Properties

Label 1110.2.h.e.961.3
Level $1110$
Weight $2$
Character 1110.961
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1110,2,Mod(961,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.h (of order \(2\), degree \(1\), 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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.3
Root \(2.37228i\) of defining polynomial
Character \(\chi\) \(=\) 1110.961
Dual form 1110.2.h.e.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.37228 q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.37228 q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.37228 q^{11} +1.00000 q^{12} +4.74456i q^{13} -2.37228i q^{14} -1.00000i q^{15} +1.00000 q^{16} -0.372281i q^{17} +1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} +2.37228 q^{21} +6.37228i q^{22} +8.74456i q^{23} +1.00000i q^{24} -1.00000 q^{25} -4.74456 q^{26} -1.00000 q^{27} +2.37228 q^{28} -3.62772i q^{29} +1.00000 q^{30} +8.37228i q^{31} +1.00000i q^{32} -6.37228 q^{33} +0.372281 q^{34} -2.37228i q^{35} -1.00000 q^{36} +(-5.74456 - 2.00000i) q^{37} +6.00000 q^{38} -4.74456i q^{39} +1.00000 q^{40} -9.11684 q^{41} +2.37228i q^{42} +4.37228i q^{43} -6.37228 q^{44} +1.00000i q^{45} -8.74456 q^{46} -1.00000 q^{48} -1.37228 q^{49} -1.00000i q^{50} +0.372281i q^{51} -4.74456i q^{52} -8.37228 q^{53} -1.00000i q^{54} +6.37228i q^{55} +2.37228i q^{56} +6.00000i q^{57} +3.62772 q^{58} +8.00000i q^{59} +1.00000i q^{60} +5.62772i q^{61} -8.37228 q^{62} -2.37228 q^{63} -1.00000 q^{64} -4.74456 q^{65} -6.37228i q^{66} +8.74456 q^{67} +0.372281i q^{68} -8.74456i q^{69} +2.37228 q^{70} -8.74456 q^{71} -1.00000i q^{72} +14.0000 q^{73} +(2.00000 - 5.74456i) q^{74} +1.00000 q^{75} +6.00000i q^{76} -15.1168 q^{77} +4.74456 q^{78} +6.74456i q^{79} +1.00000i q^{80} +1.00000 q^{81} -9.11684i q^{82} -5.48913 q^{83} -2.37228 q^{84} +0.372281 q^{85} -4.37228 q^{86} +3.62772i q^{87} -6.37228i q^{88} -2.74456i q^{89} -1.00000 q^{90} -11.2554i q^{91} -8.74456i q^{92} -8.37228i q^{93} +6.00000 q^{95} -1.00000i q^{96} +7.11684i q^{97} -1.37228i q^{98} +6.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{4} + 2 q^{7} + 4 q^{9} - 4 q^{10} + 14 q^{11} + 4 q^{12} + 4 q^{16} - 2 q^{21} - 4 q^{25} + 4 q^{26} - 4 q^{27} - 2 q^{28} + 4 q^{30} - 14 q^{33} - 10 q^{34} - 4 q^{36} + 24 q^{38} + 4 q^{40} - 2 q^{41} - 14 q^{44} - 12 q^{46} - 4 q^{48} + 6 q^{49} - 22 q^{53} + 26 q^{58} - 22 q^{62} + 2 q^{63} - 4 q^{64} + 4 q^{65} + 12 q^{67} - 2 q^{70} - 12 q^{71} + 56 q^{73} + 8 q^{74} + 4 q^{75} - 26 q^{77} - 4 q^{78} + 4 q^{81} + 24 q^{83} + 2 q^{84} - 10 q^{85} - 6 q^{86} - 4 q^{90} + 24 q^{95} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) −2.37228 −0.896638 −0.448319 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.37228 1.92132 0.960658 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.74456i 1.31590i 0.753059 + 0.657952i \(0.228577\pi\)
−0.753059 + 0.657952i \(0.771423\pi\)
\(14\) 2.37228i 0.634019i
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 0.372281i 0.0902915i −0.998980 0.0451457i \(-0.985625\pi\)
0.998980 0.0451457i \(-0.0143752\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 2.37228 0.517674
\(22\) 6.37228i 1.35857i
\(23\) 8.74456i 1.82337i 0.410893 + 0.911684i \(0.365217\pi\)
−0.410893 + 0.911684i \(0.634783\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) −4.74456 −0.930485
\(27\) −1.00000 −0.192450
\(28\) 2.37228 0.448319
\(29\) 3.62772i 0.673650i −0.941567 0.336825i \(-0.890647\pi\)
0.941567 0.336825i \(-0.109353\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.37228i 1.50371i 0.659331 + 0.751853i \(0.270840\pi\)
−0.659331 + 0.751853i \(0.729160\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −6.37228 −1.10927
\(34\) 0.372281 0.0638457
\(35\) 2.37228i 0.400989i
\(36\) −1.00000 −0.166667
\(37\) −5.74456 2.00000i −0.944400 0.328798i
\(38\) 6.00000 0.973329
\(39\) 4.74456i 0.759738i
\(40\) 1.00000 0.158114
\(41\) −9.11684 −1.42381 −0.711906 0.702275i \(-0.752168\pi\)
−0.711906 + 0.702275i \(0.752168\pi\)
\(42\) 2.37228i 0.366051i
\(43\) 4.37228i 0.666767i 0.942791 + 0.333383i \(0.108190\pi\)
−0.942791 + 0.333383i \(0.891810\pi\)
\(44\) −6.37228 −0.960658
\(45\) 1.00000i 0.149071i
\(46\) −8.74456 −1.28932
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −1.37228 −0.196040
\(50\) 1.00000i 0.141421i
\(51\) 0.372281i 0.0521298i
\(52\) 4.74456i 0.657952i
\(53\) −8.37228 −1.15002 −0.575011 0.818146i \(-0.695002\pi\)
−0.575011 + 0.818146i \(0.695002\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 6.37228i 0.859238i
\(56\) 2.37228i 0.317009i
\(57\) 6.00000i 0.794719i
\(58\) 3.62772 0.476343
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 5.62772i 0.720556i 0.932845 + 0.360278i \(0.117318\pi\)
−0.932845 + 0.360278i \(0.882682\pi\)
\(62\) −8.37228 −1.06328
\(63\) −2.37228 −0.298879
\(64\) −1.00000 −0.125000
\(65\) −4.74456 −0.588491
\(66\) 6.37228i 0.784374i
\(67\) 8.74456 1.06832 0.534159 0.845384i \(-0.320628\pi\)
0.534159 + 0.845384i \(0.320628\pi\)
\(68\) 0.372281i 0.0451457i
\(69\) 8.74456i 1.05272i
\(70\) 2.37228 0.283542
\(71\) −8.74456 −1.03779 −0.518894 0.854838i \(-0.673656\pi\)
−0.518894 + 0.854838i \(0.673656\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 2.00000 5.74456i 0.232495 0.667792i
\(75\) 1.00000 0.115470
\(76\) 6.00000i 0.688247i
\(77\) −15.1168 −1.72272
\(78\) 4.74456 0.537216
\(79\) 6.74456i 0.758823i 0.925228 + 0.379411i \(0.123873\pi\)
−0.925228 + 0.379411i \(0.876127\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 9.11684i 1.00679i
\(83\) −5.48913 −0.602510 −0.301255 0.953544i \(-0.597406\pi\)
−0.301255 + 0.953544i \(0.597406\pi\)
\(84\) −2.37228 −0.258837
\(85\) 0.372281 0.0403796
\(86\) −4.37228 −0.471475
\(87\) 3.62772i 0.388932i
\(88\) 6.37228i 0.679287i
\(89\) 2.74456i 0.290923i −0.989364 0.145462i \(-0.953533\pi\)
0.989364 0.145462i \(-0.0464667\pi\)
\(90\) −1.00000 −0.105409
\(91\) 11.2554i 1.17989i
\(92\) 8.74456i 0.911684i
\(93\) 8.37228i 0.868165i
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 1.00000i 0.102062i
\(97\) 7.11684i 0.722606i 0.932448 + 0.361303i \(0.117668\pi\)
−0.932448 + 0.361303i \(0.882332\pi\)
\(98\) 1.37228i 0.138621i
\(99\) 6.37228 0.640438
\(100\) 1.00000 0.100000
\(101\) −5.25544 −0.522936 −0.261468 0.965212i \(-0.584207\pi\)
−0.261468 + 0.965212i \(0.584207\pi\)
\(102\) −0.372281 −0.0368613
\(103\) 7.48913i 0.737925i 0.929444 + 0.368963i \(0.120287\pi\)
−0.929444 + 0.368963i \(0.879713\pi\)
\(104\) 4.74456 0.465243
\(105\) 2.37228i 0.231511i
\(106\) 8.37228i 0.813188i
\(107\) 3.25544 0.314715 0.157358 0.987542i \(-0.449703\pi\)
0.157358 + 0.987542i \(0.449703\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.11684i 0.298540i −0.988797 0.149270i \(-0.952308\pi\)
0.988797 0.149270i \(-0.0476923\pi\)
\(110\) −6.37228 −0.607573
\(111\) 5.74456 + 2.00000i 0.545250 + 0.189832i
\(112\) −2.37228 −0.224160
\(113\) 9.86141i 0.927683i −0.885918 0.463842i \(-0.846471\pi\)
0.885918 0.463842i \(-0.153529\pi\)
\(114\) −6.00000 −0.561951
\(115\) −8.74456 −0.815435
\(116\) 3.62772i 0.336825i
\(117\) 4.74456i 0.438635i
\(118\) −8.00000 −0.736460
\(119\) 0.883156i 0.0809588i
\(120\) −1.00000 −0.0912871
\(121\) 29.6060 2.69145
\(122\) −5.62772 −0.509510
\(123\) 9.11684 0.822038
\(124\) 8.37228i 0.751853i
\(125\) 1.00000i 0.0894427i
\(126\) 2.37228i 0.211340i
\(127\) −9.48913 −0.842024 −0.421012 0.907055i \(-0.638325\pi\)
−0.421012 + 0.907055i \(0.638325\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.37228i 0.384958i
\(130\) 4.74456i 0.416126i
\(131\) 9.48913i 0.829069i 0.910034 + 0.414534i \(0.136056\pi\)
−0.910034 + 0.414534i \(0.863944\pi\)
\(132\) 6.37228 0.554636
\(133\) 14.2337i 1.23422i
\(134\) 8.74456i 0.755415i
\(135\) 1.00000i 0.0860663i
\(136\) −0.372281 −0.0319229
\(137\) −2.74456 −0.234484 −0.117242 0.993103i \(-0.537405\pi\)
−0.117242 + 0.993103i \(0.537405\pi\)
\(138\) 8.74456 0.744387
\(139\) −1.62772 −0.138061 −0.0690306 0.997615i \(-0.521991\pi\)
−0.0690306 + 0.997615i \(0.521991\pi\)
\(140\) 2.37228i 0.200494i
\(141\) 0 0
\(142\) 8.74456i 0.733827i
\(143\) 30.2337i 2.52827i
\(144\) 1.00000 0.0833333
\(145\) 3.62772 0.301266
\(146\) 14.0000i 1.15865i
\(147\) 1.37228 0.113184
\(148\) 5.74456 + 2.00000i 0.472200 + 0.164399i
\(149\) 11.4891 0.941226 0.470613 0.882340i \(-0.344033\pi\)
0.470613 + 0.882340i \(0.344033\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) −11.2554 −0.915955 −0.457977 0.888964i \(-0.651426\pi\)
−0.457977 + 0.888964i \(0.651426\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0.372281i 0.0300972i
\(154\) 15.1168i 1.21815i
\(155\) −8.37228 −0.672478
\(156\) 4.74456i 0.379869i
\(157\) −16.3723 −1.30665 −0.653325 0.757077i \(-0.726627\pi\)
−0.653325 + 0.757077i \(0.726627\pi\)
\(158\) −6.74456 −0.536569
\(159\) 8.37228 0.663965
\(160\) −1.00000 −0.0790569
\(161\) 20.7446i 1.63490i
\(162\) 1.00000i 0.0785674i
\(163\) 9.11684i 0.714086i 0.934088 + 0.357043i \(0.116215\pi\)
−0.934088 + 0.357043i \(0.883785\pi\)
\(164\) 9.11684 0.711906
\(165\) 6.37228i 0.496081i
\(166\) 5.48913i 0.426039i
\(167\) 3.25544i 0.251913i −0.992036 0.125957i \(-0.959800\pi\)
0.992036 0.125957i \(-0.0402000\pi\)
\(168\) 2.37228i 0.183025i
\(169\) −9.51087 −0.731606
\(170\) 0.372281i 0.0285527i
\(171\) 6.00000i 0.458831i
\(172\) 4.37228i 0.333383i
\(173\) −7.62772 −0.579925 −0.289962 0.957038i \(-0.593643\pi\)
−0.289962 + 0.957038i \(0.593643\pi\)
\(174\) −3.62772 −0.275017
\(175\) 2.37228 0.179328
\(176\) 6.37228 0.480329
\(177\) 8.00000i 0.601317i
\(178\) 2.74456 0.205714
\(179\) 8.74456i 0.653599i −0.945094 0.326800i \(-0.894030\pi\)
0.945094 0.326800i \(-0.105970\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 22.7446 1.69059 0.845295 0.534300i \(-0.179425\pi\)
0.845295 + 0.534300i \(0.179425\pi\)
\(182\) 11.2554 0.834309
\(183\) 5.62772i 0.416013i
\(184\) 8.74456 0.644658
\(185\) 2.00000 5.74456i 0.147043 0.422349i
\(186\) 8.37228 0.613885
\(187\) 2.37228i 0.173478i
\(188\) 0 0
\(189\) 2.37228 0.172558
\(190\) 6.00000i 0.435286i
\(191\) 7.11684i 0.514957i 0.966284 + 0.257478i \(0.0828916\pi\)
−0.966284 + 0.257478i \(0.917108\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.2337i 1.31249i −0.754548 0.656245i \(-0.772144\pi\)
0.754548 0.656245i \(-0.227856\pi\)
\(194\) −7.11684 −0.510960
\(195\) 4.74456 0.339765
\(196\) 1.37228 0.0980201
\(197\) 15.4891 1.10355 0.551777 0.833992i \(-0.313950\pi\)
0.551777 + 0.833992i \(0.313950\pi\)
\(198\) 6.37228i 0.452858i
\(199\) 10.7446i 0.761662i 0.924645 + 0.380831i \(0.124362\pi\)
−0.924645 + 0.380831i \(0.875638\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) −8.74456 −0.616794
\(202\) 5.25544i 0.369771i
\(203\) 8.60597i 0.604021i
\(204\) 0.372281i 0.0260649i
\(205\) 9.11684i 0.636748i
\(206\) −7.48913 −0.521792
\(207\) 8.74456i 0.607789i
\(208\) 4.74456i 0.328976i
\(209\) 38.2337i 2.64468i
\(210\) −2.37228 −0.163703
\(211\) 23.8614 1.64269 0.821343 0.570434i \(-0.193225\pi\)
0.821343 + 0.570434i \(0.193225\pi\)
\(212\) 8.37228 0.575011
\(213\) 8.74456 0.599168
\(214\) 3.25544i 0.222537i
\(215\) −4.37228 −0.298187
\(216\) 1.00000i 0.0680414i
\(217\) 19.8614i 1.34828i
\(218\) 3.11684 0.211099
\(219\) −14.0000 −0.946032
\(220\) 6.37228i 0.429619i
\(221\) 1.76631 0.118815
\(222\) −2.00000 + 5.74456i −0.134231 + 0.385550i
\(223\) 18.3723 1.23030 0.615149 0.788411i \(-0.289096\pi\)
0.615149 + 0.788411i \(0.289096\pi\)
\(224\) 2.37228i 0.158505i
\(225\) −1.00000 −0.0666667
\(226\) 9.86141 0.655971
\(227\) 6.37228i 0.422943i 0.977384 + 0.211472i \(0.0678256\pi\)
−0.977384 + 0.211472i \(0.932174\pi\)
\(228\) 6.00000i 0.397360i
\(229\) 4.51087 0.298087 0.149043 0.988831i \(-0.452381\pi\)
0.149043 + 0.988831i \(0.452381\pi\)
\(230\) 8.74456i 0.576599i
\(231\) 15.1168 0.994615
\(232\) −3.62772 −0.238171
\(233\) 14.7446 0.965948 0.482974 0.875635i \(-0.339556\pi\)
0.482974 + 0.875635i \(0.339556\pi\)
\(234\) −4.74456 −0.310162
\(235\) 0 0
\(236\) 8.00000i 0.520756i
\(237\) 6.74456i 0.438106i
\(238\) −0.883156 −0.0572465
\(239\) 7.86141i 0.508512i −0.967137 0.254256i \(-0.918169\pi\)
0.967137 0.254256i \(-0.0818306\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 21.4891i 1.38424i 0.721784 + 0.692118i \(0.243322\pi\)
−0.721784 + 0.692118i \(0.756678\pi\)
\(242\) 29.6060i 1.90314i
\(243\) −1.00000 −0.0641500
\(244\) 5.62772i 0.360278i
\(245\) 1.37228i 0.0876718i
\(246\) 9.11684i 0.581269i
\(247\) 28.4674 1.81134
\(248\) 8.37228 0.531640
\(249\) 5.48913 0.347859
\(250\) 1.00000 0.0632456
\(251\) 4.00000i 0.252478i 0.992000 + 0.126239i \(0.0402906\pi\)
−0.992000 + 0.126239i \(0.959709\pi\)
\(252\) 2.37228 0.149440
\(253\) 55.7228i 3.50326i
\(254\) 9.48913i 0.595401i
\(255\) −0.372281 −0.0233132
\(256\) 1.00000 0.0625000
\(257\) 23.4891i 1.46521i 0.680653 + 0.732606i \(0.261696\pi\)
−0.680653 + 0.732606i \(0.738304\pi\)
\(258\) 4.37228 0.272206
\(259\) 13.6277 + 4.74456i 0.846785 + 0.294813i
\(260\) 4.74456 0.294245
\(261\) 3.62772i 0.224550i
\(262\) −9.48913 −0.586240
\(263\) 16.6060 1.02397 0.511984 0.858995i \(-0.328911\pi\)
0.511984 + 0.858995i \(0.328911\pi\)
\(264\) 6.37228i 0.392187i
\(265\) 8.37228i 0.514305i
\(266\) −14.2337 −0.872723
\(267\) 2.74456i 0.167965i
\(268\) −8.74456 −0.534159
\(269\) −28.2337 −1.72144 −0.860719 0.509080i \(-0.829986\pi\)
−0.860719 + 0.509080i \(0.829986\pi\)
\(270\) 1.00000 0.0608581
\(271\) 1.48913 0.0904579 0.0452290 0.998977i \(-0.485598\pi\)
0.0452290 + 0.998977i \(0.485598\pi\)
\(272\) 0.372281i 0.0225729i
\(273\) 11.2554i 0.681210i
\(274\) 2.74456i 0.165805i
\(275\) −6.37228 −0.384263
\(276\) 8.74456i 0.526361i
\(277\) 9.48913i 0.570146i −0.958506 0.285073i \(-0.907982\pi\)
0.958506 0.285073i \(-0.0920179\pi\)
\(278\) 1.62772i 0.0976241i
\(279\) 8.37228i 0.501235i
\(280\) −2.37228 −0.141771
\(281\) 20.9783i 1.25146i −0.780041 0.625729i \(-0.784802\pi\)
0.780041 0.625729i \(-0.215198\pi\)
\(282\) 0 0
\(283\) 30.7446i 1.82757i −0.406193 0.913787i \(-0.633144\pi\)
0.406193 0.913787i \(-0.366856\pi\)
\(284\) 8.74456 0.518894
\(285\) −6.00000 −0.355409
\(286\) −30.2337 −1.78776
\(287\) 21.6277 1.27664
\(288\) 1.00000i 0.0589256i
\(289\) 16.8614 0.991847
\(290\) 3.62772i 0.213027i
\(291\) 7.11684i 0.417197i
\(292\) −14.0000 −0.819288
\(293\) −13.1168 −0.766294 −0.383147 0.923687i \(-0.625160\pi\)
−0.383147 + 0.923687i \(0.625160\pi\)
\(294\) 1.37228i 0.0800331i
\(295\) −8.00000 −0.465778
\(296\) −2.00000 + 5.74456i −0.116248 + 0.333896i
\(297\) −6.37228 −0.369757
\(298\) 11.4891i 0.665547i
\(299\) −41.4891 −2.39938
\(300\) −1.00000 −0.0577350
\(301\) 10.3723i 0.597848i
\(302\) 11.2554i 0.647678i
\(303\) 5.25544 0.301917
\(304\) 6.00000i 0.344124i
\(305\) −5.62772 −0.322242
\(306\) 0.372281 0.0212819
\(307\) 2.51087 0.143303 0.0716516 0.997430i \(-0.477173\pi\)
0.0716516 + 0.997430i \(0.477173\pi\)
\(308\) 15.1168 0.861362
\(309\) 7.48913i 0.426041i
\(310\) 8.37228i 0.475514i
\(311\) 19.8614i 1.12624i 0.826376 + 0.563119i \(0.190399\pi\)
−0.826376 + 0.563119i \(0.809601\pi\)
\(312\) −4.74456 −0.268608
\(313\) 18.2337i 1.03063i 0.857001 + 0.515314i \(0.172325\pi\)
−0.857001 + 0.515314i \(0.827675\pi\)
\(314\) 16.3723i 0.923941i
\(315\) 2.37228i 0.133663i
\(316\) 6.74456i 0.379411i
\(317\) 16.3723 0.919559 0.459779 0.888033i \(-0.347928\pi\)
0.459779 + 0.888033i \(0.347928\pi\)
\(318\) 8.37228i 0.469494i
\(319\) 23.1168i 1.29429i
\(320\) 1.00000i 0.0559017i
\(321\) −3.25544 −0.181701
\(322\) 20.7446 1.15605
\(323\) −2.23369 −0.124286
\(324\) −1.00000 −0.0555556
\(325\) 4.74456i 0.263181i
\(326\) −9.11684 −0.504935
\(327\) 3.11684i 0.172362i
\(328\) 9.11684i 0.503393i
\(329\) 0 0
\(330\) 6.37228 0.350783
\(331\) 3.48913i 0.191780i 0.995392 + 0.0958898i \(0.0305697\pi\)
−0.995392 + 0.0958898i \(0.969430\pi\)
\(332\) 5.48913 0.301255
\(333\) −5.74456 2.00000i −0.314800 0.109599i
\(334\) 3.25544 0.178130
\(335\) 8.74456i 0.477766i
\(336\) 2.37228 0.129419
\(337\) −34.7446 −1.89266 −0.946328 0.323206i \(-0.895239\pi\)
−0.946328 + 0.323206i \(0.895239\pi\)
\(338\) 9.51087i 0.517323i
\(339\) 9.86141i 0.535598i
\(340\) −0.372281 −0.0201898
\(341\) 53.3505i 2.88909i
\(342\) 6.00000 0.324443
\(343\) 19.8614 1.07242
\(344\) 4.37228 0.235738
\(345\) 8.74456 0.470791
\(346\) 7.62772i 0.410069i
\(347\) 30.9783i 1.66300i 0.555525 + 0.831500i \(0.312517\pi\)
−0.555525 + 0.831500i \(0.687483\pi\)
\(348\) 3.62772i 0.194466i
\(349\) 11.4891 0.614999 0.307499 0.951548i \(-0.400508\pi\)
0.307499 + 0.951548i \(0.400508\pi\)
\(350\) 2.37228i 0.126804i
\(351\) 4.74456i 0.253246i
\(352\) 6.37228i 0.339644i
\(353\) 5.11684i 0.272342i −0.990685 0.136171i \(-0.956520\pi\)
0.990685 0.136171i \(-0.0434797\pi\)
\(354\) 8.00000 0.425195
\(355\) 8.74456i 0.464113i
\(356\) 2.74456i 0.145462i
\(357\) 0.883156i 0.0467416i
\(358\) 8.74456 0.462164
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 1.00000 0.0527046
\(361\) −17.0000 −0.894737
\(362\) 22.7446i 1.19543i
\(363\) −29.6060 −1.55391
\(364\) 11.2554i 0.589945i
\(365\) 14.0000i 0.732793i
\(366\) 5.62772 0.294166
\(367\) 13.6277 0.711361 0.355681 0.934608i \(-0.384249\pi\)
0.355681 + 0.934608i \(0.384249\pi\)
\(368\) 8.74456i 0.455842i
\(369\) −9.11684 −0.474604
\(370\) 5.74456 + 2.00000i 0.298646 + 0.103975i
\(371\) 19.8614 1.03115
\(372\) 8.37228i 0.434083i
\(373\) −23.4891 −1.21622 −0.608110 0.793852i \(-0.708072\pi\)
−0.608110 + 0.793852i \(0.708072\pi\)
\(374\) 2.37228 0.122668
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 17.2119 0.886460
\(378\) 2.37228i 0.122017i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) −6.00000 −0.307794
\(381\) 9.48913 0.486143
\(382\) −7.11684 −0.364129
\(383\) 35.7228i 1.82535i −0.408686 0.912675i \(-0.634013\pi\)
0.408686 0.912675i \(-0.365987\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 15.1168i 0.770426i
\(386\) 18.2337 0.928070
\(387\) 4.37228i 0.222256i
\(388\) 7.11684i 0.361303i
\(389\) 5.86141i 0.297185i 0.988899 + 0.148593i \(0.0474743\pi\)
−0.988899 + 0.148593i \(0.952526\pi\)
\(390\) 4.74456i 0.240250i
\(391\) 3.25544 0.164635
\(392\) 1.37228i 0.0693107i
\(393\) 9.48913i 0.478663i
\(394\) 15.4891i 0.780331i
\(395\) −6.74456 −0.339356
\(396\) −6.37228 −0.320219
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −10.7446 −0.538576
\(399\) 14.2337i 0.712576i
\(400\) −1.00000 −0.0500000
\(401\) 28.9783i 1.44710i −0.690269 0.723552i \(-0.742508\pi\)
0.690269 0.723552i \(-0.257492\pi\)
\(402\) 8.74456i 0.436139i
\(403\) −39.7228 −1.97873
\(404\) 5.25544 0.261468
\(405\) 1.00000i 0.0496904i
\(406\) −8.60597 −0.427107
\(407\) −36.6060 12.7446i −1.81449 0.631725i
\(408\) 0.372281 0.0184307
\(409\) 21.4891i 1.06257i 0.847194 + 0.531284i \(0.178290\pi\)
−0.847194 + 0.531284i \(0.821710\pi\)
\(410\) 9.11684 0.450249
\(411\) 2.74456 0.135379
\(412\) 7.48913i 0.368963i
\(413\) 18.9783i 0.933859i
\(414\) −8.74456 −0.429772
\(415\) 5.48913i 0.269451i
\(416\) −4.74456 −0.232621
\(417\) 1.62772 0.0797097
\(418\) 38.2337 1.87007
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 2.37228i 0.115755i
\(421\) 8.74456i 0.426184i 0.977032 + 0.213092i \(0.0683534\pi\)
−0.977032 + 0.213092i \(0.931647\pi\)
\(422\) 23.8614i 1.16156i
\(423\) 0 0
\(424\) 8.37228i 0.406594i
\(425\) 0.372281i 0.0180583i
\(426\) 8.74456i 0.423675i
\(427\) 13.3505i 0.646078i
\(428\) −3.25544 −0.157358
\(429\) 30.2337i 1.45970i
\(430\) 4.37228i 0.210850i
\(431\) 13.6277i 0.656424i −0.944604 0.328212i \(-0.893554\pi\)
0.944604 0.328212i \(-0.106446\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −11.4891 −0.552132 −0.276066 0.961139i \(-0.589031\pi\)
−0.276066 + 0.961139i \(0.589031\pi\)
\(434\) 19.8614 0.953378
\(435\) −3.62772 −0.173936
\(436\) 3.11684i 0.149270i
\(437\) 52.4674 2.50985
\(438\) 14.0000i 0.668946i
\(439\) 12.3723i 0.590497i −0.955421 0.295248i \(-0.904598\pi\)
0.955421 0.295248i \(-0.0954024\pi\)
\(440\) 6.37228 0.303787
\(441\) −1.37228 −0.0653467
\(442\) 1.76631i 0.0840149i
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −5.74456 2.00000i −0.272625 0.0949158i
\(445\) 2.74456 0.130105
\(446\) 18.3723i 0.869953i
\(447\) −11.4891 −0.543417
\(448\) 2.37228 0.112080
\(449\) 36.9783i 1.74511i −0.488515 0.872556i \(-0.662461\pi\)
0.488515 0.872556i \(-0.337539\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −58.0951 −2.73559
\(452\) 9.86141i 0.463842i
\(453\) 11.2554 0.528827
\(454\) −6.37228 −0.299066
\(455\) 11.2554 0.527663
\(456\) 6.00000 0.280976
\(457\) 5.62772i 0.263254i 0.991299 + 0.131627i \(0.0420200\pi\)
−0.991299 + 0.131627i \(0.957980\pi\)
\(458\) 4.51087i 0.210779i
\(459\) 0.372281i 0.0173766i
\(460\) 8.74456 0.407717
\(461\) 14.6060i 0.680268i −0.940377 0.340134i \(-0.889528\pi\)
0.940377 0.340134i \(-0.110472\pi\)
\(462\) 15.1168i 0.703299i
\(463\) 12.2337i 0.568548i −0.958743 0.284274i \(-0.908248\pi\)
0.958743 0.284274i \(-0.0917525\pi\)
\(464\) 3.62772i 0.168413i
\(465\) 8.37228 0.388255
\(466\) 14.7446i 0.683029i
\(467\) 26.3723i 1.22036i −0.792261 0.610182i \(-0.791096\pi\)
0.792261 0.610182i \(-0.208904\pi\)
\(468\) 4.74456i 0.219317i
\(469\) −20.7446 −0.957895
\(470\) 0 0
\(471\) 16.3723 0.754395
\(472\) 8.00000 0.368230
\(473\) 27.8614i 1.28107i
\(474\) 6.74456 0.309788
\(475\) 6.00000i 0.275299i
\(476\) 0.883156i 0.0404794i
\(477\) −8.37228 −0.383340
\(478\) 7.86141 0.359572
\(479\) 10.9783i 0.501609i −0.968038 0.250805i \(-0.919305\pi\)
0.968038 0.250805i \(-0.0806951\pi\)
\(480\) 1.00000 0.0456435
\(481\) 9.48913 27.2554i 0.432667 1.24274i
\(482\) −21.4891 −0.978803
\(483\) 20.7446i 0.943910i
\(484\) −29.6060 −1.34573
\(485\) −7.11684 −0.323159
\(486\) 1.00000i 0.0453609i
\(487\) 18.7446i 0.849397i 0.905335 + 0.424699i \(0.139620\pi\)
−0.905335 + 0.424699i \(0.860380\pi\)
\(488\) 5.62772 0.254755
\(489\) 9.11684i 0.412278i
\(490\) 1.37228 0.0619934
\(491\) 36.4674 1.64575 0.822875 0.568223i \(-0.192369\pi\)
0.822875 + 0.568223i \(0.192369\pi\)
\(492\) −9.11684 −0.411019
\(493\) −1.35053 −0.0608249
\(494\) 28.4674i 1.28081i
\(495\) 6.37228i 0.286413i
\(496\) 8.37228i 0.375927i
\(497\) 20.7446 0.930521
\(498\) 5.48913i 0.245974i
\(499\) 24.9783i 1.11818i 0.829107 + 0.559090i \(0.188849\pi\)
−0.829107 + 0.559090i \(0.811151\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 3.25544i 0.145442i
\(502\) −4.00000 −0.178529
\(503\) 21.4891i 0.958153i −0.877773 0.479076i \(-0.840972\pi\)
0.877773 0.479076i \(-0.159028\pi\)
\(504\) 2.37228i 0.105670i
\(505\) 5.25544i 0.233864i
\(506\) −55.7228 −2.47718
\(507\) 9.51087 0.422393
\(508\) 9.48913 0.421012
\(509\) −18.7446 −0.830838 −0.415419 0.909630i \(-0.636365\pi\)
−0.415419 + 0.909630i \(0.636365\pi\)
\(510\) 0.372281i 0.0164849i
\(511\) −33.2119 −1.46921
\(512\) 1.00000i 0.0441942i
\(513\) 6.00000i 0.264906i
\(514\) −23.4891 −1.03606
\(515\) −7.48913 −0.330010
\(516\) 4.37228i 0.192479i
\(517\) 0 0
\(518\) −4.74456 + 13.6277i −0.208464 + 0.598768i
\(519\) 7.62772 0.334820
\(520\) 4.74456i 0.208063i
\(521\) 5.11684 0.224173 0.112087 0.993698i \(-0.464247\pi\)
0.112087 + 0.993698i \(0.464247\pi\)
\(522\) 3.62772 0.158781
\(523\) 29.2554i 1.27925i 0.768687 + 0.639625i \(0.220911\pi\)
−0.768687 + 0.639625i \(0.779089\pi\)
\(524\) 9.48913i 0.414534i
\(525\) −2.37228 −0.103535
\(526\) 16.6060i 0.724055i
\(527\) 3.11684 0.135772
\(528\) −6.37228 −0.277318
\(529\) −53.4674 −2.32467
\(530\) 8.37228 0.363669
\(531\) 8.00000i 0.347170i
\(532\) 14.2337i 0.617109i
\(533\) 43.2554i 1.87360i
\(534\) −2.74456 −0.118769
\(535\) 3.25544i 0.140745i
\(536\) 8.74456i 0.377708i
\(537\) 8.74456i 0.377356i
\(538\) 28.2337i 1.21724i
\(539\) −8.74456 −0.376655
\(540\) 1.00000i 0.0430331i
\(541\) 27.7228i 1.19190i −0.803023 0.595948i \(-0.796776\pi\)
0.803023 0.595948i \(-0.203224\pi\)
\(542\) 1.48913i 0.0639634i
\(543\) −22.7446 −0.976063
\(544\) 0.372281 0.0159614
\(545\) 3.11684 0.133511
\(546\) −11.2554 −0.481688
\(547\) 11.3505i 0.485314i −0.970112 0.242657i \(-0.921981\pi\)
0.970112 0.242657i \(-0.0780189\pi\)
\(548\) 2.74456 0.117242
\(549\) 5.62772i 0.240185i
\(550\) 6.37228i 0.271715i
\(551\) −21.7663 −0.927276
\(552\) −8.74456 −0.372193
\(553\) 16.0000i 0.680389i
\(554\) 9.48913 0.403154
\(555\) −2.00000 + 5.74456i −0.0848953 + 0.243843i
\(556\) 1.62772 0.0690306
\(557\) 24.2337i 1.02681i −0.858145 0.513407i \(-0.828383\pi\)
0.858145 0.513407i \(-0.171617\pi\)
\(558\) −8.37228 −0.354427
\(559\) −20.7446 −0.877402
\(560\) 2.37228i 0.100247i
\(561\) 2.37228i 0.100158i
\(562\) 20.9783 0.884914
\(563\) 3.86141i 0.162739i 0.996684 + 0.0813694i \(0.0259294\pi\)
−0.996684 + 0.0813694i \(0.974071\pi\)
\(564\) 0 0
\(565\) 9.86141 0.414872
\(566\) 30.7446 1.29229
\(567\) −2.37228 −0.0996265
\(568\) 8.74456i 0.366914i
\(569\) 21.2554i 0.891074i 0.895263 + 0.445537i \(0.146987\pi\)
−0.895263 + 0.445537i \(0.853013\pi\)
\(570\) 6.00000i 0.251312i
\(571\) 41.3505 1.73047 0.865233 0.501370i \(-0.167170\pi\)
0.865233 + 0.501370i \(0.167170\pi\)
\(572\) 30.2337i 1.26413i
\(573\) 7.11684i 0.297310i
\(574\) 21.6277i 0.902724i
\(575\) 8.74456i 0.364673i
\(576\) −1.00000 −0.0416667
\(577\) 35.7228i 1.48716i −0.668647 0.743580i \(-0.733126\pi\)
0.668647 0.743580i \(-0.266874\pi\)
\(578\) 16.8614i 0.701342i
\(579\) 18.2337i 0.757766i
\(580\) −3.62772 −0.150633
\(581\) 13.0217 0.540233
\(582\) 7.11684 0.295003
\(583\) −53.3505 −2.20955
\(584\) 14.0000i 0.579324i
\(585\) −4.74456 −0.196164
\(586\) 13.1168i 0.541852i
\(587\) 1.62772i 0.0671831i −0.999436 0.0335916i \(-0.989305\pi\)
0.999436 0.0335916i \(-0.0106945\pi\)
\(588\) −1.37228 −0.0565919
\(589\) 50.2337 2.06984
\(590\) 8.00000i 0.329355i
\(591\) −15.4891 −0.637137
\(592\) −5.74456 2.00000i −0.236100 0.0821995i
\(593\) −40.2337 −1.65220 −0.826100 0.563524i \(-0.809445\pi\)
−0.826100 + 0.563524i \(0.809445\pi\)
\(594\) 6.37228i 0.261458i
\(595\) −0.883156 −0.0362059
\(596\) −11.4891 −0.470613
\(597\) 10.7446i 0.439746i
\(598\) 41.4891i 1.69662i
\(599\) 0.744563 0.0304220 0.0152110 0.999884i \(-0.495158\pi\)
0.0152110 + 0.999884i \(0.495158\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 40.3723 1.64682 0.823410 0.567447i \(-0.192069\pi\)
0.823410 + 0.567447i \(0.192069\pi\)
\(602\) 10.3723 0.422743
\(603\) 8.74456 0.356106
\(604\) 11.2554 0.457977
\(605\) 29.6060i 1.20365i
\(606\) 5.25544i 0.213488i
\(607\) 3.76631i 0.152870i −0.997075 0.0764349i \(-0.975646\pi\)
0.997075 0.0764349i \(-0.0243538\pi\)
\(608\) 6.00000 0.243332
\(609\) 8.60597i 0.348731i
\(610\) 5.62772i 0.227860i
\(611\) 0 0
\(612\) 0.372281i 0.0150486i
\(613\) −31.3505 −1.26624 −0.633118 0.774055i \(-0.718225\pi\)
−0.633118 + 0.774055i \(0.718225\pi\)
\(614\) 2.51087i 0.101331i
\(615\) 9.11684i 0.367627i
\(616\) 15.1168i 0.609075i
\(617\) −37.2554 −1.49985 −0.749924 0.661524i \(-0.769910\pi\)
−0.749924 + 0.661524i \(0.769910\pi\)
\(618\) 7.48913 0.301257
\(619\) −9.62772 −0.386971 −0.193485 0.981103i \(-0.561979\pi\)
−0.193485 + 0.981103i \(0.561979\pi\)
\(620\) 8.37228 0.336239
\(621\) 8.74456i 0.350907i
\(622\) −19.8614 −0.796370
\(623\) 6.51087i 0.260853i
\(624\) 4.74456i 0.189935i
\(625\) 1.00000 0.0400000
\(626\) −18.2337 −0.728765
\(627\) 38.2337i 1.52691i
\(628\) 16.3723 0.653325
\(629\) −0.744563 + 2.13859i −0.0296877 + 0.0852713i
\(630\) 2.37228 0.0945140
\(631\) 2.88316i 0.114777i 0.998352 + 0.0573883i \(0.0182773\pi\)
−0.998352 + 0.0573883i \(0.981723\pi\)
\(632\) 6.74456 0.268284
\(633\) −23.8614 −0.948406
\(634\) 16.3723i 0.650226i
\(635\) 9.48913i 0.376564i
\(636\) −8.37228 −0.331983
\(637\) 6.51087i 0.257970i
\(638\) 23.1168 0.915205
\(639\) −8.74456 −0.345930
\(640\) 1.00000 0.0395285
\(641\) −25.8614 −1.02146 −0.510732 0.859740i \(-0.670626\pi\)
−0.510732 + 0.859740i \(0.670626\pi\)
\(642\) 3.25544i 0.128482i
\(643\) 29.1168i 1.14826i 0.818765 + 0.574128i \(0.194659\pi\)
−0.818765 + 0.574128i \(0.805341\pi\)
\(644\) 20.7446i 0.817450i
\(645\) 4.37228 0.172158
\(646\) 2.23369i 0.0878833i
\(647\) 9.76631i 0.383953i −0.981399 0.191977i \(-0.938510\pi\)
0.981399 0.191977i \(-0.0614898\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 50.9783i 2.00107i
\(650\) 4.74456 0.186097
\(651\) 19.8614i 0.778430i
\(652\) 9.11684i 0.357043i
\(653\) 15.4891i 0.606136i 0.952969 + 0.303068i \(0.0980110\pi\)
−0.952969 + 0.303068i \(0.901989\pi\)
\(654\) −3.11684 −0.121878
\(655\) −9.48913 −0.370771
\(656\) −9.11684 −0.355953
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 46.9783 1.83001 0.915006 0.403439i \(-0.132185\pi\)
0.915006 + 0.403439i \(0.132185\pi\)
\(660\) 6.37228i 0.248041i
\(661\) 29.3505i 1.14160i 0.821088 + 0.570802i \(0.193367\pi\)
−0.821088 + 0.570802i \(0.806633\pi\)
\(662\) −3.48913 −0.135609
\(663\) −1.76631 −0.0685979
\(664\) 5.48913i 0.213019i
\(665\) −14.2337 −0.551959
\(666\) 2.00000 5.74456i 0.0774984 0.222597i
\(667\) 31.7228 1.22831
\(668\) 3.25544i 0.125957i
\(669\) −18.3723 −0.710313
\(670\) −8.74456 −0.337832
\(671\) 35.8614i 1.38441i
\(672\) 2.37228i 0.0915127i
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 34.7446i 1.33831i
\(675\) 1.00000 0.0384900
\(676\) 9.51087 0.365803
\(677\) 32.9783 1.26746 0.633729 0.773555i \(-0.281524\pi\)
0.633729 + 0.773555i \(0.281524\pi\)
\(678\) −9.86141 −0.378725
\(679\) 16.8832i 0.647916i
\(680\) 0.372281i 0.0142763i
\(681\) 6.37228i 0.244186i
\(682\) −53.3505 −2.04290
\(683\) 18.3723i 0.702996i 0.936189 + 0.351498i \(0.114328\pi\)
−0.936189 + 0.351498i \(0.885672\pi\)
\(684\) 6.00000i 0.229416i
\(685\) 2.74456i 0.104864i
\(686\) 19.8614i 0.758312i
\(687\) −4.51087 −0.172101
\(688\) 4.37228i 0.166692i
\(689\) 39.7228i 1.51332i
\(690\) 8.74456i 0.332900i
\(691\) 33.6277 1.27926 0.639629 0.768683i \(-0.279088\pi\)
0.639629 + 0.768683i \(0.279088\pi\)
\(692\) 7.62772 0.289962
\(693\) −15.1168 −0.574241
\(694\) −30.9783 −1.17592
\(695\) 1.62772i 0.0617429i
\(696\) 3.62772 0.137508
\(697\) 3.39403i 0.128558i
\(698\) 11.4891i 0.434870i
\(699\) −14.7446 −0.557691
\(700\) −2.37228 −0.0896638
\(701\) 28.9783i 1.09449i 0.836971 + 0.547247i \(0.184324\pi\)
−0.836971 + 0.547247i \(0.815676\pi\)
\(702\) 4.74456 0.179072
\(703\) −12.0000 + 34.4674i −0.452589 + 1.29996i
\(704\) −6.37228 −0.240164
\(705\) 0 0
\(706\) 5.11684 0.192575
\(707\) 12.4674 0.468884
\(708\) 8.00000i 0.300658i
\(709\) 39.8614i 1.49703i 0.663120 + 0.748513i \(0.269232\pi\)
−0.663120 + 0.748513i \(0.730768\pi\)
\(710\) 8.74456 0.328178
\(711\) 6.74456i 0.252941i
\(712\) −2.74456 −0.102857
\(713\) −73.2119 −2.74181
\(714\) 0.883156 0.0330513
\(715\) −30.2337 −1.13068
\(716\) 8.74456i 0.326800i
\(717\) 7.86141i 0.293590i
\(718\) 8.00000i 0.298557i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 17.7663i 0.661652i
\(722\) 17.0000i 0.632674i
\(723\) 21.4891i 0.799189i
\(724\) −22.7446 −0.845295
\(725\) 3.62772i 0.134730i
\(726\) 29.6060i 1.09878i
\(727\) 18.4674i 0.684917i −0.939533 0.342459i \(-0.888740\pi\)
0.939533 0.342459i \(-0.111260\pi\)
\(728\) −11.2554 −0.417154
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 1.62772 0.0602034
\(732\) 5.62772i 0.208006i
\(733\) 38.6060 1.42594 0.712972 0.701192i \(-0.247349\pi\)
0.712972 + 0.701192i \(0.247349\pi\)
\(734\) 13.6277i 0.503008i
\(735\) 1.37228i 0.0506174i
\(736\) −8.74456 −0.322329
\(737\) 55.7228 2.05258
\(738\) 9.11684i 0.335596i
\(739\) 25.3505 0.932534 0.466267 0.884644i \(-0.345599\pi\)
0.466267 + 0.884644i \(0.345599\pi\)
\(740\) −2.00000 + 5.74456i −0.0735215 + 0.211174i
\(741\) −28.4674 −1.04578
\(742\) 19.8614i 0.729135i
\(743\) 8.60597 0.315722 0.157861 0.987461i \(-0.449540\pi\)
0.157861 + 0.987461i \(0.449540\pi\)
\(744\) −8.37228 −0.306943
\(745\) 11.4891i 0.420929i
\(746\) 23.4891i 0.859998i
\(747\) −5.48913 −0.200837
\(748\) 2.37228i 0.0867392i
\(749\) −7.72281 −0.282185
\(750\) −1.00000 −0.0365148
\(751\) −3.25544 −0.118793 −0.0593963 0.998234i \(-0.518918\pi\)
−0.0593963 + 0.998234i \(0.518918\pi\)
\(752\) 0 0
\(753\) 4.00000i 0.145768i
\(754\) 17.2119i 0.626822i
\(755\) 11.2554i 0.409627i
\(756\) −2.37228 −0.0862790
\(757\) 22.5109i 0.818172i −0.912496 0.409086i \(-0.865848\pi\)
0.912496 0.409086i \(-0.134152\pi\)
\(758\) 28.0000i 1.01701i
\(759\) 55.7228i 2.02261i
\(760\) 6.00000i 0.217643i
\(761\) 10.8832 0.394514 0.197257 0.980352i \(-0.436797\pi\)
0.197257 + 0.980352i \(0.436797\pi\)
\(762\) 9.48913i 0.343755i
\(763\) 7.39403i 0.267682i
\(764\) 7.11684i 0.257478i
\(765\) 0.372281 0.0134599
\(766\) 35.7228 1.29072
\(767\) −37.9565 −1.37053
\(768\) −1.00000 −0.0360844
\(769\) 32.0000i 1.15395i −0.816762 0.576975i \(-0.804233\pi\)
0.816762 0.576975i \(-0.195767\pi\)
\(770\) 15.1168 0.544773
\(771\) 23.4891i 0.845940i
\(772\) 18.2337i 0.656245i
\(773\) 35.3505 1.27147 0.635735 0.771907i \(-0.280697\pi\)
0.635735 + 0.771907i \(0.280697\pi\)
\(774\) −4.37228 −0.157158
\(775\) 8.37228i 0.300741i
\(776\) 7.11684 0.255480
\(777\) −13.6277 4.74456i −0.488892 0.170210i
\(778\) −5.86141 −0.210142
\(779\) 54.7011i 1.95987i
\(780\) −4.74456 −0.169883
\(781\) −55.7228 −1.99392
\(782\) 3.25544i 0.116414i
\(783\) 3.62772i 0.129644i
\(784\) −1.37228 −0.0490100
\(785\) 16.3723i 0.584352i
\(786\) 9.48913 0.338466
\(787\) −45.2119 −1.61163 −0.805816 0.592166i \(-0.798273\pi\)
−0.805816 + 0.592166i \(0.798273\pi\)
\(788\) −15.4891 −0.551777
\(789\) −16.6060 −0.591188
\(790\) 6.74456i 0.239961i
\(791\) 23.3940i 0.831796i
\(792\) 6.37228i 0.226429i
\(793\) −26.7011 −0.948183
\(794\) 14.0000i 0.496841i
\(795\) 8.37228i 0.296934i
\(796\) 10.7446i 0.380831i
\(797\) 10.7446i 0.380592i −0.981727 0.190296i \(-0.939055\pi\)
0.981727 0.190296i \(-0.0609448\pi\)
\(798\) 14.2337 0.503867
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 2.74456i 0.0969744i
\(802\) 28.9783 1.02326
\(803\) 89.2119 3.14822
\(804\) 8.74456 0.308397
\(805\) 20.7446 0.731150
\(806\) 39.7228i 1.39918i
\(807\) 28.2337 0.993873
\(808\) 5.25544i 0.184886i
\(809\) 6.00000i 0.210949i 0.994422 + 0.105474i \(0.0336361\pi\)
−0.994422 + 0.105474i \(0.966364\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 13.4891 0.473667 0.236834 0.971550i \(-0.423890\pi\)
0.236834 + 0.971550i \(0.423890\pi\)
\(812\) 8.60597i 0.302010i
\(813\) −1.48913 −0.0522259
\(814\) 12.7446 36.6060i 0.446697 1.28304i
\(815\) −9.11684 −0.319349
\(816\) 0.372281i 0.0130325i
\(817\) 26.2337 0.917801
\(818\) −21.4891 −0.751350
\(819\) 11.2554i 0.393297i
\(820\) 9.11684i 0.318374i
\(821\) 13.7228 0.478929 0.239465 0.970905i \(-0.423028\pi\)
0.239465 + 0.970905i \(0.423028\pi\)
\(822\) 2.74456i 0.0957276i
\(823\) 42.9783 1.49813 0.749064 0.662498i \(-0.230504\pi\)
0.749064 + 0.662498i \(0.230504\pi\)
\(824\) 7.48913 0.260896
\(825\) 6.37228 0.221854
\(826\) 18.9783 0.660338
\(827\) 3.86141i 0.134274i 0.997744 + 0.0671371i \(0.0213865\pi\)
−0.997744 + 0.0671371i \(0.978614\pi\)
\(828\) 8.74456i 0.303895i
\(829\) 23.8614i 0.828741i −0.910108 0.414370i \(-0.864002\pi\)
0.910108 0.414370i \(-0.135998\pi\)
\(830\) 5.48913 0.190530
\(831\) 9.48913i 0.329174i
\(832\) 4.74456i 0.164488i
\(833\) 0.510875i 0.0177008i
\(834\) 1.62772i 0.0563633i
\(835\) 3.25544 0.112659
\(836\) 38.2337i 1.32234i
\(837\) 8.37228i 0.289388i
\(838\) 8.00000i 0.276355i
\(839\) −43.7228 −1.50948 −0.754740 0.656025i \(-0.772237\pi\)
−0.754740 + 0.656025i \(0.772237\pi\)
\(840\) 2.37228 0.0818515
\(841\) 15.8397 0.546195
\(842\) −8.74456 −0.301358
\(843\) 20.9783i 0.722529i
\(844\) −23.8614 −0.821343
\(845\) 9.51087i 0.327184i
\(846\) 0 0
\(847\) −70.2337 −2.41326
\(848\) −8.37228 −0.287505
\(849\) 30.7446i 1.05515i
\(850\) −0.372281 −0.0127691
\(851\) 17.4891 50.2337i 0.599519 1.72199i
\(852\) −8.74456 −0.299584
\(853\) 32.7446i 1.12115i 0.828103 + 0.560576i \(0.189420\pi\)
−0.828103 + 0.560576i \(0.810580\pi\)
\(854\) 13.3505 0.456846
\(855\) 6.00000 0.205196
\(856\) 3.25544i 0.111269i
\(857\) 18.6060i 0.635568i −0.948163 0.317784i \(-0.897061\pi\)
0.948163 0.317784i \(-0.102939\pi\)
\(858\) 30.2337 1.03216
\(859\) 30.0000i 1.02359i 0.859109 + 0.511793i \(0.171019\pi\)
−0.859109 + 0.511793i \(0.828981\pi\)
\(860\) 4.37228 0.149094
\(861\) −21.6277 −0.737071
\(862\) 13.6277 0.464162
\(863\) 47.5842 1.61979 0.809893 0.586578i \(-0.199525\pi\)
0.809893 + 0.586578i \(0.199525\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 7.62772i 0.259350i
\(866\) 11.4891i 0.390416i
\(867\) −16.8614 −0.572643
\(868\) 19.8614i 0.674140i
\(869\) 42.9783i 1.45794i
\(870\) 3.62772i 0.122991i
\(871\) 41.4891i 1.40581i
\(872\) −3.11684 −0.105550
\(873\) 7.11684i 0.240869i
\(874\) 52.4674i 1.77474i
\(875\) 2.37228i 0.0801977i
\(876\) 14.0000 0.473016
\(877\) 41.1168 1.38842 0.694209 0.719774i \(-0.255755\pi\)
0.694209 + 0.719774i \(0.255755\pi\)
\(878\) 12.3723 0.417544
\(879\) 13.1168 0.442420
\(880\) 6.37228i 0.214810i
\(881\) −43.6277 −1.46986 −0.734928 0.678146i \(-0.762784\pi\)
−0.734928 + 0.678146i \(0.762784\pi\)
\(882\) 1.37228i 0.0462071i
\(883\) 33.8614i 1.13953i 0.821809 + 0.569764i \(0.192965\pi\)
−0.821809 + 0.569764i \(0.807035\pi\)
\(884\) −1.76631 −0.0594075
\(885\) 8.00000 0.268917
\(886\) 4.00000i 0.134383i
\(887\) −10.3723 −0.348267 −0.174134 0.984722i \(-0.555712\pi\)
−0.174134 + 0.984722i \(0.555712\pi\)
\(888\) 2.00000 5.74456i 0.0671156 0.192775i
\(889\) 22.5109 0.754991
\(890\) 2.74456i 0.0919979i
\(891\) 6.37228 0.213479
\(892\) −18.3723 −0.615149
\(893\) 0 0
\(894\) 11.4891i 0.384254i
\(895\) 8.74456 0.292298
\(896\) 2.37228i 0.0792524i
\(897\) 41.4891 1.38528
\(898\) 36.9783 1.23398
\(899\) 30.3723 1.01297
\(900\) 1.00000 0.0333333
\(901\) 3.11684i 0.103837i
\(902\) 58.0951i 1.93436i
\(903\) 10.3723i 0.345168i
\(904\) −9.86141 −0.327986
\(905\) 22.7446i 0.756055i
\(906\) 11.2554i 0.373937i
\(907\) 29.7228i 0.986930i −0.869766 0.493465i \(-0.835730\pi\)
0.869766 0.493465i \(-0.164270\pi\)
\(908\) 6.37228i 0.211472i
\(909\) −5.25544 −0.174312
\(910\) 11.2554i 0.373114i
\(911\) 53.4891i 1.77217i −0.463519 0.886087i \(-0.653413\pi\)
0.463519 0.886087i \(-0.346587\pi\)
\(912\) 6.00000i 0.198680i
\(913\) −34.9783 −1.15761
\(914\) −5.62772 −0.186148
\(915\) 5.62772 0.186047
\(916\) −4.51087 −0.149043
\(917\) 22.5109i 0.743375i
\(918\) −0.372281 −0.0122871
\(919\) 33.7228i 1.11241i 0.831044 + 0.556206i \(0.187744\pi\)
−0.831044 + 0.556206i \(0.812256\pi\)
\(920\) 8.74456i 0.288300i
\(921\) −2.51087 −0.0827361
\(922\) 14.6060 0.481022
\(923\) 41.4891i 1.36563i
\(924\) −15.1168 −0.497308
\(925\) 5.74456 + 2.00000i 0.188880 + 0.0657596i
\(926\) 12.2337 0.402024
\(927\) 7.48913i 0.245975i
\(928\) 3.62772 0.119086
\(929\) 22.6060 0.741678 0.370839 0.928697i \(-0.379070\pi\)
0.370839 + 0.928697i \(0.379070\pi\)
\(930\) 8.37228i 0.274538i
\(931\) 8.23369i 0.269848i
\(932\) −14.7446 −0.482974
\(933\) 19.8614i 0.650233i
\(934\) 26.3723 0.862927
\(935\) 2.37228 0.0775819
\(936\) 4.74456 0.155081
\(937\) 5.25544 0.171688 0.0858438 0.996309i \(-0.472641\pi\)
0.0858438 + 0.996309i \(0.472641\pi\)
\(938\) 20.7446i 0.677334i
\(939\) 18.2337i 0.595034i
\(940\) 0 0
\(941\) 13.7228 0.447351 0.223675 0.974664i \(-0.428194\pi\)
0.223675 + 0.974664i \(0.428194\pi\)
\(942\) 16.3723i 0.533438i
\(943\) 79.7228i 2.59613i
\(944\) 8.00000i 0.260378i
\(945\) 2.37228i 0.0771703i
\(946\) −27.8614 −0.905852
\(947\) 51.8614i 1.68527i 0.538486 + 0.842635i \(0.318997\pi\)
−0.538486 + 0.842635i \(0.681003\pi\)
\(948\) 6.74456i 0.219053i
\(949\) 66.4239i 2.15621i
\(950\) −6.00000 −0.194666
\(951\) −16.3723 −0.530908
\(952\) 0.883156 0.0286233
\(953\) 43.9565 1.42389 0.711945 0.702235i \(-0.247814\pi\)
0.711945 + 0.702235i \(0.247814\pi\)
\(954\) 8.37228i 0.271063i
\(955\) −7.11684 −0.230296
\(956\) 7.86141i 0.254256i
\(957\) 23.1168i 0.747261i
\(958\) 10.9783 0.354691
\(959\) 6.51087 0.210247
\(960\) 1.00000i 0.0322749i
\(961\) −39.0951 −1.26113
\(962\) 27.2554 + 9.48913i 0.878751 + 0.305942i
\(963\) 3.25544 0.104905
\(964\) 21.4891i 0.692118i
\(965\) 18.2337 0.586963
\(966\) −20.7446 −0.667445
\(967\) 19.4891i 0.626728i 0.949633 + 0.313364i \(0.101456\pi\)
−0.949633 + 0.313364i \(0.898544\pi\)
\(968\) 29.6060i 0.951572i
\(969\) 2.23369 0.0717564
\(970\) 7.11684i 0.228508i
\(971\) 5.35053 0.171707 0.0858534 0.996308i \(-0.472638\pi\)
0.0858534 + 0.996308i \(0.472638\pi\)
\(972\) 1.00000 0.0320750
\(973\) 3.86141 0.123791
\(974\) −18.7446 −0.600615
\(975\) 4.74456i 0.151948i
\(976\) 5.62772i 0.180139i
\(977\) 42.6060i 1.36309i −0.731778 0.681543i \(-0.761309\pi\)
0.731778 0.681543i \(-0.238691\pi\)
\(978\) 9.11684 0.291525
\(979\) 17.4891i 0.558955i
\(980\) 1.37228i 0.0438359i
\(981\) 3.11684i 0.0995132i
\(982\) 36.4674i 1.16372i
\(983\) −0.605969 −0.0193274 −0.00966371 0.999953i \(-0.503076\pi\)
−0.00966371 + 0.999953i \(0.503076\pi\)
\(984\) 9.11684i 0.290634i
\(985\) 15.4891i 0.493525i
\(986\) 1.35053i 0.0430097i
\(987\) 0 0
\(988\) −28.4674 −0.905668
\(989\) −38.2337 −1.21576
\(990\) −6.37228 −0.202524
\(991\) 29.8614i 0.948579i −0.880369 0.474289i \(-0.842705\pi\)
0.880369 0.474289i \(-0.157295\pi\)
\(992\) −8.37228 −0.265820
\(993\) 3.48913i 0.110724i
\(994\) 20.7446i 0.657978i
\(995\) −10.7446 −0.340626
\(996\) −5.48913 −0.173930
\(997\) 48.4674i 1.53498i 0.641063 + 0.767489i \(0.278494\pi\)
−0.641063 + 0.767489i \(0.721506\pi\)
\(998\) −24.9783 −0.790673
\(999\) 5.74456 + 2.00000i 0.181750 + 0.0632772i
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 1110.2.h.e.961.3 yes 4
3.2 odd 2 3330.2.h.k.2071.1 4
37.36 even 2 inner 1110.2.h.e.961.1 4
111.110 odd 2 3330.2.h.k.2071.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.h.e.961.1 4 37.36 even 2 inner
1110.2.h.e.961.3 yes 4 1.1 even 1 trivial
3330.2.h.k.2071.1 4 3.2 odd 2
3330.2.h.k.2071.3 4 111.110 odd 2