12.7028i −0.902745 0.425558i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −10.2203 + 3.32076i −0.341055 + 0.110815i
\(899\) 0 0
\(900\) −24.2545 17.6219i −0.808484 0.587398i
\(901\) 0 0
\(902\) 7.49452 58.9989i 0.249540 1.96445i
\(903\) 0 0
\(904\) −8.64117 + 11.8935i −0.287401 + 0.395573i
\(905\) 0 0
\(906\) 0 0
\(907\) 48.2013 35.0203i 1.60050 1.16283i 0.713943 0.700204i \(-0.246908\pi\)
0.886554 0.462625i \(-0.153092\pi\)
\(908\) −34.5685 47.5795i −1.14720 1.57898i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(912\) 0.804834i 0.0266507i
\(913\) −42.3394 5.37829i −1.40123 0.177995i
\(914\) 17.2202 0.569594
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.20807 0.877711i 0.0398721 0.0289688i
\(919\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(920\) 0 0
\(921\) 0.554936 + 0.180310i 0.0182858 + 0.00594140i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.2625 25.6197i 1.15693 0.840555i 0.167539 0.985865i \(-0.446418\pi\)
0.989386 + 0.145310i \(0.0464179\pi\)
\(930\) 0 0
\(931\) −30.1195 + 9.78640i −0.987125 + 0.320736i
\(932\) 24.8360 + 8.06971i 0.813531 + 0.264332i
\(933\) 0 0
\(934\) 42.4264i 1.38823i
\(935\) 0 0
\(936\) 0 0
\(937\) 28.2540 38.8882i 0.923016 1.27042i −0.0395055 0.999219i \(-0.512578\pi\)
0.962522 0.271204i \(-0.0874217\pi\)
\(938\) 0 0
\(939\) 0.401159 + 1.23464i 0.0130913 + 0.0402910i
\(940\) 0 0
\(941\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −37.5250 27.2635i −1.22134 0.887352i
\(945\) 0 0
\(946\) 58.7171 11.1590i 1.90906 0.362812i
\(947\) −60.3804 −1.96210 −0.981050 0.193757i \(-0.937933\pi\)
−0.981050 + 0.193757i \(0.937933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 9.88576 + 30.4252i 0.320736 + 0.987125i
\(951\) 0 0
\(952\) 0 0
\(953\) 11.3413 3.68501i 0.367381 0.119369i −0.119508 0.992833i \(-0.538132\pi\)
0.486889 + 0.873464i \(0.338132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 9.57953 + 29.4828i 0.309017 + 0.951057i
\(962\) 0 0
\(963\) 22.0116 + 30.2963i 0.709313 + 0.976285i
\(964\) 57.0095 18.5235i 1.83615 0.596602i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −7.77885 + 30.1246i −0.250022 + 0.968240i
\(969\) −0.796439 −0.0255853
\(970\) 0 0
\(971\) 16.6869 51.3571i 0.535509 1.64813i −0.207039 0.978333i \(-0.566383\pi\)
0.742547 0.669793i \(-0.233617\pi\)
\(972\) −0.741478 2.28203i −0.0237829 0.0731963i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.85410 3.52671i −0.155296 0.112829i 0.507423 0.861697i \(-0.330598\pi\)
−0.662720 + 0.748867i \(0.730598\pi\)
\(978\) 1.48382i 0.0474474i
\(979\) 40.6409 43.2185i 1.29889 1.38127i
\(980\) 0 0
\(981\) 0 0
\(982\) −5.78861 + 17.8155i −0.184722 + 0.568515i
\(983\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(984\) 1.29037 0.937511i 0.0411356 0.0298868i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0.494712 1.52257i 0.0156992 0.0483172i
\(994\) 0 0
\(995\) 0 0
\(996\) −0.672785 0.926009i −0.0213180 0.0293417i
\(997\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(998\) 48.7748 + 15.8479i 1.54394 + 0.501656i
\(999\) 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 88.2.k.a.51.2 yes 8
3.2 odd 2 792.2.bp.a.667.1 8
4.3 odd 2 352.2.s.a.271.1 8
8.3 odd 2 CM 88.2.k.a.51.2 yes 8
8.5 even 2 352.2.s.a.271.1 8
11.2 odd 10 968.2.k.d.699.2 8
11.3 even 5 968.2.k.b.723.1 8
11.4 even 5 968.2.k.d.475.2 8
11.5 even 5 968.2.g.a.483.7 8
11.6 odd 10 968.2.g.a.483.3 8
11.7 odd 10 968.2.k.c.475.1 8
11.8 odd 10 inner 88.2.k.a.19.2 8
11.9 even 5 968.2.k.c.699.1 8
11.10 odd 2 968.2.k.b.403.1 8
24.11 even 2 792.2.bp.a.667.1 8
33.8 even 10 792.2.bp.a.19.1 8
44.19 even 10 352.2.s.a.239.1 8
44.27 odd 10 3872.2.g.b.1935.3 8
44.39 even 10 3872.2.g.b.1935.4 8
88.3 odd 10 968.2.k.b.723.1 8
88.5 even 10 3872.2.g.b.1935.3 8
88.19 even 10 inner 88.2.k.a.19.2 8
88.27 odd 10 968.2.g.a.483.7 8
88.35 even 10 968.2.k.d.699.2 8
88.43 even 2 968.2.k.b.403.1 8
88.51 even 10 968.2.k.c.475.1 8
88.59 odd 10 968.2.k.d.475.2 8
88.61 odd 10 3872.2.g.b.1935.4 8
88.75 odd 10 968.2.k.c.699.1 8
88.83 even 10 968.2.g.a.483.3 8
88.85 odd 10 352.2.s.a.239.1 8
264.107 odd 10 792.2.bp.a.19.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.k.a.19.2 8 11.8 odd 10 inner
88.2.k.a.19.2 8 88.19 even 10 inner
88.2.k.a.51.2 yes 8 1.1 even 1 trivial
88.2.k.a.51.2 yes 8 8.3 odd 2 CM
352.2.s.a.239.1 8 44.19 even 10
352.2.s.a.239.1 8 88.85 odd 10
352.2.s.a.271.1 8 4.3 odd 2
352.2.s.a.271.1 8 8.5 even 2
792.2.bp.a.19.1 8 33.8 even 10
792.2.bp.a.19.1 8 264.107 odd 10
792.2.bp.a.667.1 8 3.2 odd 2
792.2.bp.a.667.1 8 24.11 even 2
968.2.g.a.483.3 8 11.6 odd 10
968.2.g.a.483.3 8 88.83 even 10
968.2.g.a.483.7 8 11.5 even 5
968.2.g.a.483.7 8 88.27 odd 10
968.2.k.b.403.1 8 11.10 odd 2
968.2.k.b.403.1 8 88.43 even 2
968.2.k.b.723.1 8 11.3 even 5
968.2.k.b.723.1 8 88.3 odd 10
968.2.k.c.475.1 8 11.7 odd 10
968.2.k.c.475.1 8 88.51 even 10
968.2.k.c.699.1 8 11.9 even 5
968.2.k.c.699.1 8 88.75 odd 10
968.2.k.d.475.2 8 11.4 even 5
968.2.k.d.475.2 8 88.59 odd 10
968.2.k.d.699.2 8 11.2 odd 10
968.2.k.d.699.2 8 88.35 even 10
3872.2.g.b.1935.3 8 44.27 odd 10
3872.2.g.b.1935.3 8 88.5 even 10
3872.2.g.b.1935.4 8 44.39 even 10
3872.2.g.b.1935.4 8 88.61 odd 10